Some aspects of suspension flows: Stokes to turbulent flows by Kyongmin Yeo Sc.M., Applied Mathematics, Brown University, USA, 2007 Sc.M., Mechanical Engineering, Yonsei University, Korea, 2005 Sc.B., Civil Engineering, Yonsei University, Korea, 1999 Thesis Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Division of Applied Mathematics at Brown University May 2011 c Copyright 2011 ° by Kyongmin Yeo This dissertation by Kyongmin Yeo is accepted in its present form by the Division of Applied Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy Date . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................... Martin R. Maxey, Director Recommended to the Graduate Council Date . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................... George E. Karniadakis, Reader Date . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................... Jeffrey F. Morris, Reader Approved by the Graduate Council Date . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................... Peter M. Weber Dean of the Graduate School iii The Vita of Kyongmin Yeo Kyongmin Yeo was born on 22 February, 1977, in Seoul, Korea. He was an undergraduate student at Yonsei University, graduating with a B.Sc. in Civil Engineering on February 1999. Upon finishing his military service on June 2001 as a First Lieutenant, he started his graduate study in the Department of Civil Engineering at Yonsei University. Later, he changed his major to Mechanical Engineering and earned an M.Sc. in Mechanical Engineer- ing from Yonsei University on February 2005. Subsequently, he came to Brown University to study towards a Ph.D. in Applied Mathematics under the direction of Professor Martin R. Maxey. He attained an Sc.M. in Applied Mathematics in 2007. Honors and Awards • Sigma Xi Award for excellence in research 2011 • Dissertation Fellowship, Brown University 2011 • Outstanding Master Thesis in Fluid Engineering, KSME 2005 Bibliography Peer-reviewed Journal Papers 1. K. Yeo & M.R. Maxey, “Force-coupling simulations of dense finite-inertia suspensions in a linear shear flow,” in preparation 2. K. Yeo & M.R. Maxey, “Numerical simulations of concentrated suspensions of monodis- perse particles in a Poiseuille flow,” J. Fluid Mech. in revision. 3. K. Yeo & M.R. Maxey, “Anomalous diffusion in wall-bounded suspensions of non- Brownian particles under steady shear,” Europhys. Lett., 92, 24008 (2010) 4. K. Yeo, B.-G. Kim & C. Lee, “On the near-wall characteristics of acceleration in turbulence,” J. Fluid Mech., 659, 405 (2010) 5. K. Yeo & M.R. Maxey, “Rheology and ordering transitions of non-Brownian suspen- sions in a confined shear flow: effects of external torques,” Phys. Rev. E, 81, 062501 (2010) iv 6. K. Yeo & M.R. Maxey, “Ordering transitions of non-Brownian suspensions in con- fined steady shear flow,” Phys. Rev. E, 81, 051502 (2010). 7. K. Yeo & M.R. Maxey, “Dynamics of concentrated suspensions of non-colloidal par- ticles in Couette flow,” J. Fluid Mech., 649, 205 (2010) 8. K. Yeo & M.R. Maxey, “Simulation of concentrated suspensions using the force- coupling method,” J. Comput. Phys., 229, 2401 (2010) 9. K. Yeo, S. Dong, E. Climent & M.R. Maxey “Modulation of homogeneous turbulence seeded with finite size bubbles or particles,” Int. J. Multiphase flow, 36, 221 (2010) 10. K. Yeo, B.-G. Kim & C. Lee, “Eulerian and Lagrangian statistics in stably stratified turbulent channel flows,” J. Turbulence, 10, 17 (2009) 11. J. Jung, K. Yeo & C. Lee, “Behavior of heavy particles in isotropic turbulence,” Phys. Rev. E, 77, 016307 (2008) 12. E. Climent, K. Yeo, M.R. Maxey & G.E. Karniadakis “Dynamic self-assembly of spinning particles,” J. Fluid Eng., 129, 379 (2007). 13. A.M. Reynolds, K. Yeo & C. Lee, “Anisotropy of acceleration in turbulent flows,” Phys. Rev. E, 70, 017302 (2004) 14. C. Lee, K. Yeo & J.-I. Choi, “Intermittent nature of acceleration in near wall turbu- lence,” Phys. Rev. Lett., 92, 144502 (2004) 15. J.-I. Choi, K. Yeo & C. Lee, “Lagrangian statistics in turbulent channel flow,” Phys. Fluids, 16, 779 (2004) 16. S.-U. Choi, H. Kang & K. Yeo, “Flow and sediment transport in emerging vegetated zone,” Ecology Civil Eng., 6, 87 (2003) Conference Presentations 1. K. Yeo & M.R. Maxey, “Anomalous diffusion of non-colloidal suspensions in a Cou- ette flow,” APS DFD 63rd Annual Meeting, CA, U.S., 2010 2. K. Yeo & M.R. Maxey, “Numerical simulations of concentrated, non-colloidal sus- pensions in Poiseuille flows,” SOR 82nd Annual Meeting, NM, U.S., 2010 v 3. K. Yeo & M.R. Maxey, “Order transition in non-colloidal Couette suspension flows:effects of external torques,” APS DFD 62nd Annual Meeting, MN, U.S., 2009 4. K. Yeo & M.R. Maxey, “Numerical simulation of concentrated suspensions of non- colloidal particles in Couette flow,” SOR 81st Annual Meeting, WI, U.S., 2009 5. K. Yeo & M.R. Maxey, “Dynamic self-assembly of non-colloidal particles in Couette flow,” SOR 81st Annual Meeting, WI, U.S., 2009 6. L.-P. Wang, H. Gao, L.-S. Luo, Y. Peng, K. Yeo & M.R. Maxey, “Comparing particle- resolved simulation methods for moving particles in a viscous fluid,” APS DFD 61st Annual Meeting, TX, U.S., 2008 7. J. Lim, K. Yeo & C. Lee, “On the modification of near-wall structures in stably stratified turbulence,” APS DFD 58th Annual Meeting, IL, U.S., 2005 8. J. Jeong, K. Yeo & C. Lee, “Dispersion of heavy particles in isotropic turbulence,” APS DFD 58th Annual Meeting, IL, U.S., 2005 9. C. Lee, J. Jeong & K. Yeo, “Acceleration, enstrophy and dissipation in isotropic turbulence,” APS DFD 58th Annual Meeting, IL, U.S., 2005 10. K. Yeo & C. Lee, “On the modification of near-wall structures in stably stratified turbulence,” APS DFD 57th Annual Meeting, WA, U.S., 2004. 11. K. Yeo & C. Lee, “Modification of near-wall turbulent characteristics under stable stratification,” The 6th KSME-JSME Thermal and Fluid Engineering Conference, Jeju, Korea, 2004. 12. K. Yeo & C. Lee, “On the acceleration in inhomogeneous turbulence,” APS DFD 56th Annual Meeting, NJ, U.S., 2003. 13. A.M. Reynolds, K. Yeo & C. Lee, “On the anisotropy of Lagrangian accelerations in turbulence,” APS DFD 56th Annual Meeting, NJ, U.S., 2003. 14. K. Yeo & C. Lee,“Acceleration statistics in turbulent channel flow,” 5th Asian Com- putational Fluid Dynamics, Busan, Korea, 2003. vi Acknowledgments This thesis is dedicated to the memory of my grandfather, in whose final days I could not be with, and to my grandmother, who is losing memory of me. First of all, I would like to thank my advisor Professor Martin Maxey for his guidance, support, and particularly patience. I cannot express enough how much I owe him. He has been always kind and supportive in person while keeping high standard in research. Thanks to his help, I could enjoy my life and remain active in research. I would like to express my gratitude to Professor George Karniadakis for his encouragement over the years. In particular, I would like to acknowledge that the huge amount of computing time he provided me in my early graduate years was very helpful for me developing my specialty in high performance computing. I wish to thank Professor Jeffrey Morris for his service as a reader and for his time and effort coming all the way from New York. Although we have not met often, his short lecture at Brown University and discussions with him were very inspiring. I am indebted to the faculty members and staffs for the high quality classes and the warm, welcoming atmosphere here in the Division of Applied Mathematics. Especially, I thank Madeline Brewster for helping me in many ways throughout my graduate study and for her coffee which is one of the major contributors to the completion of this thesis. I would like to thank all of the members of the CRUNCH group. It has been always my pleasure chitchatting with them. Particularly, I would like to express my deep gratitude to Igor Pivkin, Eric Keaveny, Wenxiao Pan, and Huan Lei for their insightful discussions and friendship. I am sure our paths will cross again someday. I will never forget the Korean gang for turning a potentially (almost surely?) dull graduate student life into a pleasant one; Hyoungsu (Boss), Minseok (All-round sports player), and Heyrim (Sleeping beauty). Last, but foremost, I cannot thank enough to my wife, Sung Won, for her love and encouragement over the course of this long graduate study. It would have been more difficult if it were not for her love. I would like to express my deepest gratitude to all of my family members in Korea for their continuous support and encouragement. It would have not been possible if it were not for their support. I love all of you! This work was supported in part by funding from DARPA under an award ATO-Friction Drag Technologies Program and by NSF under an award DUE-0734234. Support of HPC vii resources from the Arctic Region Supercomputing Center at the University of Alaska, Fair- banks as part of the DoD HPCMP and from the National Institute for Computational Sciences provided by NSF under TG-CTS090097. viii Contents Acknowledgments vii 1 Introduction 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Numerical methods 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Force-coupling method: Stokes flow . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Review of the force-coupling method . . . . . . . . . . . . . . . . . . 10 2.2.2 Quadrupole expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 Modification of the FCM envelopes near a wall . . . . . . . . . . . . 28 2.2.4 Discretized equation of FCM . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Lubrication correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.2 Particle-wall lubrication correction . . . . . . . . . . . . . . . . . . . 44 2.3.3 Fictitious inertia for dynamic simulation . . . . . . . . . . . . . . . . 45 2.4 Solution procedure for finite-inertia suspensions . . . . . . . . . . . . . . . . 47 2.5 Flow solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5.1 Tri-periodic domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5.2 Bounded domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.6.1 Periodic domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.6.2 Channel flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.7 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ix 2.7.1 Results I: particles in an infinite domain . . . . . . . . . . . . . . . . 66 2.7.2 Results II: homogeneous suspensions in an periodic domain . . . . . 74 2.7.3 Results III: single particle in a channel . . . . . . . . . . . . . . . . . 85 2.7.4 Results IV: a particle pair under a finite fluid inertia . . . . . . . . . 89 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3 Stokes flow: Concentrated suspensions in Couette flow 94 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.2 Wall effects on the rheology of concentrated suspensions . . . . . . . . . . . 96 3.2.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2.2 Wall effects on particle concentration . . . . . . . . . . . . . . . . . . 98 3.2.3 Relative viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.2.4 Normal stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2.5 Normal stresses and continuum models . . . . . . . . . . . . . . . . . 113 3.2.6 Micro-structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.3 Effects of confinement on the mobility of the particles . . . . . . . . . . . . 119 3.4 Ordering transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.4.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.4.2 Wall-induced ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.4.3 Effects of the channel height on the order structures . . . . . . . . . 140 3.5 Effects of external torques on rheology and ordering transitions . . . . . . . 145 3.5.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.5.2 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4 Stokes flow: Concentrated suspensions in Poiseuille flow 161 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.3 Volume fraction and velocity statistics . . . . . . . . . . . . . . . . . . . . . 165 4.3.1 Volume fraction profile . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.3.2 Velocity statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.4 Particle stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.4.1 Normal stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 x 4.4.2 Shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 4.4.3 Pair-distribution function . . . . . . . . . . . . . . . . . . . . . . . . 188 4.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5 Finite-Reynolds-number flow: Hydrodynamic interaction between spin- ning particles 192 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6 Finite-Reynolds-number flow: Concentrated suspensions in a linear shear flow 206 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.3 Velocity statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6.4 Particle stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.4.1 Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.4.2 Normal stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.5 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7 Turbulent flow: Modulation of isotropic turbulence seeded with finite size bubbles or particles 235 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.2 Simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7.2.1 Inertial particle simulation . . . . . . . . . . . . . . . . . . . . . . . 239 7.2.2 Turbulent flow simulation . . . . . . . . . . . . . . . . . . . . . . . . 242 7.2.3 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 243 7.3 Turbulence modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.4 Lagrangian statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 8 Concluding Remarks 262 8.1 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 xi 8.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 A Analytical resistance functions 270 xii List of Tables 2.1 FCM resistance functions for a particle pair . . . . . . . . . . . . . . . . . . 40 2.2 Particle-wall FCM resistance functions . . . . . . . . . . . . . . . . . . . . . 45 2.3 Relative velocity of two spheres in a straining flow when E = 1. . . . . . . . 70 2.4 FCM resistance functions (X M , Y M , Z M ) . . . . . . . . . . . . . . . . . . . 71 2.5 Upstream ∆y I and downstream center-to-center distances ∆y F for Rref = 2.001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.6 Simulation parameters of the random static simulations . . . . . . . . . . . 76 2.7 Simulation parameters of the dynamic simulations . . . . . . . . . . . . . . 78 2.8 Diffusion coefficients in y and z directions . . . . . . . . . . . . . . . . . . . 85 3.1 Simulation parameters. Ny is the number of elements in the vertical direction. Every elements has the same length EL = 2.5 and quadrature points Qy = 9. 97 3.2 P i estimated in the core region ΩC and the wall The deviatoric stress hσ12 region ΩW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.3 The relative viscosity for different Rref at φ = 0.4. . . . . . . . . . . . . . . 109 3.4 Channel height Hy /a, characteristic gap width η, number of particle layers N , and order parameter C6 for φ = 0.52. . . . . . . . . . . . . . . . . . . . . 141 4.1 The translational velocity of a particle at different vertical locations Y . The channel height is Hy /a = 20. The particle velocity is normalized by the centerline velocity uc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 xiii 4.2 Simulation parameters. Φ is the bulk volume fraction and Np is the number of particles. Nx and Nz are the number of Fourier modes in the streamwise and spanwise directions, respectively. Ny is the number of spectral elements in the wall-normal direction. The length of each spectral elements are the same, lE = Hy /Ny . The same numbers of quadrature points QE and the spectral modes PE are used for every elements; QE = 10 and PE = 8. . . . 164 4.3 The coefficients of the fitting curves for velocity fluctuations. . . . . . . . . 175 4.4 The fitting coefficients for the particle-phase contribution to the total stress hχp σ12 i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.1 Simulation parameters of the sustained homogeneous isotropic turbulence (single phase flow). Parameter definitions are given in sections 7.2.2 and 7.2.3. The standard Kolmogorov velocity, length and time scales are vK , η and τK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.2 Parameters of the flow simulations: Bubbles (B, mB = 0), Neutrally-buoyant particles (N, mP = mF ) and Solid particles (S, mP = 1.4mF ). Particle radius a = 0.1305 (1) or a = 0.091 (2). All simulations include force monopole (M) and dipole terms (D), except S2-M which is based on the monopole only. Parameter definitions are given in sections 7.2.2 and 7.2.3; NP is the number of particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 7.3 Additional effective viscosity in turbulent two-phase flow simulations. . . . . 252 7.4 Lagrangian data following the particles or bubbles for: rms velocity fluctua- tion, V 0 ; flatness factor for the velocity fluctuations, FV ; rms fluctuation in the acceleration, A0 ; flatness factor for the acceleration fluctuations, FA ; rms angular velocity, Ω0 . The last column gives the value of C0∗ , estimated from the maximum values in Fig. 7.7. . . . . . . . . . . . . . . . . . . . . . . . . 254 8.1 Changes in the particle shear stress with the channel height. . . . . . . . . . 264 A.1 Near field forms of scalar resistance functions . . . . . . . . . . . . . . . . . 271 xiv List of Figures 2.1 Streamlines of (a) exact solution and (b) FCM solution. . . . . . . . . . . . 17 2.2 u1 along (a) x1 and (b) x2 axes. . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 u1 in (a) x1 and (b) x2 directions. . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 The residual 2-norm history for bi-disperse suspension. . . . . . . . . . . . . 36 2.5 A of the exact solution (dashed line), FCM (solid The resistance function X11 line), and SD-FTS (dash-dot line). . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Configuration of the test problem. . . . . . . . . . . . . . . . . . . . . . . . 61 2.7 Effects of the tolerance level of the preconditioned conjugate gradient solver to the accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.8 Effects of the integration width to the accuracy. . . . . . . . . . . . . . . . . 63 2.9 Convergence of LC-FCM as a function of the grid resolution. . . . . . . . . 64 2.10 The ratio of the angular velocity to the translational velocity as a function of the maximum order of the polynomial in each elements. The dash line is the analytical solution from Dance & Maxey [50]. . . . . . . . . . . . . . . . 64 2.11 Convergence of h−type and p−type refinement. Inset shows the convergence of p−type refinement in a log-linear plot. . . . . . . . . . . . . . . . . . . . 65 2.12 Comparison of angular velocity for a pair of equal spheres with horizontal separation. ◦, Ganatos et al. (1978); N, FCM-LUB; –, FCM-MD. . . . . . . 67 2.13 Illustration of the horizontal chain of 7 spheres. . . . . . . . . . . . . . . . . 67 2.14 Comparison of (a) the drag coefficient λ = F/6πµaU and (b) angular velocity. ◦, Ganatos et al. (1978); 4, Durlofsky et al. (1987); +, FCM-LUB; ¤, Dance & Maxey (2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.15 Illustration of a particle-pair in a pure straining flow. . . . . . . . . . . . . . 69 xv 2.16 (a) Relative trajectory of the centers of two equal spheres and (b) separation distances for different initial positions, ∆y I . The length scale of the contact force Rref = 2.001a. From top to bottom: ∆y I = 0.7, 0.6, 0.4, and 0.2 . . . 71 2.17 Relative trajectory of the centers of two equal spheres for different initial positions, ∆y I . The length scale of the contact force is Rref = 2.0001a. From top to bottom: ∆y I = 0.4 and 0.2 . . . . . . . . . . . . . . . . . . . . 73 2.18 The effective viscosity functions (a) α and (b) β in the functions of the volume fraction φ for the simple cubic lattice. The circle indicates FCM results. The dashed and solid lines are, respectively, the high and low concentration asymptotic solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.19 Shear viscosity in terms of volume fraction. Solid line, Krieger & Dougherty (1959); dashed line, Stokes-Einstein estimate; dash-dot line, Batchelor & Green (1972); M, Ladd (1990); ◦, FCM-LUB; ¤, FCM-MD. . . . . . . . . . 77 2.20 Computation time of one time step for φ = 0.3. . . . . . . . . . . . . . . . . 79 2.21 Time history of the deviatoric stress tensor. The top and bottom curves correspond, respectively, to φ = 0.4 and 0.3. . . . . . . . . . . . . . . . . . . 80 2.22 Projection of the pair-distribution function onto the x − y plane. The darker the contour, the higher the probability is. . . . . . . . . . . . . . . . . . . . 81 2.23 Velocity auto-correlation functions: (a) ρV2 (t) and (b) ρV3 (t) for φ = 0.3 and 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.24 Mean squared displacement normalized by a2 in the velocity gradient (solid line) and the vorticity directions (dashed line): (a) φ = 0.3 and (b) φ = 0.4. 83 2.25 Geometry for a sphere placed between two walls. . . . . . . . . . . . . . . . 85 2.26 Drag coefficient λt from °, Ganatos et al. (1980); 4, FCM with wall lubri- cation; ¤, FCM with monopole and dipole. . . . . . . . . . . . . . . . . . . 86 2.27 Translational velocity of a sphere V in a Couette flow: Solid line, Ganatos et al. (1982); ¥, Singh & Nott (2000); °, present FCM. . . . . . . . . . . . 88 2.28 Relative trajectories in the x − y plane for a particle pair in a linear shear flow at Rep = 1.0. The initial vertical separation distances are ∆y/a = 0.5 (dash-dot), 0.6 (dashed), and 0.7 (solid). . . . . . . . . . . . . . . . . . . . . 89 2.29 Relative trajectories in the x − z plane for a particle pair in a linear shear flow at Rep = 0.1 (a), 0.3 (b), and 0.5 (c). . . . . . . . . . . . . . . . . . . . 91 xvi 3.1 Concentration profiles for C2S (solid), C3S (dashed), and C4S (dash-dot). . 98 3.2 The pair-distribution function projected onto (a) x − y and (b) y − z planes for C4S. Light contours represent high probability. . . . . . . . . . . . . . . 100 3.3 Concentration profiles for C2L (solid), C3L (dashed), C4La (dash-dot), and C4Lc (dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.4 The pair-distribution functions for C4La projected onto x − y plane for par- ticles in (a) 5.5 < y/a < 6.5 and (b) 8 < y/a < 12. Light contours represent high probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.5 Concentration profiles for C4S (solid), C4La (dashed), and C4H (dash-dot). 104 3.6 Average particle velocities for C3L (solid) and C4La (dash-dot). The dashed line is the linear velocity profile of Couette flow without suspensions. . . . . 105 3.7 Relative viscosity µr : solid line, [115]; dashed line, Eilers fit; ¤, C2S, C3S, and C4S; M, C2L, C3L, and C4La; O, C4Lb; ◦, C4Lc; ¦, C4H; +, [233]. . . 106 3.8 Normalized normal stress differences (a) −N1 /τ and (b) −N2 /τ . ♦, C2S, C3S and C4S; ¤, C2L, C3L and C4La; O, C4Lc; ∗, C4H; ◦, [197]. . . . . . . 111 3.9 (a) Ratio of N2 to N1 in ΩD ; ♦, C2S, C3S and C4S; ¤, C2L, C3L and C4La; O, C4Lc; ∗, C4H. (b) Ratio of N2 to N1 in ΩC (¤) and ΩW (M) for C2L, C3L and C4La. ◦ denotes [197]. . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.10 Anisotropy parameters (a) λ2 and (b) λ3 . ♦, C2L, C3L and C4La; ◦, C4H. 113 3.11 The normal stress viscosity in terms of φ. ¤, C2L, C3L and C4La; ♦, C4Lb; M, C4Lc; ◦, C4H; [160]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.12 The pair distribution function for particles of which center is located in 5 ≤ y/a ≤ 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.13 The pair distribution function for particles of which center is located in 2 ≤ y/a ≤ 5. Solid line, 1 − θ/π; dashed line, θ/π − 1. . . . . . . . . . . . . . . . 117 3.14 The pair distribution function for particles of which center is located in 1 ≤ y/a ≤ 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 xvii 3.15 (a) Area fraction profile for the bulk volume fraction φ = 0.4. Considering the symmetry, φA is shown only for the lower half of the channel. The wall- normal displacements as a function of time are shown for particles which are initially in (b) Zone I, (c) Zone II + III, and (d) Zone IV. (e) shows a representative trajectory in y − z plane for a particle whose initial location is in Zone I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.16 (a) Variances of the wall-normal and spanwise displacements of the particles in Zone I at t∗ = 0. The inset shows the variances divided by tν . (b) The probability density functions (PDF) of the standardized wall-normal displacements at different time instances. The dashed line is ∼ exp(−0.8y ∗ ). 123 3.17 (a, b) PDFs of the wall-normal velocity V2 of the particles initially located in Zone I. The velocity PDFs are shown for t∗ = 200, 300, and 400. The dashed line in (a) is the Gaussian distribution. In (b), B is the velocity PDF for t∗ = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.18 Variances of the (a) wall-normal and (b) spanwise displacements of particles in Zones II – IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.19 The area fraction profiles at different time instances for the particles whose initial locations are in (a) Zone II, (b) Zone III, and (c) Zone IV. . . . . . . 128 3.20 The area fraction profiles (a) and variances of the wall-normal displacement (b) for φ = 0.25 (solid), 0.3 (dashed), 0.35 (dash-dot), and 0.40 (dash-dot- dot), Hy /a = 30. In (b), the circles are for φ = 0.40 and Hy /a = 40. . . . . 130 3.21 (a) Transient behavior of the relative viscosity normalized by the relative viscosity in the stationary state. The channel height is fixed; Hy /a = 20. Snapshots (end view) for φ = 0.52 and Hy /a = 20 at γt ˙ = 1 (b) and 50 (c). For visualization, the particle radius is reduced to 1/2 of the actual size. . . 134 3.22 2-dimensional pair distributions in the velocity-gradient–vorticity (y − z) plane for Hy /a = 20 obtained in (a,b) ΩW and (c,d) ΩC . . . . . . . . . . . . 135 3.23 Area fraction profiles for Hy /a = 20. . . . . . . . . . . . . . . . . . . . . . . 137 3.24 Order parameter C6 in (a) ΩW and (b) ΩC . . . . . . . . . . . . . . . . . . . 137 3.25 Snapshots (end view) for φ = 0.52; (a) Hy /a = 20 and (b) Hy /a = 30. For visualization, the particle radius is reduced to 1/2 of the actual size. . . . . 138 3.26 The relative viscosity for various φ and Hy /a. . . . . . . . . . . . . . . . . . 139 xviii 3.27 Snapshots (end view) for φ = 0.52; (a) Hy /a = 9, (b) Hy /a = 10, (c) Hy /a = 11. For visualization, the particle radius is reduced to 1/2 of the actual size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.28 The order parameter as a function of Hy /a for φ = 0.52 and 0.54. . . . . . . 142 3.29 The normalized mean-square vertical displacements h[Y2 (t) − Y2 (0)]2 i/a2 for φ = 0.52. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.30 The particle pressure Π (¨) and the order parameter C6 (•) as a function of Hy /a for (a) φ = 0.52 and (b) 0.54. . . . . . . . . . . . . . . . . . . . . . . . 144 3.31 Order structures in the horizontal plane for (a) φ = 0.52 and Hy /a = 9, (b) φ = 0.60 and Hy /a = 9, and (c) φ = 0.60 and Hy /a = 10. Green (lighter) particles are in the lower layer and red (darker) particles are in the upper layer.146 3.32 Snapshots (end view) for φ = 0.48 (a,b) and φ = 0.52) (c,d). For visualiza- tion, the particle radius is reduced to 1/2 of the actual size. . . . . . . . . . 149 3.33 The order parameter C6 as a function of the non-dimensional torque T ∗ . . . 150 3.34 The shear (a) and apparent viscosities (b) as functions of the non-dimensional torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.35 Variations in the vortex viscosity µv as a function of volume fraction for different values of the non-dimensional torque in a confined shear flow. Also shown are results of Monte Carlo simulations for randomly seeded suspensions in a homogeneous uniform shear flow, compared to prior results [143]. . . . 154 3.36 Effects of the torque on the angular velocity in the vorticity direction for φ = 0.52 and Hy /a = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.1 The area fraction profiles for (a) P3S and (b) P4S. The solid lines are the present simulation results and the circles are from Lyon & Leal [139]. . . . . 165 4.2 The area fraction profiles near the wall for (a) P3S and (b) P4S. The solid lines are the present simulation results and the circles are from Hampton et al. [84]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.3 The area fraction profiles for P2L (solid), P3L (dashed), and P4L (dash-dot). The symbols are the experiments by Gilchrist [79]: ◦, Φ = 0.298; 4, Φ = 0.411.168 xix 4.4 The local volume fraction profiles for (a) Φ = 0.3 and (b) Φ = 0.4; •, P3L and P4L; 4, Gilchrist [79]; , Yapici et al. [227]. The dashed line in (a) is the area fraction (φ) profile of P3L. . . . . . . . . . . . . . . . . . . . . . 169 4.5 The averaged particle velocity for (a) P3S and (b) P4S. The solid lines are the present simulation results and the symbols are from [139]. . . . . . . . . 171 4.6 The average particle-phase velocity profiles for Hy /a = 40. Dotted line, P2L; dashed line, P3L; solid line, P4L. . . . . . . . . . . . . . . . . . . . . . . . . 171 4.7 Apparent viscosity estimated from the mean flux: ¤ P3S and P4S; ◦ P2L, P3L, and P4L; ∗ Nott & Brady [164]. . . . . . . . . . . . . . . . . . . . . . 172 4.8 The particle-phase velocity fluctuations normalized by the mean particle- phase velocity Qp for (a) streamwise, (b) wall-normal, and (c) spanwise com- ponents: Solid line, P2L; dashed line, P3L; dash-dot line P4L. . . . . . . . . 174 4.9 The particle-phase velocity fluctuations normalized by the fitting curves for (a) P2L, (b) P3L, and (c) P4L: solid line, vx ; dashed line, vy ; dash-dot line, vz .176 4.10 The average angular velocity normalized by γ˙ c = uc /h. Solid line, P2L; dashed line, P3L; dash-dot line, P4L. Long-dash line is the rate-of-rotational of clear flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 1 dhV i 4.11 The angular velocity (solid line) and 2 dy (dashed line) normalized by γ˙ c = uc /h with fitting curves (dotted line) for (a) P2L, (b) P3L, and (c) P4L. . . 179 4.12 The phasic-average of the particle stresses normalized by f D h for (a) P2L, p p (b) P3L, and (c) P4L. Solid line, hχp σ11 i; dashed line, hχp σ22 i; dash-dot line, p hχp σ33 i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.13 The (a) first and (b) second normal stress differences normalized by f D h. Solid line, P2L; dashed line, P3L; dash-dot line, P4L. . . . . . . . . . . . . . 182 4.14 The average particle pressure normalized by the local shear rate γ˙ L (a) and the normal viscosity model by Morris & Boulay [160] (b): ¤, P2L; 4, P3L; ◦, P4L. (c) The volume average of the particle pressure scaled by the apparent viscosity as a function of the bulk volume fraction. The dashed line is Ψ = e−0.8 Φ2.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 xx 4.15 (a) The particle shear stress, χp σ12 , normalized by f D h and shown with corresponding fitting curves. The long-dashed line represents the total shear stress normalized by f D h. From top to bottom, the lines indicate P4L, P3L, and P2L. (b) The effective viscosity as a function of the local volume fraction: ¤, P2L; 4, P3L; ◦, P4L. Solid line is an empirical relation by Krieger & Dogherty [115] and dashed line is Eilers’ fit. . . . . . . . . . . . . . . . . . . 186 4.16 Decomposition of hχp σ12 i into the contributions from the hydrodynamic stresslet (solid line) and the interparticle force (dashed line) for P4L. All the variables are normalized by f D h. . . . . . . . . . . . . . . . . . . . . . . 188 4.17 The pair-distribution functions for the reference particles in (a) 7 ≤ y/a ≤ 13 and(b) 18 ≤ y/a ≤ 19. The lighter the contour, the higher the probability. . 189 5.1 The azimuthal velocity in the equatorial plane normalized by the particle radius a and the angular velocity ω. The solid line is the analytical solution at zero Reynolds number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.2 The non-dimensional torque coefficient M as a function of the Reynolds num- ber Reω . The solid and hollow symbols are, respectively, FCM simulations without and with the stresslet. . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.3 Secondary flow in the x − z plane for a sphere spinning about the z axis at Reω = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.4 Flow field in the x − z plane for a pair of co-rotating spheres at Reω = 2. The spheres are rotating about the z axis. Inset shows the detailed velocity field for the sphere located at x/a = −3. . . . . . . . . . . . . . . . . . . . . 198 5.5 Hydrodynamic repulsion force between two co-rotating spheres as a function of the distance between the particles 2R. . . . . . . . . . . . . . . . . . . . . 199 5.6 Precession angular velocity Ω for a pair of co-rotating spheres in terms of the separation distance R: (a) comparison of the FCM results at Reω = 0.25 with the Stokes-limit estimate and (b) FCM results for Reω = 0.25, 2, and 8. The lines in (b) are the fitting curves. . . . . . . . . . . . . . . . . . . . 200 5.7 Comparison of the radial velocity ur obtained from FCM-TS with the ana- lytical solution by Bickley [17]. . . . . . . . . . . . . . . . . . . . . . . . . . 201 xxi 5.8 Ratio of the magnitude of the non-linear interaction of FCM-TS to that of the analytical Stokes solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.9 Comparison of the radial velocity ur obtained from FCM-TS with a nonlinear- interaction-correction with the analytical solution by Bickley [17]. . . . . . . 204 6.1 Root-mean-square velocity fluctuation normalized by aγ; ˙ (a) φ = 0.20, (b) φ = 0.30, and (c) φ = 0.40. . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.2 Auto-correlation functions for the (a) velocity-gradient- and (b) vorticity- direction velocities for φ = 0.30. The time-lag τ is normalized by the shear rate γ. ˙ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.3 ˙ 2 in the (a) velocity-gradient and (b) The self-diffusivities normalized by γa vorticity directions as functions of Reγ˙ : ¥, φ = 0.20; N, φ = 0.30; •, φ = 0.40. 212 6.4 Mean-square displacements normalized by the particle radius a2 for φ = 0.20 in the (a) velocity-gradient and (b) vorticity directions. . . . . . . . . . . . 213 6.5 The probability density functions (PDF) of particle velocity for (a, b) φ = 0.20 and (c, d) φ = 0.40; (a,c) are PDFs of the particle velocity in the velocity-gradient direction and (b, d) are PDFs for the vorticity direction. The dashed line is the Normal distribution. Each pdf is normalized by its standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.6 The changes in the effective viscosity with φ and Reγ˙ . The dashed line is the Eilers’ fit [203]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.7 The normalized particle shear stress as a function of Reγ˙ . The particle stress is normalized by its value at Reγ˙ = 0.005. ¥, φ = 0.20; N, φ = 0.30; •, φ = 0.40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.8 The standardized probability density functions of the shear component of the particle stresslet for (a) φ = 0.20 and (b) φ = 0.40: ¥, Reγ˙ = 0.005; N, Reγ˙ = 0.5; ¨, Reγ˙ = 1.0; •, Reγ˙ = 2.0. The dashed line is the Normal distribution. The inset shows the positive tails of the pdfs for Reγ˙ = 0.005 and Reγ˙ = 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.9 The particle pressure normalized by the effective viscosity: ¥, φ = 0.20; N, φ = 0.30; •, φ = 0.40. The hollow symbols are the numerical results of Sierou & Brady [197] at zero Reynolds number. . . . . . . . . . . . . . . . . . . . 220 xxii 6.10 Normal stresses normalized by µ0 γ˙ as functions of Reγ˙ for (a) φ = 0.2, (b) φ = 0.3, and (c) φ = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.11 First (a) and second (b) normal stress differences normalized by µ0 γ˙ as func- tions of Reγ˙ : ¥, φ = 0.20; N, φ = 0.30; •, φ = 0.40. . . . . . . . . . . . . . 222 6.12 The pair-distribution function in the plane of shear for (a, b) φ = 0.2 and (c, d) φ = 0.4. The line denotes g(r, θ) = 1. . . . . . . . . . . . . . . . . . . . 224 6.13 The near-contact pair-distribution function in the plane of shear for (a) φ = 0.2, (b) φ = 0.3, and (c) φ = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . 225 ∗ for (a) φ = 0.2, 6.14 The near-contact pair-distribution function weighted by S12 ∗ i. (b) φ = 0.3, and (c) φ = 0.4. The weighted gnb is normalized by hS12 . . 227 ∗ for (a) φ = 0.2, 6.15 The near-contact pair-distribution function weighted by S11 ˙ 3. (b) φ = 0.3, and (c) φ = 0.4. The weighted gnb is normalized by µ0 γa . . 228 ∗ for (a) φ = 0.2, 6.16 The near-contact pair-distribution function weighted by S22 ˙ 3. (b) φ = 0.3, and (c) φ = 0.4. The weighted gnb is normalized by µ0 γa . . 230 ∗ for (a) φ = 0.2, 6.17 The near-contact pair-distribution function weighted by S33 ˙ 3. (b) φ = 0.3, and (c) φ = 0.4. The weighted gnb is normalized by µ0 γa . . 231 6.18 The three-dimensional near-contact pair-distribution function gn b(θ, ψ) weighted ∗ for φ = 0.3; (a) S ∗ , (b) S ∗ , (c) S ∗ , and (d) S ∗ . S ∗ g (θ, ψ) is shown by Sij 11 22 33 12 ij nb in the y − z plane. The flow direction is into the paper; the upper hemisphere represents the compressional axis and the low hemisphere is in the extensional axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.1 Energy (a) and dissipation (b) spectra (grid 1283 ). Solid line, single phase flow; dashed line with circles, bubbles B1; dash dot line with triangles, neutral particles N1; dotted line with diamonds, solid particles S1. . . . . . . . . . . 247 7.2 Energy (a) and dissipation (b) spectra (grid 1923 ). Solid line, single phase flow; dashed line with circles, bubbles B2; dash dot line with triangles, solid particles S2; dotted line with diamonds, solid particles S2-M; Long dash line, data set S2 from [208]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 7.3 Dissipation spectra D(k) plotted against wavenumber scaled as kη: solid line, single phase flow; dashed line, neutral particles N1; dash dot line, neutral particles N2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 xxiii 7.4 Determination of the effective viscosity. (a) Solid line, dissipation rate spectra (N1); dashed line, dipole energy transfer (N1); dash dot line, corresponds to νadd /ν = 0.150. (b) Solid line, dissipation rate spectra (N2); dashed line, dipole energy transfer (N2); dash dot line corresponds to νadd /ν = 0.105. . . 252 7.5 Velocity auto-correlation function. Solid line with circles, B1; dashed line with crosses, B2; dash dot line with triangles, N1; long dash line with dia- monds, S1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.6 Probability density function of particle acceleration. Legend for symbols: circle, B1; triangle, B2; nabla, N1; diamond, S1. . . . . . . . . . . . . . . . . 257 7.7 Acceleration auto-correlation function. (a) Solid line, B1; dashed line, N1; dash dot line, S1. (b) Solid line, B2; dashed line, N2; dash dot line, S2. . . . 258 7.8 Normalized velocity structure function. Solid line with solid circles, B1; solid line with solid triangles, N1; solid line with solid diamonds, S1; dashed line with open circles, B2; dashed line with open triangles, N2; dashed line with open diamonds, S2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 xxiv Chapter 1 Introduction 1.1 Introduction Solid-fluid multiphase flow is ubiquitous in nature as well as in technological applications. The multiphase flow systems of current interest range from microscale mixing and mass transport, such as blood flow, microbial swimming, suspensions of hard spheres or soft materials in microfluidic devices, to large scale environmental flows, e.g. sediment transport in river or ocean, bioagents or pollutant dispersion in atmospheric boundary layer and ocean mixed layer, and droplet formation in wet clouds, to name a few. Due to its immense range of applications, study of the multiphase flow has been an important subject of many disciplines. One of the major branches of the study is “suspension flows” led by Chemical and Biomechanical Engineers. The study of suspension flows has been focused largely on rheology and self-diffusion of solid particles suspended in a very viscous liquid, i.e. in the limit of Stokes flows [5, 159, 203]. Particle suspensions in the Stokes limit have been a subject of great interest in the soft matter physics community, where usually systems under thermodynamic equilibrium conditions, i.e. without flow, are studied in the framework of statistical mechanics [129, 184]. “Turbulent dispersed flow” has attracted attention of the Mechanical/Environmental Engineering and Atmospheric Science communities [12, 20, 209]. In turbulent dispersed flows, particular interest has been paid to the dispersion of passive tracer or point particles of which radius is assumed to be much smaller than the smallest lengthscale of turbulence, Kolmogorov lengthscale. Major objectives of this thesis are (1) to develop a numerical method which can be easily applied to study these wide range of solid- fluid multiphase flows and (2) to contribute to our understanding on fundamental physics of 1 2 solid-fluid interactions in a wide range of flow parameters; from the zero-Reynolds-number Stokes suspensions to high-Reynolds-number turbulent suspensions. Suspensions of micron or submicron particles in a viscous liquid are prevalent in many biological and engineering systems. For example, the diameter of human red blood cell is about 6 − 7µm and usually the lengthscale of bacteria is O(1) ∼ O(10)µm. Typical size of colloids in many technological applications is < O(1)µm and the particles of current interest in microfluidic lab-on-a-chip devices, MEMS/Bio-MEMS, and food processing are usually in the range of O(1) ∼ O(10)µm. The particle Reynolds number is defined as ˙ 2 ρf γD Rep = µf , in which ρf and µf are, respectively, density and viscosity of the liquid, γ˙ is the local shear rate, and D is the characteristic lengthscale of the particle, usually the particle radius. In many of the aforementioned systems, Rep is in the range of < O(10−3 ), in which case the hydrodynamics is governed by viscous forces, i.e. Stokes flow. Another important flow parameter is these microscale systems is the P´eclet number, which is the ˙ 3 6πµf γa ratio of the advection by the flow to the thermal diffusion, P e = kB T , and kB T is the thermal energy. In many flowing suspensions of O(1)µm particles, the timescale of flow- induced self-diffusion is much smaller than that of the thermal diffusion. Here, we consider only non-Brownian or non-colloidal suspension flows, suspensions in the limit of large P e (P e > 103 ), where the hydrodynamic interaction determines the suspension dynamics. As the Stokes equations are linear partial differential equations, in principle, hydro- dynamic interaction between the suspended particles can be found analytically. However, in practice, analytical computation of multibody interaction becomes mathematically too complicated even for a couple of particles. If we confine our interest to suspensions of solid particles in a Newtonian fluid, thanks to the pioneering works by Batchelor [13, 14, 15], the mechanics of suspension flows up to semi-dilute systems, in which the hydrodynamic interaction of a particle-pair is considered, is now well understood. However, still our un- derstanding on the dynamics of concentrated suspensions is far from complete. Following Stickel & Powell [203], we define concentrated suspensions as the suspension flows in which the average distance between the suspended particles is smaller than the particle radius, where both short-range lubrication and interparticle forces and long-range multibody hydro- dynamic interactions are important in the dynamics. Typically, we assume the suspension is concentrated if the volume fraction of solid phase is larger than 10% of the total vol- ume, φ > 0.1. In contrast to the theories in the dilute to semi-dilute limit, concentrated 3 suspensions of hard spheres in a Newtonian liquid exhibit non-Newtonian behaviors, such as the existence of normal stress differences and transient responses of rheology upon step changes of γ. ˙ Over the last three decades, the bulk behavior of suspension flows has been studied extensively notably by the research groups led by Acrivos [5, 6] and Brady [23, 24] (see [159, 203] for review). While there is a large volume of literature devoted to the bulk behavior of concentrated suspensions in a homogeneous flow, i.e. far from a solid boundary, still there are only a very limited number of studies about the suspension dynamics under a geometric confinement, in which the presence of an impenetrable boundary introduces inhomogeneity to the system. If the size of a container is much larger than the characteristic lengthscale of the suspension, the suspension field in the core of the container can be assumed as locally homogeneous. However, in many microfluidic systems of current interest, the confinement size is on the order of 10-particle radii and the effects of the confinement suspension dynamics is signifi- cant. For example, Zarraga et al. [241] have shown, in their experiments in a parallel plate geometry, that the effective viscosity is a function of the channel height if the gap width is less than forty-particle radii. Theoretical analysis of confined suspensions becomes much more complicated than that in homogeneous suspensions, as the suspension field becomes a function of the distance from the wall. T¨ozeren & Skalak [210], whose contribution has been largely underappreciated compared to the importance to the theoretical analysis of inhomogeneous suspensions, were one of the first who pointed out that the ergodic prin- ciple of equating volume averages to ensemble averages is no longer valid in wall-bounded suspensions. Both experiments and numerical simulations of wall-bounded concentrated suspension flows are very challenging. It is only recently that three-dimensional simula- tions of wall-bounded concentrated suspensions have become available. Using the novel lattice-Boltzmann approach developed by Nguyen & Ladd [162], there are several studies of concentrated suspensions in wall-bounded Couette flows [116, 119, 215]. Yet, most studies have been focused on the bulk properties and the effects of the wall-induced inhomogeneity on the suspension dynamics has remained largely unexplored. In this thesis, we present a new numerical method to simulate wall-bounded concentrated suspension flows and study the effects of the confinement on the suspension dynamics both in shear and pressure-driven flows. Study of turbulent dispersed flows has been one of the major research topics in the fluid 4 mechanics community due to its relevance to a wide range of applications, such as predic- tion of dispersion of airborne pollutant in atmospheric boundary layer, cloud microphysics, sediment transport in river or ocean, and particle transport in human respiratory system. In turbulent dispersed flows of interest, the volume fraction of the particle phase is usually in the semi-dilute regime (φ < 0.1). While the suspension dynamics are determined by the hydrodynamic interaction between the suspended particles in Stokes-flow suspensions, in turbulent suspension flows, any long-range hydrodynamic interaction between the par- ticles is screened by the fluid inertia and the dynamics is governed mostly by the inertial responses of the suspended particles to the small scale turbulent coherent structures. One of the important parameters in turbulent dispersed flows, which represents the inertial re- sponse of the particles, is the turbulent Stokes number St defined as the ratio of the particle relaxation timescale τp = (2ρp + ρf )a2 /9µf to the turbulent eddy turnover time Te = u0 2 /². Here, ρp and ρf are, respectively, density of the particle and the fluid, u0 2 is root-mean square of velocity fluctuations, and ² is the dissipation rate. Now, it is well known that, de- pending on St, the suspended particles tend to be accumulated in some regions of a specific flow topology [145, 202, 220], which simultaneously modifies the turbulent flows. Since an analytical solution is not available for turbulent flows, most of our current knowledge on the physics of turbulent dispersed flows comes from experiments and numerical simulations. Direct numerical simulation, which solves the Navier-Stokes equations without any tur- bulence model, has been providing valuable information on the physics of turbulent flows. However, in many of the previous direct numerical studies of turbulent dispersed flows, the radius of the suspended particles is assumed to be much smaller than the Kolmogorov scale and the hydrodynamic interaction between the particles is neglected. In this point particle approach, the momentum transfer between the particle and fluid phases is usually computed from a simple drag force based on the slip velocity. Hwang & Eaton [94, 95] showed in their experiments that, while the point-particle method captures some of the basic features of the particle-turbulence interaction qualitatively, the turbulence modulation by the particle phase is significantly underestimated. With the rapid increase in the computation power, particle-resolved direct numerical simulations of the turbulence modulation by suspensions of finite-size solid particles have become plausible more recently [138, 208, 243]. In this thesis, we study turbulence modulation by suspensions of three different types of finite-size particles; heavy particles, neutrally buoyant particles, and bubbles (massless particles) by 5 employing the force-coupling method. 1.2 Outline In Chapter 2, the force-coupling method (FCM) is briefly reviewed. A higher-order FCM multipole (force quadrupole) is discussed. We present the semi-discretized equations of the force-coupling method and show that FCM is equivalent to a matrix-free method to solve a grand-mobility problem. Built upon the matrix relation, we develop the lubrication- corrected FCM (LC-FCM). A solution procedure to use LC-FCM for finite-Reynolds-number suspensions is presented. Here, LC-FCM is tested against various configurations and the results are compared with the previous theoretical, experimental, and numerical results. Most of the results in Chapter 2 are from Yeo & Maxey [233] Chapter 3 describes the fundamental mechanisms of suspensions of non-colloidal, neutrally- buoyant spheres in a wall-bounded Couette flow at zero Reynolds number. We report the basic effects of the inhomogeneity induced by the solid boundary on suspension dynamics by investigating the particle density fluctuation, rheological parameters, and suspension microstructure as functions of the distance from the wall. A fundamental issue of the ap- propriate lengthscale of concentrated suspensions is discussed through the changes in the diffusive behavior of the suspended particles. We report on the wall-induced ordering tran- sitions of the suspended particles. Chapter 3 is based on Yeo & Maxey [229, 230, 231, 232]. LC-FCM simulations of concentrated suspensions in a Poiseuille flow are performed and the results are shown in Chapter 4. The ensemble averaged particle-phase velocity and local volume fraction profiles are compared with the previous experiments. Rheological param- eters are calculated from a phase-averaging procedure and compared with the rheological models in suspension models. The results are reported in Yeo & Maxey [234]. We consider hydrodynamic interaction between a pair of coplanar spheres co-rotating in response to a rotating magnetic field at finite Reynolds numbers in Chapter 5. It is shown that a pair of rotating coplanar spheres at finite-Reynolds-number flows experiences a repulsive hydrodynamic force in the axis connecting their centers due to the secondary flow induced by the nonlinear interaction. The FCM solution of the secondary flow is compared with the regular-perturbation solution of Bickley [17]. Most of the results in Chapter 5 are from the work with Professor Climent [39]. 6 Chapter 6 reports the LC-FCM results of concentrated suspensions of neutrally buoyant particles in a homogeneous linear shear flow under finite fluid inertia. The changes of the velocity statistics, such as the velocity fluctuations, velocity auto-correlation functions, and self-diffusion, and the rheological parameters, e.g. particle stresses, are thoroughly ˙ 2 ρf γa investigated as functions of the particle Reynolds number, Reγ˙ = µf . The behavior of the particle stresses are discussed by studying the changes in the pair-distribution function and a weighted pair-distribution function with Reγ˙ . In Chapter 7, we report the results of the FCM simulations of turbulence modulation by suspensions of finite size particles of different densities; heavy and neutrally buoyant particles, and (massless) bubbles. We present both Eulerian statistics of the mixture and Lagrangian statistics of the particle phase. The Eulerian statistics for the suspensions of heavy particles are compared with the previous numerical simulations by ten Cate et al. [208]. Chapter 7 is based on Yeo et al. [228]. Finally, Chapter 8 devoted to concluding remarks and brief discussions. Future research directions are also suggested in this chapter. Chapter 2 Numerical methods 2.1 Introduction Numerical simulations of monodisperse suspension flows in uniform shear have had a large impact on the characterization and modeling of low Reynolds number suspensions. The principal tools have been Stokes Dynamics (SD)[24], multipole expansions [35, 121, 187], and boundary integral methods [175]. Lattice-Boltzmann methods (LBM) bridge both finite Reynolds number and low Reynolds number systems and have also contributed [162, 163]. The force-coupling method (FCM) [135, 144] and subsequent developments, bridge both Stokes flows and finite, low Reynolds number systems. FCM may be used to simulate large systems of particles in both open and wall-bounded flows in fully three-dimensional configurations. For the accurate simulation of Stokes flow, a numerical method should be able to capture both the long-range multi-body interactions and the short-range lubrication interactions. Especially, the singular nature of the lubrication forces hinders the development of numerical schemes. For example, the most straightforward and exact numerical method would be the direct numerical simulation with an arbitrary Lagrangian-Eulerian technique [91, 92]. However, considering that the lubrication forces for the normal and tangential motions between a particle pair are, respectively, ∼ 1/² and log ², in which a² denotes the separation distance between two particles and a is the particle radius, the grid spacing should be smaller than at least 10−3 a ∼ 10−4 a to resolve the lubrication forces. Correspondingly, the time step size also should be very small, of the order of 10−3 ∼ 10−4 , making long term simulations of the suspension dynamics too costly. 7 8 In Stokes flow, the hydrodynamic interactions are determined solely by the instanta- neous configuration of particles. Using the special properties of Stokes flow, several numer- ical methods have been developed focusing on computing hydrodynamic interactions rather than resolving the whole flow field. The multipole expansion is a representative approach to consider hydrodynamic interactions between particles. However, Cichocki & Felderhof [35] reported that the number of multipole moments needed to resolve the hydrodynamic inter- action becomes impractically large as the gap between particles becomes so small that the lubrication force plays an important role. Durlofsky et al. [63] developed the Stokesian Dy- namics method, which is a low order multipole representation, supplemented by short-range lubrication forces, to compute the position and movement of suspended particles. In the Stokesian Dynamics, they assumed the lubrication interaction can be added in a pair-wise manner in the resistance formulation, which has become a standard approach for incorpo- rating the lubrication forces [36, 121, 162]. Sierou & Brady [196] developed the Accelerated Stokesian Dynamics (ASD) method and performed the simulations of up to 1000 particles [198]. The boundary element method (BEM) is another distinguishing numerical method for Stokes flows, see for example [175]. BEM can calculate the hydrodynamic interactions in particulate suspensions with greater accuracy. However, even here some form of lubrication or contact forces must be included between rigid particles. Ingber et al. [97] have devel- oped a traction-corrected BEM which can accurately calculate the lubrication interaction. Although BEM can simulate wall-bounded suspensions and suspension of particles with arbitrary shapes [176, 177], due to the high computational cost most simulations are done in 2-dimensions or with relatively small numbers of particles. Since Nguyen & Ladd [162] implemented the lubrication interaction into the lattice- Boltzmann simulation, LBM has become popular for the suspensions in the low to finite Reynolds number flows [118, 163, 215]. Compared to SD, LBM is computationally inexpen- sive and a rigid wall boundary can be included without any special treatment, while present techniques for SD use an image method [18, 207] or wall particles [164, 199] to represent a rigid wall. Maxey & Patel [144] developed the force-coupling method (FCM) for Stokes flow by replacing the Dirac delta function in the standard multipole expansion [185] by a localized force envelope. The force-coupling method has been verified [49, 136, 135] and applied in 9 many suspension flows; for example, a sedimentation problem [51], bimodal suspensions [2], turbulent flows [224], and biological flows [108]. Since multi-body hydrodynamic inter- actions are accounted for by solving the Stokes equation, the computational cost of FCM depends on the choice of the Stokes solver. In a periodic domain, the long-range hydro- dynamic interactions can be calculated in O(Np logNp ) operations using a Fourier spectral method, in which Np is the number of particles. Dance & Maxey[51] performed the numer- ical simulations of particle sedimentation with up to 10,000 particles. The force-coupling method can be implemented with any existing flow solver by adding functions to integrate and distribute the force envelope. Recently, Liu et al. [131] showed that the force-coupling method can simulate suspensions of ellipsoidal particles by appropriately rescaling the force envelopes. In the force-coupling method, the translational and angular velocities of a particle are estimated by the local average of the fluid velocity weighted by the corresponding force envelopes. This approach has a computational advantage by reducing the number of grid points necessary to resolve a particle. Once the resolution is fine enough to resolve the force envelope, FCM can accurately reproduce the far-field solution. Typically, it requires only a/∆x ' 3 to resolve a particle in which ∆x is the grid spacing. This is less than other methods, such as the immersed boundary method or the lattice-Boltzmann simulations. On the other hand, it has a disadvantage that the near field solution is not correctly resolved. When two particles are close, the force-coupling method cannot reproduce the lubrication effects [135]. As a result, the force-coupling method has been used mainly for low volume fractions, φ < 0.1. Dance & Maxey [50] developed a method to incorporate the lubrication effects into the force-coupling method based on the exact solution of the viscous lubrication interactions of a particle-pair. They employed a predictor-corrector type approach. First, the far- field interaction is calculated by the standard FCM to estimate the lubrication force and then the lubrication force is used as a feedback force in the mobility formulation. By construction, their method can reproduce the exact particle velocities for a particle-pair interaction. At low volume fractions in which most lubrication interactions are from particle doublets, the ‘lubrication barrier’ would be sufficient [2]. However, it is difficult to generalize the approach to include multi-body interactions, which generally require the addition of lubrication interactions to a resistance formulation [63]. 10 The main goal of this chapter is to develop an efficient method to incorporate the lubrication forces for general volume fractions into the force-coupling method. First, a brief review of the force-coupling method is given. For the particles near a wall, the force envelopes are modified to correctly account for particle-wall hydrodynamic interactions. A procedure to include an FCM quadrupole is proposed. We derive the discretized equations of FCM and show that the force-coupling method in Stokes flow can be expressed in terms of a mobility matrix problem. A new efficient and robust iterative scheme to calculate the stresslet is introduced. The lubrication correction method for the force-coupling method is developed by applying the pair-wise additivity approximation. The force-coupling procedure for Navier-Stokes suspensions is then discussed. The numerical methods used to solve the Stokes equations in a tri-periodic and a wall-bounded domain are explained. The numerical simulations of suspensions of monodisperse spheres in an infinite domain and a channel are performed to verify the present lubrication correction method. 2.2 Force-coupling method: Stokes flow 2.2.1 Review of the force-coupling method The equation of the fluid motion with the force-coupling method is ∇p(x) = µ∇2 u(x) + f (x), (2.1) ∇ · u = 0, (2.2) in which p is pressure, µ is viscosity of the fluid, u is fluid velocity. The force density f is defined as Np ½ ¾ X n n n ∂ n fi (x) = Fi ∆M (x − Y ) + Gij ∆D (x − Y ) , (2.3) ∂xj n=1 where Fi and Gij are the force monopole and force dipole moments, respectively. ∆M and ∆D are the FCM force envelopes defined as µ ¶ 1 x2 ∆M (x) = 2 )3/2 exp − 2 , (2.4) (2πσM 2σM µ ¶ 1 x2 ∆D (x) = 2 )3/2 exp − 2 . (2.5) (2πσD 2σD 11 The length scales σM and σD are chosen to satisfy the overall energy budget for the flow [135, 144], a √ = π, (2.6) σM a ¡ √ ¢1/3 = 6 π . (2.7) σD These expressions for the length scales ensure that the exact Stokes drag is obtained and further that corresponding degenerate force quadrupoles and the linear Faxen terms for finite particle size and local flow variations are estimated to good approximation with no additional terms. Once u(x) is obtained, the velocity of each particle V n (t) is found by the weighted volume integral of u(x) [144], Z n V = u(x)∆M (x − Y n )d3 x. (2.8) Similarly, the angular velocity of each particle Ωn is calculated from Z 1 ∂uk Ωni = ²ijk ∆D (x − Y n )d3 x. (2.9) 2 ∂xj In dynamic simulation, the location of a particle is determined by numerically integrating the equation of particle motion, dY n = V n. (2.10) dt Here, we use the third-order Adam-Bashforth method to solve equation (2.10). The monopole coefficient F n is the force of the particle on the fluid, which is the sum of forces on the particle other than the hydrodynamic interaction, and include gravity, short-range inter-particle surface forces, magnetic forces, and random Brownian forces. The dipole coefficient Gij consists of symmetric and anti-symmetric parts. The anti- symmetric part Tij is related to the torque exerted by the particle on the fluid, 1 Tij = ²ijk Tkext , (2.11) 2 in which T ext in turn denotes an external torque on the particle. The symmetric part Sij 12 corresponds to a stresslet acting on the fluid. Sij is found from the condition that the contribution of the stresslet to the total rate of work on the fluid is zero [135]. In other words, Sij is chosen to satisfy the constraint, Z µ ¶ 1 ∂ui ∂uj Eij = + ∆D (x − Y n (t))d3 x = 0, (2.12) 2 ∂xj ∂xi for each particle. Since Sij depends on the rate-of-strain, it has to be determined by an iterative method. Lomholt et al. [136] and Dance & Maxey [50] described steepest-decent iterative methods to accomplish this. In section 2.2.4 a conjugate gradient procedure to solve equation (2.12) efficiently is introduced. 2.2.2 Quadrupole expansion In this section, we extend the force-coupling method to include a higher-order multipole. In Lomholt & Maxey [135], it is shown that the force-coupling method with force dipole gives a good approximation to the exact Stokes solution in uniform straining flows. As such, the force-coupling method can be used reliably for non-uniform flows, if the lengthscale of the spatial variation of the fluid velocity is large enough compared to the particle diameter that the strain rate can be assumed to be uniform on the particle scale. However, in some special cases (see, e.g., chapter 9 of Keaveny [107]) or for large particles in a quadratic flow, a higher-order multipole needs to be considered. 2.2.2.1 Traceless quadrupole Consider the Stokes equations with a singularity and its derivatives at the origin, ∞ X ∂p ∇p(x) = µ∇2 u(x) + Fξ1 ···ξp δ(x). (2.13) ∂xξ1 · · · ∂xξp p=0 The fluid field induced by the singularities can be obtained by the Green’s function for Stokes flows and its derivatives. The Green’s function for the Stokes equations, or so called the Oseen tensor, is given by µ ¶ δij xi xj Gij (x) = + 3 , (2.14) r r 13 in which r = |x|. The fundamental solution of the Stokes equations, so called Stokeslet, is given by 1 ui (x) = Gij Fj . (2.15) 8πµ Then, the solution of (2.13) can be written as the sum of the Stokeslet and its derivatives, ∞ 1 X ∂p ui (x) = Fj ξ1 ···ξp Gij . (2.16) 8πµ ∂xξ1 · · · ∂xξp p=0 The flow field induced by an object in Stokes flows can be obtained by a combination of these singularity solutions. For example, a sphere moving in a quiescent flow is given by the sum of the Stokeslet and the degenerate quadrupole, µ ¶ a2 2 Gij ui (x) = 1 + ∇ Fj , (2.17) 6 8πµ in which a is the particle radius. Consider a sphere in an undisturbed flow (u∞ ) given by u∞ i = Sijk xj xk , (2.18) in which the third-order tensor Sijk is chosen to satisfy ∞ ∂Eij = C. (2.19) ∂xk ∞ is the strain rate E = 1/2(∂ u + ∂ u ) and C is a constant. To satisfy the Here, Eij ij j i i j divergence-free (∇ · u∞ = 0) and traceless (∇2 u∞ = 0) conditions, the third-order tensor Sijk should satisfy Siij = Siji = 0, Sijj = 0. And, from the symmetry of the strain rate, Sijk = Sikj . 14 The disturbance flow generated by a sphere in u∞ is given by a traceless force quadrupole and a degenerate force sextupole, ¡ ¢ 2 Gij, kl (r) uD i (x) = 1 + α∇ Qjkl , (2.20) 8πµ in which Qjkl denotes the quadrupole strength and α is a constant to be determined from the Fax´en’s law. The quadrupole is given by ∂ 2 Gij 1 = (−δij δkl + δjk δil + δik δjl ) ∂xk ∂xl r3 3 − (−δij xk xl + δjk xi xl + δik xj xl + δil xj xk + δjl xi xk + δkl xi xj ) r5 15 + xi xj xk xl , (2.21) r7 and, similarly, the degenerate-sextupole is ∂ 2 Gij 6 ∇2 = − (δij δkl + δjk δil + δik δjl ) ∂xk ∂xl r5 30 + (δij xk xl + δjk xi xl + δik xj xl + δil xj xk + δjl xi xk + δkl xi xj ) r7 210 − xi xj xk xl . (2.22) r9 As the sphere is not moving, the no-slip boundary condition on the sphere surface is uD (r = a) = −u∞ (r = a). To match the no-slip boundary condition, the terms in uD of which order is higher than those in u∞ need to vanish, which in turn determines the constant α as α = a2 /14, and so µ ¶ a2 2 Gij, kl uD i (x) = 1+ ∇ Qjkl for r ≥ a. (2.23) 14 8πµ Matching the no-slip boundary condition, the quadrupole strength Qijk can be related to Sijk as · Qjkl (−10δij δkl + 4δjk δil + 4δik δjl ) 7a3 ¸ 6 − 2 (−6δij xk xl + δjk xi xl + δik xj xl + δil xj xk + δjl xi xk + δkl xi xj ) (2.24) a = −8πµSijk xj xk . 15 Applying the divergence-free, traceless, and symmetry conditions yields 7 Qijk = − πµa5 Sijk . (2.25) 3 The corresponding Stokes equations with the finite force quadrupole may be written as ∂p ∂2∆ = µ∇2 ui + Qijk , (2.26) ∂xi ∂xj ∂xk in which µ ¶ 1 r2 ∆(r) = exp − 2 . (2.27) (2πσ 2 )3/2 2σ Here, the lengthscale σ is to be determined. Similar to the standard multipole expansion, the disturbed velocity given by the FCM quadrupole can be written as uD F CM i (x) = Gij, kl (x)Qjkl , (2.28) F CM is the FCM Oseen tensor given by Maxey & Patel [144]. By differentiating in which Gij the FCM Oseen tensor, the FCM force quadrupole is given as · ¸ F CM 1 dA(r) xk xl 1 dA(r) d2 A(r) Gij, kl (x) = δij δkl + (δik δjl + δjk δil )B(r) − δij 2 − r dr r r dr dr2 1 dB(r) + (δjk xi xl + δik xj xl + δilxj xk + δjl xi xk + δkl xi xj ) µ ¶ r dr xi xj xk xl 1 dB(r) d B(r) 2 − − . (2.29) r2 r dr dr2 And, ·µ ¶ µ ¶ µ ¶ ¸ dA 1 3 ξ 6 exp(−ξ 2 /2) = − 1 + 2 erf √ − 4ξ + √ , (2.30) dr 8πµr2 ξ 2 ξ 2π 2 ·µ ¶ µ ¶ µ ¶ 2 ¸ d A 1 6 ξ 12 3 exp(−ξ /2) = 1 + 2 erf √ − + 6ξ + 2ξ √ , (2.31) dr2 4πµr3 ξ 2 ξ 2π ·µ ¶ µ ¶ ¸ 1 3 ξ 6 exp(−ξ 2 /2) B = 1 − 2 erf √ + √ , (2.32) 8πµr3 ξ 2 ξ 2π ·µ ¶ µ ¶ µ ¶ ¸ dB 1 15 ξ 30 exp(−ξ 2 /2) = − 3 − 2 erf √ + + 4ξ √ , (2.33) dr 8πµr4 ξ 2 ξ 2π 2 ·µ ¶ µ ¶ µ ¶ ¸ d B 3 15 1 ξ 15 1 3 exp(−ξ 2 /2) = 1− erf √ + + 3ξ + ξ √ , (2.34) dr2 2πµr5 2 ξ2 2 ξ 3 2π 16 F CM has a finite value, in which ξ = r/σ. At r = 0, Gij, kl F CM 1 Gij, kl (0) = √ (δik δjl + δjk δil − 4δij δkl ). (2.35) 30πµσ 3 2π Considering that a traceless quadrupole does not contribute to the total rate-of-work, we have Z ∂ 2 uFi CM ∆(r)d3 x = −2Sijk . (2.36) ∂xj ∂xk From the Fourier transform of the Stokes equations (2.26), the fluid velocity can be written as µ ¶ 1 ki kj kk kl b ˆFi CM (k) = − u δij − 2 ∆(k)Qjkl , (2.37) µ k k2 b is where the hat denotes a Fourier coefficient, k is a wavenumber, i.e. ∆ µ 2 2¶ b 1 k σ ∆(k) = 3 exp − . (2.38) (2π) 2 Then, Z ∂ 2 uFi CM Γijk = ∆(r)d3 x ∂xj ∂xk Z µ ¶ Qpqr ki kp kj kk kq kr = δ ip − exp(−k 2 σ 2 )d3 k. (2.39) (2π)3 µ k2 k2 Consider a 2-dimensional flow, such that S111 = −S122 , (2.40) S122 = S212 . (2.41) Then, Z · ¸ Q111 k22 2 1 4 Γ122 = k − k2 − 2 (k1 − 3k1 k2 ) exp(−k 2 σ 2 )d3 k. 2 2 2 (2.42) (2π)3 µ k2 1 k 17 (a) (b) 4 4 2 2 y/a y/a 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 x/a x/a Figure 2.1: Streamlines of (a) exact solution and (b) FCM solution. Direct integration of (2.42) gives 1 3/2 5 −1 Γ122 = − (π σ µ) Q111 = 2S111 . (2.43) 70 The FCM lengthscale σ for force quadrupole can be computed by substituting (2.25) into (2.43), ³ a ´5 √ = 60 π. (2.44) σ To verify the result, the flow field induced by a sphere in a two-dimensional uniform viscous force field is computed from the FCM Green’s function. The undisturbed field is given by u∞ i = Sijk xj xk , (2.45)   1 0 0     S1jk = 0 −1 0 ,   0 0 0   0 −1 0     S2jk = −1 0 0 , (2.46)   0 0 0 S3jk = 0. 18 (a) 4 (b) 1 Stokes 3 FCM 0 2 -1 u1 u1 1 -2 0 -3 -1 -4 0 0.5 1 1.5 2 0 0.5 1 1.5 2 x1/a x2/a Figure 2.2: u1 along (a) x1 and (b) x2 axes. Figures 2.1 shows the streamline obtained from the exact and the FCM solutions. It is shown that the FCM solution agrees well with the exact solution. The flow velocity in x1 direction is shown in 2.2. It is shown that FCM can reproduce the far-field solution almost exactly. However, similarly to the results in Lomholt & Maxey [135], there is some discrepancy between the FCM and exact solutions near the particle surface (r < 1.25a). Using a regularized multipole, FCM resolves a Fax´en term only partially [144]. It seems that the error may be caused by the degenerate sextupole being resolved only partially. In the case considered here, a single sphere in an infinite domain, it is straightforward to calculate the quadrupole strength analytically. However, in general, we need an iterative solver to compute multibody hydrodynamic interactions or in the case where Sijk is not uniform. Here, we suggest an iterative procedure to find Q. We hope to find Qijk from the constraint Z ¡ F CM ¢ Γ + Γ∞ ∆(x)d3 x = 0, (2.47) in which the third-order tensor ΓFijkCM = ∂j ∂k uFi CM . Define a linear operator Ijk , Z µ ¶ ∂2 Ijk Q = SQ ∆(x)d3 x, (2.48) ∂xj ∂xk in which S is a Stokes operator to compute the fluid velocity from the given FCM quadrupole. 19 Let r k be a residual by k-th iteration, ˜ ∞, r k = IQk + Γ (2.49) R ˜∞ = in which Γ Γ∞ ∆dx. Assume (k + 1)-th approximation can be obtained from the following relation, Qk+1 = Qk + λr k , (2.50) in which λ is a constant. For the convergence of the sequence, we want to find λ minimizing ˜ ∞ k2 . λ can be found from a L2 -norm, kIQk+1 + Γ ∂ ˜ ∞ k2 = ∂ ˜ ∞ k2 = ∂ k(IQk + Γ ˜ ∞ ) + λIr k k2 kIQk+1 + Γ kI(Qk + λr k ) + Γ ∂λ ∂λ ∂λ ∂ = kr k + λIr k k2 . (2.51) ∂λ Hence, r k · Ir k λ=− , (2.52) Ir k · Ir k and Qk+1 = Qk + λr k , (2.53) ˜ ∞ = I(Qk + λr k ) + Γ r k+1 = IQk+1 + Γ ˜ ∞ = r k + λIr k . (2.54) Note that the iterative solver is essentially a steepest-decent method and the iteration converges only if the linear operator I is positive (semi-)definite, which is true for the Stokes operator. So far, we have considered only the traceless quadrupole. The degenerate quadrupole case will be considered in 2.2.2.2. In the numerical tests, it is found that the required number of iteration is similar to that of the dipole. 2.2.2.2 Degenerate quadrupole Consider a force-free and torque-free particle in a parabolic flow, u∞ = (A(x22 + x23 ), 0, 0), (2.55) 20 in which A is a constant. Assume a particle is placed at the origin x = (0, 0, 0)T . Since the flow is axisymmetric, it is convenient to use a cylindrical coordinate. Let ur and uθ be the radial and azimuthal components of u, respectively. Then,   u∞ 2 2 1 ∂ψ ∞ r = Ar sin θcosθ = r2 sinθ ∂θ , (2.56)  u∞ = −Ar2 sin3 θ = − 1 ∂ψ∞ , θ rsinθ ∂r in which ψ ∞ is the streamfunction of u∞ . Solving (2.56) for ψ ∞ gives 1 ψ ∞ = Ar4 sin4 θ. (2.57) 4 Similarly, using the streamfunction, the Stokes equation for the disturbance velocity uD reduces to [87] E 4 ψ D = 0, (2.58) in which E is an operator defined as µ ¶ 2∂2 sinθ ∂ 1 ∂ E = 2+ 2 . (2.59) ∂r r ∂θ sinθ ∂θ The solution of (2.58) can be found by splitting into the homogeneous ψhD and particular solutions ψpD ,   E 2 ψ D = 0, h (2.60)  E 2 ψ D 6= 0. p As the streamfunction subjected to the operator E 2 represents the vorticity, ψhD and ψpD correspond to the flow by irrotational and rotational singularities in the multipole solution, respectively. The general solution for ψhD is given by ∞ X ¡ ¢ ψhD (r, ξ) = An rn + Bn r−n+1 Gn (ξ), (2.61) n=0 in which A and B are constants, ξ = cosθ, and Gn (ξ) is the Gegenbauer function. The first 21 few terms of the Gegenbauer function are 1 1 G2 (ξ) = (1 − ξ 2 ) = sin2 θ, 2 2 1 1 G3 (ξ) = ξ(1 − ξ ) = cosθsin2 θ, 2 2 2 1 1 G4 (ξ) = (1 − ξ )(5ξ − 1) = sin2 θ(4 − 5sin2 θ), 2 2 8 8 1 1 G5 (ξ) = ξ(1 − ξ 2 )(7ξ 2 − 3) = cosθsin2 θ(4 − 7sin2 θ). 8 8 The solution for ψpD is given by ∞ X ¡ ¢ ψpD (r, ξ) = Cn rn+2 + Dn r−n+3 Gn (ξ). (2.62) n=2 Therefore, ∞ X ¡ ¢ ψ D (r, ξ) = An rn + Bn r−n+1 + Cn rn+2 + Dn r−n+3 Gn (ξ). (2.63) n=2 The coefficients An , Bn , Cn , and Dn in (2.63) can be determined by applying the no- slip boundary condition on the particle surface. In a cylindrical coordinate, the no-slip boundary conditions are uD ∞ r + ur = Vr & uD ∞ θ + uθ = Vθ on r = a, (2.64) in which the particle velocity V = (2a2 A/3, 0, 0)T . By comparing the order of the terms, it is obvious that terms other than n = 2 and 4 in (2.63) should vanish. And, as the velocity is zero at r → ∞, all the terms with a positive exponent (rk , k ≥ 0) should be eliminated. Then, ¡ ¢ ψ D = B2 r−1 G2 (ξ) + B4 r−3 + D4 r−1 G4 (ξ). (2.65) Applying the no-slip boundary condition, 4 5 B2 = a A, 15 7 B4 = −a7 A, D4 = a5 A. 5 22 Comparing with the multipole solution, B2 , B4 , and D4 terms represent, respectively, the degenerate quadrupole, the degenerate sextupole, and the quadrupole. Now, the fluid ve- locity is 4 ³ a ´3 ur = cosθa2 A : degenerate quadrupole (D.Q) 15 r 7 ³ a ´3 + cosθ(2 − 5sin2 θ)a2 A : quadrupole (Q) (2.66) 10 r 1 ³ a ´5 − cosθ(2 − 5sin2 θ)a2 A, : degenerate sextupole (D.S) 2 r 2 ³ a ´3 uθ = cosθa2 A : D.Q 15 r 7 ³ a ´3 + sinθ(2 − 5sin2 θ)a2 A : Q (2.67) 40 r 3 ³ a ´5 − sinθ(2 − 5sin2 θ)a2 A. : D.S 8 r In Cartesian coordinates, µ ¶ 1 ˜ 2 ³ a ´3 x u ˜1 = − 1 − 3 21 V1 5 r r µ ¶ 21 ˜21 x ˜21 x x ˜22 5 x˜42 ³ a ´3 − 1−3 2 +5 4 − V1 (2.68) 20 r r 4 r4 r µ ¶ 3 ˜21 x x˜21 x ˜22 15 x ˜42 ³ a ´5 + 3−5 2 +5 4 − V1 , 4 r r 4 r4 r 3x ˜2 ³ a ´3 ˜1 x u ˜2 = V1 5 rµ2 r ¶ 21 x ˜1 x˜2 25 x ˜32 ³ a ´3 ˜1 x − −3 2 + V1 (2.69) 20 r 4 r4 r µ ¶ 3 x ˜1 x ˜2 35 x ˜32 ³ a ´5 ˜1 x + −5 2 + V1 , 4 r 4 r4 r where the tilde denotes a variable in the plane of axisymmetry. As the solution consists of the degenerate quadrupole, quadrupole, and degenerate sex- tupole, the fluid velocity may be written as Gij ¡ ¢ Gij, kl ui = ∇2 Hj + 1 + βa2 ∇2 Qjkl , (2.70) 8πµ 8πµ 23 in which β is a constant. By comparing (2.70) with (2.68 - 2.69), µ ¶ Gij a2 Gij, kl ui = ∇2 Hj + 1 + ∇2 Qjkl , (2.71) 8πµ 14 8πµ 8 4 H1 = − πµa5 A = − πµa3 V1 , H2 = H3 = 0, (2.72) 15 5   2A 0 0     Q1jk = 0 −A 0  ,   0 0 −A   0 −A 0     Q2jk = −A 0 0 , (2.73)   0 0 0   0 0 −A     Q3jk =  0 0 0 .   −A 0 0 Here, 7 7 A= πµa5 A = πµa3 V1 . 15 10 Now, consider the velocity field by the FCM degenerate quadrupole (uDQ ). Define Z ∂ 2 uDQ ΓDQ ijk = i ∆d3 x. (2.74) ∂xj ∂xk From the convolution theorem, Z µ ¶ 1 ki kp kj kk ΓDQ ijk = δip − 2 2 exp(−σDQ k 2 )d3 kHp (2.75) (2π)3 k µ = τijkp Hp , in which σDQ is the lengthscale of the FCM degenerate quadrupole and τijkp is a fourth-order isotropic tensor. In general, a fourth-order isotropic tensor can be written as τijkp = α1 δij δkp + α2 δik δjp + α3 δip δjk , (2.76) 24 in which αi is the coefficient to be determined. First, from the divergence-free condition, ΓDQ DQ DQ 111 + Γ212 + Γ313 = (α1 + 3α2 + α3 )H1 = 0, (2.77) and, from the symmetry (Γijk = Γikj ), α1 = α2 . Therefore, the fourth-order tensor reduces to τijkp = α (δij δkp + δik δjp − 4δip δjk ) . (2.78) By comparing the direct integration of (2.75) and the coefficient of τijkp , Z 1 3 τijji = 2k 2 exp(−σDQ 2 k 2 )d3 k = (µσDQ 5 π 3/2 )−1 (2.79) (2π)3 8 = α (δij δij + δij δij − 4δii δjj ) = −30α, 5 α = −(80µσDQ π 3/2 )−1 . (2.80) Therefore,   −2B 0 0     ΓDQ 1jk =  0 −4B 0 ,   0 0 −4B   0 B 0     ΓDQ 2jk = B 0 0 , (2.81)   0 0 0   0 0 B     ΓDQ 3jk = 0 0 0,   B 0 0 5 π 3/2 )−1 H . in which B = −(80µσDQ 1 Similarly, for the FCM quadrupole, Z µ ¶ 1 ki kp kj kk kq kr ΓQ ijk = δip − 2 2 2 3 exp(−σQ k )d kQpqr (2.82) (2π)3 k µk 2 = τijkpqr Qpqr , (2.83) in which σQ is the lengthscale of the FCM quadrupole and τijkpqr is a 6th-order isotropic 25 tensor. Although σQ is derived in the previous section, we will treat σQ as an unknown coefficient for now. A sixth-order isotropic tensor can be written as τijkpqr = δip (α1 δqj δrk + α2 δqk δrj + α3 δqr δkj ) + δiq (α4 δpj δrk + α5 δpk δrj + α6 δpr δkj ) + δir (α7 δpj δqk + α8 δpk δqj + α9 δpq δkj ) (2.84) + δij (α10 δpk δqr + α11 δpq δkr + α12 δpr δkq ) + δik (α13 δpj δqr + α14 δpq δjr + α15 δpr δjq ), in which αi ’s are unknown coefficients. Again, applying symmetry and divergence-free conditions, the number of the unknown coefficients can be reduced to three; τijkpqr = α1 δip (δqj δrk + δqk δrj ) + α2 [δpj (δiq δrk + δir δqk ) + δpk (δiq δrj + δir δqk ) − 2δip δkj δqr ] (2.85) + α3 [δqr (δij δpk + δik δpj ) − 4δip δkj δqr ]. From the constraint, Γ∞ + ΓDQ + ΓQ = 0, (2.86) we have ΓQ 212 = 2(α1 + 2α2 )Q122 = −B, (2.87) ΓQ 111 = −4(α1 + α2 − α3 )Q122 = 2B, (2.88) which indicates α2 = −α3 . Then, the sixth-order tensor becomes τijkpqr = α1 δip (δqj δrk + δqk δrj ) + α2 [δpj (δiq δrk + δir δqk ) + δpk (δiq δrj + δir δqj ) − δqr (δij δpk + δik δpj ) (2.89) + 2δip δkj δqr ]. 26 Comparing with the direct integration of τijkpqr , Z 1 k2 2 2 3 3 5 3/2 −1 τijkijk = 3 2 exp(−σQ k )d k = (µσQ π ) (2.90) (2π) µ 8 = 36α1 + 36α2 , and assuming σDQ = σQ = σ, the unknown coefficients α1 and α2 are 46 α1 = (µπ 3/2 σ 5 )−1 , 3360 11 α2 = − (µπ 3/2 σ 5 )−1 . 3360 And,   2B 0 0     ΓQ 1jk =  0 −B 0 ,   0 0 −B   0 −B 0     ΓQ 2jk = −B 0 0 , (2.91)   0 0 0   0 0 −B     ΓQ 3jk =  0 0 0 .   −B 0 0 By comparing (2.81) with (2.91) and from (2.86), the lengthscale σ can be determined as −5B = −2A, (2.92) a = (60π 1/2 )1/5 . (2.93) σ It is shown that the σ for the degenerate quadrupole flow is the same with that for the traceless quadrupole flow. The FCM and exact solutions are compared in figure 2.3. Figure 2.3 (a) shows u1 in the x1 direction and (b) is u1 in the x2 direction. It is shown that in both cases, the FCM solution approximates the far-field solution with a good accuracy. In u1 (x2 ), it is shown that the FCM solution underestimates the fluid velocity when r/a ≤ 1.25. As a final remark, in the standard FCM procedure [144], the particle velocity is obtained 27 (a) (b) 1.5 1.5 FCM FCM EXACT EXACT 1 1 u1 u1 0.5 0.5 0 0 -0.5 0 1 2 3 4 0 1 2 3 4 x1 x2 Figure 2.3: u1 in (a) x1 and (b) x2 directions. R by a weight average, V = u∆M d3 x, in which ∆M is the monopole envelope. As mentioned in the beginning of this section, the procedure works well if the particle diameter is small compared to the scale of the spatial variation of u. However, if we include the degenerate quadrupole to resolve the parabolic flow, the standard procedure no longer works. Consider the FCM for a sphere in a parabolic flow, ∂p 0=− + µ∇2 ui + Hi ∇2 ∆DQ + traceless Quadrupole. (2.94) ∂xi As the traceless quadrupole does not contribute to the translational motion of the particle, in the standard FCM procedure, the particle velocity is computed from Z Z µ ¶ à ! σ 2 + σ 2 Hj ki kj DQ M (uFi CM + u∞ 3 i )∆M d x = − δij − 2 exp −k 2 d3 k (2π)3 µ k 2 3˜ + Vi (2.95) π in which V˜ is the exact particle velocity. Then, the FCM particle velocity is H 3 V =− µ 2 2 ¶3/2 + π V˜ = 1.282V˜ . (2.96) σ +σ 12µ π DQ2 M It is shown that the standard FCM procedure to compute the particle velocity results in a non-negligible error if the degenerate quadrupole is included. A more general approach to include higher-order multipoles is a subject of further investigation. 28 2.2.3 Modification of the FCM envelopes near a wall For wall-bounded flows, (2.1) is solved numerically with the appropriate no-slip boundary conditions for rigid walls to obtain the incompressible Stokes flow field u, which gives far- field hydrodynamic interactions. The force envelopes, ∆M and ∆D , while narrowly confined to each particle do extend beyond the physical size of the particle. As a result, it is possible that when a particle is close to a wall the envelope overlaps the boundary. Similarly, in evaluating the integrals for the particle velocity (2.8) or for the rate of strain (2.12) and the angular velocity (2.9) the range of integration goes outside of the physical flow domain. In previous simulations of wall-bounded flows based on FCM [51, 132, 133, 135], if a particle is close to a wall then the FCM envelope is truncated at the wall boundary in (2.1) and integration is confined to the physical flow domain. For a particle in contact with a rigid boundary, the truncated volume in (2.8) is approx- imately 3.8% of the total and similarly for (2.12) is 1.4%. The effects of this truncation procedure can be included and accounted for at low volume fractions in the estimates of the particle-wall lubrication forces [50]. This procedure is self-consistent. However, it presents a minor problem for evaluating results directly related to an imposed external shear flow. As an illustration, consider a flow bounded by a rigid wall at x2 = 0 so that the flow domain ΩD is the semi-infinite region ΩD = (−∞, ∞) × (0, ∞) × (−∞, ∞) and the wall is located on the boundary ∂ΩD . A particle in a linear shear flow u∞ = (γx ˙ 2 , 0, 0)T , in which γ˙ is the shear rate, would usually give the result from FCM that Z ∞ ∞ V = u (Y ) = u∞ (x)∆M (x − Y )d3 x. (2.97) In (Y ) The domain of numerical integration is In (Y ) = {x : x ∈ R3 , |x−Y | < a×n}, representing a sphere of larger radius na where n is a positive real number. In order to obtain full accuracy, n is usually chosen such that n ≥ 2.5. If a particle is close to a wall such that In (Y ) * ΩD , then the FCM envelope is truncated at the wall boundary, i.e. the integration is performed only in In (Y ) ∩ ΩD . This results in V ∞ 6= u∞ (Y ). As a remedy to this problem, the FCM envelope is modified for particles near a wall (Y2 < a × n) and an image envelope is introduced. That is for the force monopole term, ∆W all M (x − Y ) = ∆M (x − Y ) − ∆M (x − Y Img ), (2.98) 29 in which Y Img = (Y1 , −Y2 , Y3 )T . Similarly, the dipole envelope is given as, ∆W D all (x − Y ) = ∆D (x − Y ) + ∆D (x − Y Img ). (2.99) The modified wall envelopes ensure that Z ∞ ∞ V = u (Y ) = u∞ (x)∆W all 3 M (x − Y )d x, (2.100) In (Y )∩ΩD and Z ∞ Eij = e∞ ij (Y ) = e∞ W all ij (x)∆D (x − Y )d3 x, (2.101) In (Y )∩ΩD as well as ensuring that the particle angular velocity matches the fluid vorticity (2.9) for the imposed external flow. This modified procedure is also self-consistent. Coincidentally, the image envelope ∆W M all would correspond to the lowest order image Stokeslet as given by [18], where they apply image methods to solve for Stokes flow in the presence of a rigid boundary. However, the reasons for the formulation of the wall-envelopes are different and the underlying numerical solution to (2.1) already ensures that the no-slip conditions are satisfied. 2.2.4 Discretized equation of FCM Due to the linearity of the Stokes equation, there are linear relations between the force moments and the flow parameters [28, 110]. Finding the force moments from the prescribed flow parameters is called the resistance problem while in the mobility problem the flow parameters are the unknowns which are to be found from the given force moments. In this section, the semi-discretized equation of the force-coupling method is presented conceptually in the form of the grand mobility matrix and a more efficient conjugate gradient method to calculate Sij is described. For simplicity, let the computational domain ΩD be a cubic domain in R3 . ΩD can be a bounded domain with some boundary conditions on ∂ΩD or an unbounded domain with periodic boundary conditions. We consider the fluid of a constant viscosity µ extends in the whole domain including the volume occupied by the particles and the fluid velocity and 30 pressure satisfy equations (2.1)–(2.2) in ΩD . In equation (2.8), V n can be approximated by a Gaussian quadrature rule, Ny Nz Nx X X X V n = u(xi,j,k )∆M (xi,j,k − Y n )wix wjy wkz , (2.102) i=1 j=1 k=1 in which Nx , Ny , and Nz are, respectively, the number of grid points in the x1 , x2 , and x3 directions (or x, y, z), xi,j,k is the position vector of the (i, j, k)-th grid point, and wiα is the weight of the Gaussian quadrature in the α direction. In matrix form, V = DM W u, (2.103) in which the terms are V : (3Np ) vector which contains the particle velocities     Vj1 V    1  2    Vj  V = V2  , Vj =   ..   (2.104)    .  V3   Np Vj u : (3N g) vector for the fluid velocity at each grid points (N g = Nx × Ny × Nz ).     u1 uj (x1,1,1 )        ..  u = u2  , uj =  .  (2.105)     u3 uj (x Nx ,Ny ,Nz ) W : (3N g) × (3N g) diagonal matrix for the weighting coefficients of the Gaussian quadrature.     w1x w1y w1z 0 ··· 0 w 0 0      .. ..     0 . .   W = 0 w 0, w =   (2.106)   .. ..   . . 0  0 0 w   0 ··· 0 x wy wz wNx Ny Nz 31 DM : (3Np ) × (3N g) matrix of ∆M at each grid points.   d 0 0     DM = 0 d 0 , (2.107)   0 0 d   ∆ (x1,1,1 − Y 1 ) · · · ∆M (xNx ,Ny ,Nz − Y 1 )  M   .. ..  d =  . .    ∆M (x1,1,1 − Y Np ) · · · ∆M (x Nx ,N y ,Nz N −Y ) p It is worthwhile to note that, because the force envelope is a rapidly decaying function, we can assume ∆M (x − Y ) = 0 if |x − Y | > 3a, which makes d a sparse matrix. Hence, the matrix-matrix multiplication DM W u is done in O(Np ) operations. Through linearity, the flow parameters can be represented by the sum of their compo- nents. First, the fluid velocity u generated by the given force monopole is T u = SDM F, (2.108) in which S is a (3N g) × (3N g) matrix determined by the choice of numerical scheme to solve the Stokes equations and F is the 3Np vector containing the monopole coefficients of each particle.     f1 Fj1        .  F = f2  , fj =  ..  (2.109)     Np f3 Fj In terms of these, the particle velocity V from the FCM monopole calculation is T V = DM W SDM F = MF V F , (2.110) in which MF V is the FCM mobility matrix relating the translational velocity to the force monopole. In the Stokes problem, the symmetric positive definiteness (SPD) of the mobility matrix can be proven by the reciprocal theorem [110]. The same property is preserved in FCM. It is trivial to show that the positive semi-definiteness of S, which is true for many numerical schemes such as the spectral or the spectral element methods, implies the positive 32 semi-definiteness of MF V . Secondly, the mobility matrix for the rate-of-strain (2.12) in response to the symmetric force dipole (stresslet) is similarly given by T MSE = −DD W SDD , (2.111) in which DD is a (5Np ) × (3N g) matrix to calculate the derivatives of the dipole Gaussian envelope ∆D (x − Y ),   dx 0 −dz     dy dx 0      DD = dz 0 dx  , (2.112)      0 dy −dz    0 dz dy   ∂ ∂ ∆ (x1,1,1 − Y 1 ) · · · ∂x ∆D (x Nx ,Ny ,Nz − Y 1)  ∂x D   .. ..  dx =  . . .   ∂ 1,1,1 − Y Np ) · · · ∂ Nx ,Ny ,Nz ∂x ∆D (x ∂x ∆D (x − Y Np ) Here, a derivative of u is calculated by Z Z ∂ui ∂ ∆D (x − Y )d3 x = − ui ∆D (x − Y )d3 x. (2.113) ∂xj ∂xj Equation (2.113) holds when either u or ∆D has compact support in ΩD . In a bounded domain, although ∆D is not compact in ΩD , u vanishes on ∂ΩD . Therefore, equation (2.113) holds also for the bounded domain. Instead of calculating all 9 components of the rate of strain Eij , only 5 independent components are evaluated   E11 − E33      2E12    ˜   E =  2E13  . (2.114)     E22 − E33    2E23 33 Similarly, let DT be a (3Np ) × (3N g) matrix whose entries are   0 dz −dy 1    DT = −d 0 dx  . (2.115) 2 z  dy −dx 0 Thirdly, the mobility matrix for the angular velocity of particles in response to torque is given as MT Ω = DT W SDTT . (2.116) The definitions of the other mobility matrices are as follows: MT V = DM W SDTT , MSV = DM W SDD T T , MSΩ = DT W SDD . (2.117) Consistent with reciprocal theorem, the FCM mobility matrices have the following symme- try relations, MT V = MFT Ω , MSV = −MFT E , MSΩ = −MTTE . (2.118) Applying constraints for the stresslet coefficients (2.12), the FCM grand mobility matrix M F CM is constructed as        V −V∞ F MF V MT V MSV F                 Ω − Ω∞  = M F CM T  =  MF Ω MT Ω MSΩ  T  , (2.119)        −E˜∞ S MF E MT E MSE S in which a sub-matrix MAB of M F CM is a mobility matrix to calculate a value B from a given coefficient A. Ω and T are (3Np ) vectors containing the angular velocity and torques, respectively. S is a (5Np ) vector of the 5 independent stresslet coefficients, S11 , S12 , S13 , S22 , S23 . V ∞ and Ω∞ are, respectively, the translational and angular velocities by imposed ˜ ∞ is the rate-of-strain vector by the imposed field. Note that the FCM grand field. E mobility matrix in equation (2.119) is not symmetric positive semi-definite, as may be seen from equation (2.118). To make it SPD, the signs of the third row need to be changed. Although FCM is formulated here in matrix form, in the actual computation, the grand 34 mobility matrix need not be constructed. The computation consists of three steps: 1. Calculate the force density at each grid point. T z = DM F + DTT T + DD T S. (2.120) The number of operations in this step is O(Np ). 2. Solve the Stokes equation. u = Sz. (2.121) If the computational domain is periodic in all 3 directions, a Fourier-spectral method can be used to solve the Stokes equation. Then, the Stokes equation can be solved in O(N g log(N g)) operations. If the domain size is increased keeping the volume fraction constant, Np increases linearly in proportion to N g. Hence, the number of operations in this step is O(Np log(Np )). 3. Finally, V and Ω are computed by integrating u (equations 2.8, 2.9), which requires O(Np ) operations. As a whole, the computational cost of FCM is O(Np log(Np )). A key aspect of solving the flow problem is the determination of the stresslet coefficients. In equation (2.119), F and T are known while S is to be determined from the constraints for dipole moments. In other words, ˜ ∞ − MF E F − MT E T = MSE S −E (2.122) must be solved to obtain S. Previously, Dance & Maxey[50] suggested a steepest-decent method to solve equation (2.122). At low volume fractions, this iterative method converges within a few iterations. However, it is found that the method converges very slowly at high volume fractions [37]. As it is shown in equation (2.111) that −MSE is SPD, a conjugate gradient method can be used to solve the system [56]. The solution procedures are as follows: 1. Solve the Stokes equation with F , T and an initial estimate S 0 to calculate the 35 residual r 0 and a vector p0 . ˜ ∞ + DD W S(D T F + D T T + D T S 0 ), r0 = E M T D p0 = r 0 . 2. Solve the Stokes equation with p as a dipole coefficient, ζ k = −DD W SDD T k p . 3. Update S, r, and p using the standard conjugate gradient procedure. Repeat the process until converges, i.e. ||r|| < δ for a tolerance δ. αk = r k · r k /ζ k · pk , S k+1 = S k + αk pk , r k+1 = r k − αk ζ k , β k = r k+1 · r k+1 /r k · r k , pk+1 = r k+1 + β k pk . (2.123) 4. Once converged, V and Ω are found from equation (2.119). In that the particle velocity is calculated by solving the Stokes equation instead of constructing the grand mobility matrix, the force-coupling method is a matrix-free method to solve the mobility problem. Representing the force-coupling method as a matrix-free method gives more options to improve the method. For example, it was observed that in the bimodal suspensions finding S takes more iterations than in the mono-disperse suspensions to reach a converged result[2, 37]. In the limit of a dilute suspension, MSE is a diagonal matrix such that n 20 Sij = πµa3n Eij n , (2.124) 3 in which an is the radius of the particle n. Due to the factor a3n in the diagonals of MSE , the condition number of MSE will increase approximately as the cube of the ratio of the 36 10 1 CG PCG 100 |r| 10-1 10-2 0 2 4 6 8 Num. of Iteration Figure 2.4: The residual 2-norm history for bi-disperse suspension. radius of the largest particle to the radius of the smallest particle, K(MSE ) ∼ (aL /aS )3 , which makes it difficult to invert MSE using CG even at low volume fractions. In this case, a preconditioned conjugate gradient method (PCG) is more effective [56].From the result in the dilute limit, we can choose the preconditioner as a diagonal matrix of which elements are 3/20πµa3n . Figure 2.4 shows the residual L2 -norm history for a bimodal suspension in a Poiseuille flow. The size ratio (aL /aS ) is 2 and the volume fractions of large and small particles are 9% (Np = 64) and 1% (Np = 57), respectively. The particles are randomly distributed. As expected, PCG converges faster than CG. 2.3 Lubrication correction 2.3.1 General formulation M F CM resolves terms up to the dipole moments and the many-body interactions are natu- rally resolved by solving the Stokes equation. However, due to the lack of high-order terms, the force-coupling method cannot fully resolve the particle-particle interaction when the 37 separation distance between two nearby particles is small [135, 144]. Dance & Maxey [50] developed an efficient lubrication model by comparing the exact re- sistance relations to the numerical resistance relations of FCM for particle-pair and particle- wall interactions. The lubrication model is suited to dilute system in which most lubrication interactions come from particle doublets. To account for the multi-body lubrication interac- tions, the lubrication force estimated by the sum of each particle-pair interactions is added to the mobility problem as a feedback force. However, it turned out that simply adding the lubrication force to the mobility problem in a pairwise manner is not enough to resolve the many-body hydrodynamic interactions accurately [50, 63]. Since Durlofsky et al. [63] suggested a lubrication approximation scheme for their Stoke- sian Dynamics simulation, their approach has been successfully applied in many subsequent SD simulations [24, 197, 198] as well as in other simulation methods [36, 121, 162]. Instead of adding high-order multipole moments in the resistance matrix, they added the near field interaction to the inverse of the grand mobility matrix in a pairwise manner as an approximation to the exact grand resistance matrix. The near field resistance function is calculated by subtracting the two-body SD resistance matrix (R2SD ) from the exact one (R2 ) obtained from the lubrication theory. Considering that L = R2 − R2SD contains the contributions from only high-order multipole moments which decay rapidly in space, the pairwise additivity of the lubrication correction is a reasonable approximation. In this sec- tion, we develop the lubrication correction method for the force-coupling method based on the pairwise addition of the lubrication matrix. The FCM resistance relation is     ¡ F CM ¢−1 V − V ∞ F M   =  , (2.125) −E˜∞ S in which     V F V =  , F =  . (2.126) Ω T 38 Following [63], the resistance matrix with the lubrication correction is     ³¡ ¢−1 ´ V − V∞ F M F CM +L   =  , (2.127) −E˜∞ S in which the lubrication correction matrix L is   RL L RΩF L REF  VF   L  L = RVL T L RΩT RET . (2.128)   RVL S L RΩS L RES RAB is the resistance matrix to calculate B from given A and RL is the difference between the exact two-body resistance matrix and the FCM resistance matrix, RL = R2B − R2B F CM . Let   RVL F L RΩF R= . (2.129) RVL T L RΩT Multiplying M F CM on both sides of equation (2.127) and rearranging give      MFV F tot I + MFV R −MSV V − V∞  =  , (2.130) ˜∞ −MFE F tot − E −MFE R MSE S tot in which F tot = F + REF L E∞, (2.131) S tot = S + RES L E ∞ − RVS L (V − V ∞ ). (2.132) Note that the dipole coefficient S tot obtained by solving equation (2.130) is different from the stresslet of the particle S. Equation (2.130) is not SPD. So that a GMRES or Bi-conjugate gradient method is required to solve the system. However, often these converge slowly or in some cases the convergence is not even guaranteed. Instead, in order to make the matrix SPD, equation (2.130) is solved for F lub = R(V − V ∞ ) instead of V − V ∞ . Rewriting equation (2.130) 39 yields      MFV F tot R−1 + MF V −MSV F lub  =  . (2.133) ˜∞ MFE F tot + E MFE −MSE S tot The signs in the second row are changed to make the matrix symmetric. The resistance relation for the force and torque of a particle pair is   V 1 − V ∞ (Y 1 )        F1 A11 A12 B 11 −B 12 G11 −G12 V 2 − V ∞ (Y 2 )       2    F   A12 A11 B 12 −B 11 G12 −G11   Ω1 − Ω∞ (Y 1 )    = µ  .  1   2 ∞ (Y 2 )  T   (B 11 )T (B 12 )T C 11 C 12 H 11 H 12   Ω − Ω         T2 −(B 12 )T −(B 11 )T C 12 C 11 H 12 H 11  −E ∞    −E ∞ (2.134) Exploiting the axisymmetry of the two sphere configuration, the resistance tensors can be written in terms of several scalar functions. Following the notation in Kim & Karrila [110], Aαβ A A ij /6πa = Xαβ di dj + Yαβ (δij − di dj ), αβ Bij /6πa2 = Yαβ B ²jik dk , αβ Cij /8πa3 = Xαβ C di dj + Yαβ C (δij − di dj ), 1 Gαβ kij /6πa 2 G = Xαβ (di dj − δij )dk + Yαβ G (di δjk + dj δik − 2di dj dk ), 3 αβ Hkij /8πa3 = Yαβ H (²ikl dl dj + ²jkl dl di ), in which d = r/|r| and r = Y β − Y α . The analytic forms of the scalar functions are given in [50, 101, 110]. To calculate the FCM resistance function, first the two-particle mobility matrices for various configurations and separation distances were constructed from the FCM Oseen operator given in [135, 144]. Then, the mobility matrices were inverted to obtain the values of the resistance functions for each separation distance. Finally, the resistance functions 40 Table 2.1: FCM resistance functions for a particle pair C0 C1 C2 C3 C4 A X11 2.3593 −2.6950 3.4626 −2.6058 0.8310 A X12 −1.7187 2.7545 −3.4658 2.6047 −0.8308 C X11 1.0151 −0.0419 0.0576 −0.0426 0.0132 C X12 −0.1241 0.1783 −0.1524 0.0816 −0.0205 G X11 0.9670 −2.0276 2.4960 −1.7932 0.5540 G X12 −1.1651 2.0898 −2.5005 1.7946 −0.5563 A Y11 1.3351 −0.6292 0.7689 −0.5602 0.1754 A Y12 −0.6114 0.7157 −0.7830 0.5571 −0.1737 B Y11 −0.1998 0.4427 −0.5613 0.4066 −0.1256 B Y12 0.3532 −0.5293 0.5749 −0.3972 0.1238 C Y11 1.1253 −0.3090 0.3944 −0.2779 0.0832 C Y12 0.0965 −0.1827 0.2050 −0.1378 0.0409 G Y11 0.0816 −0.2134 0.2889 −0.2144 0.0668 G Y12 −0.1111 0.2423 −0.2936 0.2065 −0.0630 H Y11 0.0165 −0.0669 0.1117 −0.0909 0.0294 H Y12 0.1695 −0.2605 0.2386 −0.1362 0.0361 were found by a non-singular asymptotic matching, R = C0 + C1 ² + C2 ²2 + C3 ²3 + C4 ²4 , (2.135) in which ² is the separation distance between two particles, a² = |r| − 2a. Coefficients of the polynomials are given in table 2.1. A is a resistance An example of the FCM resistance function is shown in figure 2.5. X11 function relating the force and the translational velocity for the translational motion along A for the force-torque-stresslet the sphere-sphere axis [110]. The corresponding values of X11 version of SD (SD-FTS) is obtained from [96]. For this resistance function, it is shown that the force-coupling method shows the similar far field approximation with SD-FTS. It is shown that FCM resistance function is almost indistinguishable from the exact resistance function when r/a > 2.5. Hence, the cut-off distance for the lubrication interaction is set to rc = 2.8a. Solving equation (2.133) using a conjugate gradient method requires inner and outer conjugate gradient solvers to invert both R and the full system simultaneously. However, due to the large condition number, it is not trivial to compute R−1 using an iterative solver. For example, in a squeezing configuration of a particle pair, only the resistance function 41 50 40 30 20 10 X11 A 2 2.2 2.4 r/a A of the exact solution (dashed line), FCM (solid Figure 2.5: The resistance function X11 line), and SD-FTS (dash-dot line). 42 X A is considered      Fk1 A XA X11 1 12  Vk    = 6πµa  . (2.136) Fk2 A XA X12 11 Vk 2 The leading-order singular term in X A is 1/² and the difference between diagonal (X11 A) A ) terms are O(10−3 ), which makes the condition number of the matrix and off-diagonal (X12 K(R) ∼ 1/² × 103 for ² small. Even when ² = 10−3 , K(R) is O(106 ). At high volume fractions, K(R) becomes too large for the matrix to be inverted using the standard conjugate gradient method. Here, we present a preconditioned conjugate gradient method in which R−1 is calculated recursively without an iterative solver. Let P be a preconditioner for equation (2.133) defined as   R 0 P −1 = , (2.137) 0 κI 20 3 in which κ is a scale factor, κ = 3 πµa . The procedure is as follows 1. Initialization: Initialize vectors r, ψ, and φ from an initial estimate F 0 = RV 0 and S 0 .     r 0 tot 0 0 M (F − F ) + MSV S − V 0  f  =  FV , (2.138) rs0 tot 0 0 ˜ MFE (F − F ) + MSE S + E ∞     ψf0 Rr 0   = P −1 r 0 =  f  , (2.139) ψs0 κrs0 φ0 = ψ 0 . (2.140) 2. Iteration: 43 For n = 0, 1, · · · ψn · rn αn = (2.141) φn · Aφn       F n+1 Fn φn   =   + αn  f  (2.142) S n+1 Sn φns r n+1 = r n − αn Aφn (2.143) ψ n+1 = P −1 r n+1 (2.144) ψ n+1 · r n+1 βn = (2.145) ψn · rn φn+1 = ψ n+1 + β n φn (2.146) where   R−1 + MFV −MSV A= . (2.147) MFE −MSE 3. Repeat step 2 until ||r n+1 ||2 ≤ δ for a tolerance level δ. In step 2, Aφn involves computing R−1 φnf . From the initialization step, it is obvious that R−1 φ0f = R−1 ψf0 = rf0 . (2.148) Similarly, for n ≥ 1, there is a general recursive solution, R−1 φnf = R−1 (ψfn + β n−1 φn−1 f ) = rfn + β n−1 (R−1 φfn−1 )   n−1 X n−1 Y = rfn +  β j  rfi . (2.149) i=0 j=i Once F lub is found, V can be computed from either V = R−1 F lub , (2.150) or V = V ∞ + MF V (F tot − F lub ) + MSV S tot . (2.151) 44 Note that at low volume fractions, there are many particles which do not have particles in their neighborhood. In that case, the number of degrees of freedom of equation (2.150) is 6 × Nc and not 6 × Np , in which Nc denotes the number of particles which have at least one particle within the cut-off distance (rc = 2.8a). Therefore, to calculate V for those particles which do not have any neighboring particles, we need to solve equation (2.151). 2.3.2 Particle-wall lubrication correction Close to a rigid, no-slip boundary, additional viscous lubrication corrections are required for the motion of a particle relative to the wall boundary. This modification for FCM is given by [50] in the simpler context of dilute suspensions. We summarize briefly, and with more general notation, the steps for adding these corrections in the resistance matrix. The resistance relation of a single particle moving in a flow near a wall is       V − V ∞ (Y ) F A B G     = µ    Ω − Ω∞ (Y )  . (2.152) T BT C H   −E ∞ Following the notation in [110], the resistance tensors are Aij /6πa = X A di dj + Y A (δij − di dj ), Bij /6πa2 = Y B ²jik dk , Cij /8πa3 = X C di dj + Y C (δij − di dj ), Gkij /6πa2 = Y G (di δjk + dj δik − 2di dj dk ), Hkij /8πa3 = Y H (²ikl dl dj + ²jkl dl di ), in which d is the unit vector from the particle center to a wall. The exact values for the various wall resistance functions are summarized in Appendix A, where they are given as asymptotic series in the gap width a² between the particle and the wall. Second, the non-singular estimates of the corresponding wall resistance functions from FCM are determined numerically for a single particle. This computation is done in the domain ΩD = (0, Lx ) × (0, Ly ) × (0, Lz ), where periodic boundary conditions are applied in the x1 and x3 directions with the no-slip boundary conditions (u = 0) on x2 = 0 and Ly . A Fourier spectral method is used in the horizontal (x1 , x3 ) directions and a spectral element 45 Table 2.2: Particle-wall FCM resistance functions C0 C1 C2 C3 C4 X A 8.1108 -26.117 62.532 -84.151 46.472 YA 2.3514 -3.6192 8.5516 -13.557 9.8027 Y B -0.2295 1.2393 -3.7731 6.6984 -5.0789 XC 1.1239 -0.3084 0.3610 -0.1688 -0.0030 YC 1.4205 -1.4217 3.0026 -4.1003 2.6037 YG 0.5629 -1.7656 4.0292 -6.1515 4.3758 Y H -0.0447 -0.1588 1.0918 -2.3320 1.7748 method is employed in the vertical direction (x2 ). Details of the numerical method are given in 2.5.2. In order to estimate the FCM resistance functions, the numerical simulations were performed in a large channel, Lx /a = Ly /a = Lz /a = 60, so as to minimize the effects of the periodicity and the presence of the upper wall. A sphere is located at Lx /a = Lz /a = 30. The hydrodynamic drag and torque were computed, varying the distance from the lower wall for the different configurations of particle motion or flow. The results were then matched to a regular asymptotic expansion for the FCM resistance function for small values of the gap a² between the particle and the lower wall; R = C0 + C1 ² + C2 ²2 + C3 ²3 + C4 ²4 . The coefficients are shown in table 2.2. The particle-wall resistance function X C is finite at the wall, ² = 0, and is matched satisfactorily by the FCM result so that in practice, special treatment of this term is not necessary. The lubrication correction is implemented in the same way as described in 2.3.1. First, R is constructed by summing all the particle-pair interactions. Since the lubrication force arises from the relative motion between particles, R2B is written in the relative velocity formulation [34, 162]. Then, the resistance tensors for the particle-wall lubrication are added to R. 2.3.3 Fictitious inertia for dynamic simulation In section 2.3.1, we have shown that the lubrication force can be computed without inverting the resistance matrix R. There are two formulae to compute V (2.150) and (2.151) and the difference in V calculated by between two formulae are negligibly small; usually less 46 than 0.1%. In practice, (1) solving (2.150) can be much faster than (2.151), as it consists of only matrix-vector multiplications, and (2) it was found that the preconditioned conjugate gradient solver converges faster when the velocity obtained by (2.150) is used as an initial estimate. Yet, inverting the ill-conditioned matrix R is not an easy task. Here, we introduces a fictitious inertia to the force balance equation (2.127) to accelerate the convergence for dynamic simulations. Adding fictitious particle accelerations on both sides of (2.127) yield         ³¡ ¢−1 ´ ˜ V λ d ˜ V F λ d ˜ V M F CM +L   +  dt  =   +  dt  , (2.153) ˜ −E ∞ 0 S 0 in which V˜ = V −V ∞ and λ is a constant. Simply, we can use an implicit time discretization for the acceleration term on L.H.S and an explicit discretization for the acceleration on R.H.S. If a second order temporal discretization is used,  n+1   ³¡ ¢−1 ´n+1 ˜ V λ V˜ n+1 − V˜ n−1 M F CM +L   +  = −E ˜∞ 2δt 0  n+1   F ˜ n λ 3V − 4V ˜ n−1 +V˜ n−2   + . (2.154) S 2δt 0 If we assume V is smooth in time, it is trivial to show that the error caused by the fictitious inertia is O(λδt2 ). As the time step δt in the simulations of concentrated suspensions is usually very small δt ∼ 10−3 /γ˙ – 10−4 /γ, ˙ the error due to the fictitious inertia is many times less than the tolerance of the conjugate gradient solver. It is also found that, in practice, it is sufficient to choose the coefficient λ ∼ O(δt), which makes the error even smaller ∼ O(δt3 ). Choosing λ = 2αδt and rearranging (2.154), the final equation for the lubrication force is     MFV F 0 (R + αI)−1 + MFV −MSV F lub  =  , (2.155) ˜∞ MFE F 0 + E MFE −MSE S tot 47 in which α is an arbitrary constant of O(1), I is the identity matrix, and F 0 = F tot + α(3V n − 3V n−1 + V n−2 ). (2.156) Once F lub is computed, the particle velocities are computed from V = (R + αI)−1 F lub . (2.157) Equation (2.155) can be solved by the same preconditioned conjugate gradient procedure as for (2.133). 2.4 Solution procedure for finite-inertia suspensions The force-coupling method was developed for Stokes flows [144, 135]. Nevertheless, it has been verified and successfully employed for finite-Reynolds-number and turbulent flows [136, 39, 211, 225, 228]. Although the FCM with both the mono- and dipole forces has been used for Navier-Stokes flows, the solution procedure has not been clearly shown in any of the previous papers. In this section, the force-coupling procedure for the Navier-Stokes equations with the lubrication correction shown in section 2.3.1 is described. The Navier-Stokes equations with the force-coupling method are ∇·u = 0 (2.158) ∂u ρf = −∇p − ρf (u · ∇)u + µ∇2 u ∂t Np X + Fk ∆M (x − Yk ) + (Gk · ∇)∆D (x − Yk ), (2.159) k in which ρf is the density of the fluid and Np is the total number of particles. From the equation of motion of particles, the monopole strength F is given by µ ¶ dV F = −(mp − mf ) − g − F lub + F P , (2.160) dt where g is the gravitational acceleration, F P is the interparticle force, mp is the mass of the suspended particle, and mf denotes the mass of fluid which occupies the same volume of the suspended particle. In the case of suspensions of neutrally buoyant particles, the first 48 term on R.H.S of (2.160) is zero. However, to construct a well-conditioned system, we will leave the particle acceleration term for now and disregard the gravitational acceleration. To solve the Navier-Stokes equations, usually a semi-implicit temporal discretization is employed; the viscous terms are treated implicitly and an explicit scheme is used for the non-linear terms. Using the Adam-Bashforth family schemes, a semi-discretized equation of (2.159) can be written as, Je X Ji X un+1 − un ρf = −∇pn+1 − αi H n+1−i + µ βi (∇2 u)n+1−i δt i=1 i=0 Np X + Fk ∆M (x − Yk ) + (Gk · ∇)∆D (x − Yk ), (2.161) k in which the coefficients αi and βi and the order of explicit Je and implicit discretization Ji depend on the choice of the scheme, H is the nonlinear term (Hi = −ρuj ∂j ui ). We have not specified a scheme for the force-coupling terms, yet. Note that, the Nabla operators – the gradient and Laplacian operators in the nonlinear and viscous terms, respectively – need to be included in the temporal discretization. In a fixed computational domain, it is obvious that ∇n+1 = ∇n . However, in a coordinate system deforming with time (see section 2.5.1), care should be taken on treating the Nabla operator as ∇n un 6= ∇n+1 un . The dipole term consists of the couplet and stresslet. When there is no external torque, the couplet is only due to the lubrication torque given in section 2.3.1. The stresslet acts as a constraint to satisfy the kinetic energy balance that the contribution to the total rate of work is zero, which corresponds to the rigidity constraint given in (2.12) and (2.122). The monopole coefficient includes the lubrication force, which is proportional to the velocity of the particle RV. As the lubrication force is proportional to ∼ 1/², in concentrated suspensions, the system becomes numerically too stiff to use an explicit method. Here, we use a fully implicit scheme for the FCM force terms. Then, then the FCM force terms in (2.161) are Np X Fkn+1 ∆M (x − Ykn+1 ) + (Gn+1 k · ∇)∆D (x − Ykn+1 ). (2.162) k Here, the monopole force consists of the interparticle force, lubrication force, and particle accelerations. The interparticle force only depends on the configuration of the particles at 49 t = (n + 1)δt, which can be easily evaluated. To obtain the lubrication force implicitly, we need to invert the FCM resistance matrix as done in section 2.3.1. This will be explained in detail later in this section. To make the resistance matrix well conditioned, the particle acceleration term is separated into two parts, µ ¶n+1 dV V n − V n−1 V n+1 − V n −(mp − mf ) = mf − mp . (2.163) dt δt δt Even for neutrally-buoyant particles (mf = mp ), we will treat the first and the second terms on R.H.S separately, similarly to the fictitious inertia in the previous section. The solution procedure of the semi-discretized equation (2.161) is as follows 1. Move the particles to the new location using an explicit scheme, J X Y n+1 = Y n + ξi V n−i , i=0 in which ξi is a coefficient of the explicit scheme used. 2. Calculate the force envelopes and the interparticle forces at t = (n + 1)δt. 3. Calculate an intermediate velocity from the explicit terms, µ Je Ji n δt X n+1−i X ˆ=u u + αi H +µ βi (∇2 u)n+1−i ρf i=1 i=1 Np " # ¶ X Vkn − Vkn−1 P n+1 + mf + Fk ∆M (x − Yk ) . δt k 4. Compute an intermediate pressure from the continuity equation, µ ¶ 2 ∗ ˆ u ∇ p = −ρf ∇ · , δt δt ˜ = u u ˆ− ∇p∗ . ρf Note that the Nabla operators are for t = (n + 1)δt. After this step, the intermediate ˜ satisfies the divergence-free condition. velocity u 50 5. Compute un+1 by solving ³ρ ´ ρf f − µβ0 ∇2 un+1 = ˜ − ∇p∗∗ + f , u δt δt in which ∇2 p∗∗ = ∇ · f , Np hm i X p f = − (Vkn+1 − Vkn ) + Fklub ∆M (x − Ykn+1 ) δt k + (Gn+1 k · ∇)∆D (x − Ykn+1 ). In the last step of the force-coupling procedure, the FCM terms on the R.H.S. are functions of u at t = (n + 1)δ and, hence, an iterative solver is necessary to solve the equation. The equation of un+1 in the last step is elliptic and the only difference compared to the Stokes equations is that the Laplacian operator in the Stokes equations is replaced by a Helmholtz operator. Hence, a preconditioned conjugate gradient procedure similar to the one shown in sections 2.3.1 and 2.3.3 can be used. In fact, the equation in the last step is easier to solve than the Stokes equations, as a spatial discretization of the Helmholtz operator often results in a better conditioned matrix than that of the Laplacian operator. In a matrix form, the final equation becomes        MFV F 0 V˜ (R + 1 δt Λ) −1 + MFV −MSV F lub  + =  , (2.164) MFE F 0 + E ∞ E˜ MFE −MSE S n+1 in which F 0 = F tot +(1/δt)ΛV n , Λ is the diagonal matrix for the mass and moment of inertia ˜ are, respectively, the velocities and strain rate estimated from the of the particle. V˜ and E ˜ The particle stresslet can be computed by (2.132), replacing S tot intermediate velocity u. by S n+1 . Note that the mobility matrix corresponds to solving the Helmholtz equation, not the Laplace equation in the Stokes flows. In Stokes flows, the particle velocity is calculated by inverting the resistance matrix, as we are only interested in the motion of the particles. However, in Navier-Stokes flows, the flow field at t = (n + 1)δt is necessary to compute the nonlinear terms. So, we need to first compute un+1 from F lub and S n+1 and, then, estimate V n+1 by the weighted integration of un+1 as shown in (2.8, 2.9). 51 As a final remark, the overall temporal accuracy of the scheme given in this section is ∼ O(δt), as usual in other schemes to simulate concentrated suspensions [162]. Because the time step size δt is usually very small in the simulations of concentrated suspensions, δt ∼ 10−3 /γ, ˙ the first order accuracy seems not be an issue in resolving essential dynamics of suspension flows. Nevertheless, we propose a higher-order temporal discretization. Usu- ally, in a semi-implicit method, the Crank-Nicolson method is used for the implicit terms. However, as we want to keep the stresslet term as a constraint at t = (n + 1)δt, a backward multi-step method would be more appropriate for our purpose. The stiffly-stable scheme proposed by Karniadakis et al. [103] for the Navier-Stokes equations would be a reason- able choice for the time integration. Using the stiffly-stable scheme, the semi-discretized equation is given as PJi −1 JX e −1 γ0 un+1 − q=0 αq un−q ρf = −∇pn+1 − βi H n−i + µ∇2 un+1 + f n+1 , (2.165) δt i=0 in which Np " µ ¶n+1 µ ¶n+1 # X dVk dVk f n+1 = mf − mp − Fklub + F P ∆M (x − Ykn+1 ) dt dt k + (Gn+1 k · ∇)∆D (x − Ykn+1 ). (2.166) The particle acceleration terms can be treated as before, but using a high-order polynomial. The pressure is calculated by using the same splitting procedure shown in the first-order scheme. In a tri-periodic domain, the pressure can be computed by an algebraic equation in the Fourier space. However, in a wall bounded domain, the pressure boundary condition needs to be treated carefully. See Karniadakis & Sherwin [104] for high-order pressure boundary conditions. The coefficients γ0 , αi , and βi can be found in Karniadakis et al. [103] and Karniadakis & Sherwin [104]. Even though the higher-order formulation is shown for an arbitrary order, generally it is not recommended to use more than the second-order scheme. First, as shown by Dahlquist [48], the scheme is A-stable only up to second order and, second, in a wall-bounded domain, the accuracy of the solver is anyway limited by the pressure boundary condition. 52 2.5 Flow solvers 2.5.1 Tri-periodic domain One of the simplest and yet most interesting non-colloidal suspension flows is homogeneous suspensions in a linear shear flow in an infinite domain. To model such an infinite domain, usually a tri-periodic domain is used in numerical simulations. It should be noted that, when a periodic boundary condition is used, it is assumed that the dimension of the domain in the periodic direction L is sufficiently large that the Eulerian correlation of the variables of interest decays before L/2. In other words, an integral lengthscale l should be much smaller than L/2. One of the difficulties in imposing the periodic boundary condition in a tri-periodic domain is that the periodic image of a variable is a function of the time in shear flow. That is, f (x1 , x2 , x3 ) = f (x1 + n1 H1 + n2 H2 γt, ˙ x2 + n2 H2 , x3 + n3 H3 ), in which x1 , x2 , and x3 denote, respectively, the velocity, velocity-gradient, and vorticity directions, Hi is the size of the domain in i−direction, n1 , n2 , and n3 are arbitrary constants, and γ˙ is the shear rate. Lees & Edward [125] devised a bi-periodic domain concept to account for this time dependence, which is the celebrated ‘Lees-Edward’ boundary condition. To solve the Navier- Stokes equations in such a tri-periodic domain with an imposed strain-rate, Rogallo [183] proposed a moving coordinate system, in which a coordinate system is linearly related with an inertial frame of reference. In this section, we show a Fourier spectral method to solve the Stokes equations with the force-coupling method in the moving coordinate system. Following Rogallo [183], the moving coordinate system is defined as, Xi = Bij xj , (2.167) in which X is the coordinate system deforming with the imposed strain-rate, x is the frame of reference, and B is a second order tensor. In the case of linear shear flow, B is   1 −γt ˙ 0     B = 0 1 0 . (2.168)   0 0 1 53 Then, the Stokes equations in the moving coordinate system are ∂p ∂ 2 ui Bji = µBkj Blj + fi (X), (2.169) ∂Xj ∂Xk ∂Xl or ∂p = µ∇02 u1 + f1 , (2.170) ∂X1 ∂p ∂p −T + = µ∇02 u2 + f2 , (2.171) ∂X1 ∂X2 ∂p = µ∇02 u3 + f3 , (2.172) ∂X3 where T is the strain T = γt ˙ and ∂2 ∂2 ∂2 ∂2 ∇02 = [1 + T 2 ] + + − 2T . ∂X12 ∂X22 ∂X32 ∂X1 ∂X2 Now, the periodic boundary conditions for a cubic domain D = [0, H1 ] × [0, H2 ] × [0, H3 ] are u(0, X2 , X3 ) = u(H1 , X2 , X3 ) ∀(X2 , X3 ) ∈ [0, H2 ] × [0, H3 ], u(X1 , 0, X3 ) = u(X1 , H2 , X3 ) ∀(X1 , X3 ) ∈ [0, H1 ] × [0, H3 ], u(X1 , X2 , 0) = u(X1 , X2 , H3 ) ∀(X1 , X2 ) ∈ [0, H1 ] × [0, H2 ]. The FCM force f in equation (2.169) is Np · ¸ X ∂ fi (X) = Fin ΞM (X n −Y )+ Gnij Bkj n ΞD (X − Y ) , (2.173) ∂Xk n=1 in which Ξ and Y are, respectively, the FCM envelope and the position vector of the particle center in the moving coordinate system. The FCM envelope in the moving coordinate system is · ¸ 1 (∆X1 + T ∆X2 )2 + ∆X22 + ∆X32 Ξ(∆X) = exp − , (2πσ)3/2 2σ 2 54 and its derivatives can be computed by µ ¶ ∂Ξ ∆X1 + T ∆X2 = − Ξ(∆X), ∂X1 σ2 ∂Ξ 1 = − 2 [(∆X1 + T ∆X2 )T + ∆X2 )] Ξ(∆X), ∂X2 σ ∂Ξ ∆X3 = − 2 Ξ(∆X), ∂X3 σ in which ∆X = X − Y . The relation between the strain-rate tensor Eij computed in X and that in the labora- tory frame eij is e11 = E11 , ∂u1 e12 = E12 − T , ∂X1 e13 = E13 , ∂u2 e22 = E22 − T , ∂X1 T ∂u3 e23 = E23 − , 2 ∂X1 e33 = E33 . Similarly, the vorticities in the moving coordinate Ω and in the laboratory frame ω are ∂u3 ω1 = Ω1 − T , ∂X1 ω2 = Ω2 , ∂u1 ω3 = Ω3 + T . ∂X1 The fluid velocity can be decomposed into the mean and fluctuating components, u = U + u0 , (2.174) in which U = γx ˙ 2 . The mean momentum equation can be obtained by substituting (2.174) into the Stokes equations in the frame of reference and averaging over the domain, ¿ À ∂p = µh∇2 Ui + ∇2 u0i i + hfi i. (2.175) ∂xi It is obvious that the second-order derivative terms on R.H.S of (2.175) disappear in the 55 mean momentum equation. The dipole terms in f also disappears in the mean momentum R equation because ∂Ξ/∂xj d3 x = 0. In the case of neutrally buoyant particles, the volume average of the monopole force should be zero. However, if the suspended particles are denser or lighter than the fluid, there is a mean monopole force, which corresponds to the density difference times gravity, to satisfy the mean momentum equation, the net force should be supported by the mean pressure gradient in (2.175). In the Fourier spectral method, we solve only for the fluctuating components and this process is taken care of by setting the mean component (zero wavenumber component) to be zero. Expanding functions in a truncated Fourier series and using Galerkin projection, the Stokes equations in (2.170–2.172) become u1 + fˆ1 , ik1 pˆ = −µ[|k|2 + T 2 k12 − 2T k1 k2 ]ˆ (2.176) i(k2 − T k1 )ˆ u2 + fˆ2 , p = −µ[|k|2 + T 2 k12 − 2T k1 k2 ]ˆ (2.177) u3 + fˆ3 , ik3 pˆ = −µ[|k|2 + T 2 k12 − 2T k1 k2 ]ˆ (2.178) in which ψˆ denotes a Fourier coefficient of a function ψ, X ψ(X) = ˆ ψ(k) exp(ik · X), (2.179) and k is wavenumber; ki = 2πn/Hi , n is an integer. Define Ki = Bji kj . Then, (2.176 – 2.178) can be written as ˆi + fˆi . iKi pˆ = −µ|K|2 u (2.180) The pressure in (2.180) can be eliminated by taking divergence of (2.180) and applying the ˆ = 0. Then, the fluid velocity is computed from continuity equation, K · u · ¸ 1 Ki Kj ˆ u ˆi = δij − fj . (2.181) µ|K|2 |K|2 Because the FCM envelopes are analytical, in theory, fˆ can be computed in the Fourier space as in the Fourier-Galerkin method. However, in practice, due to the load balancing problem between processors in parallel computing, a pseudo-spectral approach, in which f is computed in the physical space and then transformed into the Fourier space, is used. 56 In the moving coordinate system, as the deformation of the computational mesh gets larger, more grid points are needed to integrate the force envelope with the same accuracy as for a non-deformed mesh system. To prevent this problem, remeshing is performed at every T = 0.5H1 /H2 . This is done by setting T = −0.5H1 /H2 at T = 0.5H1 /H2 . For details about the remeshing, see [183]. 2.5.2 Bounded domain Suspensions in a confined flow is of great interest due to its relevance to many engineering processes. However, the dynamics of concentrated suspensions under a confinement is a virtually unexplored field. Here, we consider suspensions in a simple, yet important geom- etry; namely, a suspension flow bounded two parallel walls. We will briefly review a hybrid spectral/spectral-element method to solve the Stokes equations in a channel geometry. Consider a cubic domain D = (0, Hx ) × (0, Hy ) × (0, Hz ). The computational domain is bounded by two parallel walls located at x2 = 0 and x2 = Hy and assumed to be periodic in the horizontal directions (x1 , x3 ). Hence, the boundary conditions are u(0, x2 , x3 ) = u(Hx , x2 , x3 ) ∀(x2 , x3 ) ∈ (0, Hy ) × (0, Hz ), u(x1 , x2 , 0) = u(x1 , x2 , Hz ) ∀(x1 , x2 ) ∈ (0, Hx ) × (0, Hy ), u(x1 , 0, x3 ) = (Vlow , 0, 0)T ∀x ∈ ∂Dlow , u(x1 , Hy , x3 ) = (Vupp , 0, 0)T ∀x ∈ ∂Dupp , in which Dlow and Dupp are, respectively, the Dirichlet boundaries at x2 = 0 and Hy . Here, the walls are assumed to be mobile only in the x1 direction and Vlow and Vupp are the velocities of the lower and upper walls, respectively. In a Couette flow, the nominal shear rate γ˙ is set by the Dirichlet boundary condition, γ˙ = (Vupp − Vlow )/Hy . However, in practice, owing to the linearity of the Stokes equations, only the disturbance velocity field by hydrodynamic interaction between particles needs to be computed and the final solution can be obtained simply by superposing the undisturbed velocity field. Hence, in solving the Stokes equations, the homogeneous boundary condition is used for the Dirichlet boundaries. That is, Vlow and Vupp are set to zero. 57 The variational formulation of the Stokes equations is to find (u, p) ∈ X × M, such that −(p, ∇v) + (∇v, µ∇u) = (v, f ), ∀v ∈ X , (2.182) (q, ∇ · u) = 0, ∀q ∈ M, (2.183) R in which (·, ·) is an inner product (f, g) = D f gdD and the appropriate function spaces for u and p are X = H01 (D) and M = L20 (D), respectively. Here, H01 (D) is a space of functions S which are in the Sobolev space H1 and vanish at ∂D = ∂Dlow ∂Dupp and L20 (D) is a space R of functions which are in L2 (D) with zero average ( D f dD = 0). Note that, due to the inf-sup condition, different spaces are used for velocity and pressure. See [8, 29, 140] for details about the inf-sup condition. A Fourier spectral method is used in x1 and x3 directions and the spectral element method is used in x2 direction. Expanding the variables in Fourier series in x1 and x3 directions, the weak form of the Stokes equations (2.182 – 2.183) for each wavenumber (k1 , k3 ) ∈ (−∞, ∞) × (−∞, ∞) are ik1 (ˆ p, φ) + µ(dy u u1 , φ) = (fˆ1 , φ), ˆ1 , dy φ) + µk 2 (ˆ (2.184) −(ˆ p, dy φ) + µ(dy u u2 , φ) = (fˆ2 , φ), ˆ2 , dy φ) + µk 2 (ˆ (2.185) ik3 (ˆ p, φ) + µ(dy u u3 , φ) = (fˆ3 , φ), ˆ3 , dy φ) + µk 2 (ˆ (2.186) (ik1 u ˆ 1 + dy u ˆ2 + ik3 u ˆ3 , ψ) = 0, (2.187) in whichˆdenotes the Fourier coefficient, dy = d/dy, and φ and q are test functions for the velocity and pressure, respectively. For the spectral element discretization, D is partitioned into several elements. Let Ii be the interval in the i−direction, such that D = I1 × I2 × I3 . The interval I2 is partitioned into the union of disjoint intervals, such as K [ I2 = I2e , (2.188) e=1 in which I2e = (ye−1 , ye ) and 0 = y0 < y1 < · · · < yK−1 < yK = Hy . Then, the proper spaces for the spectral element discretization are ˆ ∈ Xh (I2 ) = H01 (I2 ) ∩ PN,K (I2 ), u (2.189) pˆ ∈ Mh (I2 ) = L2 (I2 ) ∩ PN −2,K (I2 ). (2.190) 58 Here, Pn,K (I) = {ψ ∈ L2 (I); ψ|Ik ∈ Pn (Ik ), k = 1, · · · , K} and Pn (Ik ) is the space of all polynomials of which degree is less than or equal to n. In the spectral element method, a variable can be written as Ndof −1 K X N X X i i g(x) = g˜ Φ (x) = g˜e, p φep (ξ), (2.191) i=0 e=1 p=0 in which g˜ is the expansion coefficient, Φ and φpe are, respectively, the global and local modes, Ndof is the total number of the degree of freedom, and ξ is the local coordinate in the standard element I st = (−1, 1). The local coordinate for an element e is simply x2 − ye−1 ξ=2 − 1, x2 ∈ I2e . (2.192) ye − ye−1 Here, we use the modal polynomial expansion. Hence, g˜e, p is the coefficient of the mode φep , not the physical value of g(x) at a collocation point. The modal expansion basis φep (ξ) can be found in Karniadakis & Sherwin [104]. In an element I2e , the inner products in (2.184 – 2.185) are computed by using a Gaus- sian quadrature. Here, we use the Gauss-Lobatto-Legendre (GLL) quadrature. The GLL quadrature is exact for a polynomial φ ∈ P2Q−3 , in which Q is the order of the GLL quadrature. The local operations of some of the inner products in (2.184 – 2.186) are given by, Z "N # 1 X u, φej ) (ˆ = ˜e, i φei (ξ) u φej (ξ)J e dξ −1 i=0   N X Q−1 X = Je ˜e, i  u φei (ξq )φej (ξq )wq  , (2.193) i=0 q=0 Z " N # 1 d X dφej (ξ) e−1 ˆ, dy φej ) (dy u = ˜e, i φei (ξ) u J dξ −1 dξ dξ i=0   N X Xµ Q−1 dφei ¶ µ dφej ¶ = J e−1 ˜e, i  u wq  , (2.194) dξ ξq dξ ξq i=0 q=0 Z "M # 1 X p, φej ) = (ˆ p˜e, i ψie (ξ) φej (ξ)J e dξ −1 i=0   M X Q−1 X = Je p˜e, i  ψie (ξq )φej (ξq )wq  , (2.195) i=0 q=0 59 in which ξq is the q-th GLL quadrature point, wq is the weight of the GLL quadrature, φe and ψ e are, respectively, the local basis functions for velocity and pressure, N, M are the orders of φ and ψ, respectively (M = N − 1). The Jacobian J is simply J = (ye−1 − ye )/2. Similarly, a local operation in (2.187) is   N X Q−1 X u, ψje ) = J e (ˆ ˜e, i  u φei (ξq )ψje (ξq )wq  . (2.196) i=0 q=0 In the modal expansion, we can choose N different from Q and the velocity and pressure share the same quadrature points. Considering the accuracy of the GLL quadrature rule, we use N ≤ Q − 2. A global operation can be easily constructed from the sum of the corresponding local operations shown above. In a matrix form, (2.184 – 2.185) can be written as ˜ P + µ(L + kM )U1 = BW F1 , ik1 M (2.197) ˜ + µ(L + kM )U2 = BW F2 , −DP (2.198) ˜ P + µ(L + kM )U3 = BW F3 , ik3 M (2.199) in which U and P are vectors of the global modal coefficients of velocity and pressure, respectively, and F is a vector of the force fˆ at each grid points. The dimensions of U , P , and F are ((N − 1)K + 1), ((M − 1)K + 1) and ((Q − 1)K + 1), respectively. B is a matrix ˜ be the matrix for ψ. That is, in which contains φ at each grid points and, similarly, let B a local element e, e Bij = φei (ξj ), for 0 ≤ i ≤ N & 0 ≤ j ≤ Q − 1, ˜e B = ψie (ξj ), for 0 ≤ i ≤ M & 0 ≤ j ≤ Q − 1. ij Let d and d˜ be the matrices of the derivatives of φ and ψ at each quadrature points, respectively; µ ¶ 1 dφei deij = , for 0 ≤ i ≤ N & 0 ≤ j ≤ Q − 1, Je dξ ξj µ ¶ 1 dψie d˜eij = , for 0 ≤ i ≤ M & 0 ≤ j ≤ Q − 1. Je dξ ξj 60 W is a diagonal matrix of the weights of GLL quadrature, Wije = J e δij wie , for 0 ≤ i, j ≤ Q − 1. Then, the matrices in (2.197 – 2.199) can be written as M e = B e W e B eT , Le = de W e deT , M˜ e = B e W e B˜e T , ˜ e = B e W e d˜e T . D Assembling local matrices to build a global matrix is straightforward. For details about constructing a global matrix, see Karniadakis & Sherwin [104]. The continuity equation (2.187) can be written as ˜ T U1 + BW ik1 M ˜ T U3 = 0. ˜ dT U2 + ik3 M (2.200) Define a Helmholtz operator as H = L + kM . Rearranging (2.197–2.199) and substituting into (2.200) yield b −1 D (DH ˜ + k2 M ˜ T H −1 M ˜ )P ˜ T H −1 BW F1 + DH = −[ik1 M b −1 BW F2 ˜ T H −1 BW F3 ], +ik3 M (2.201) b = BW in which D ˜ dT . In (2.201), the pressure is decoupled from the velocity. Note that P calculated by (2.201) can be thought as a Lagrangian multiplier to satisfy the incompressibility and is different from the physical pressure. In the Uzawa algorithm, first the pressure is computed from (2.201) and, then, the fluid velocity are calculated from (2.197 – 2.199) by using P obtained in the first step. A preconditioned conjugate gradient solver to solve (2.201) is explained in [104, 140]. 2.6 Verification In the force-coupling method, there are three major parameters, which determine the nu- merical accuracy. First, as the far-field interaction is computed by solving the Stokes equa- 61 Figure 2.6: Configuration of the test problem. tions in the force-coupling method, naturally the order of accuracy of the Stokes solver plays an important role in the accuracy of the solution. Second, for the computational efficiency, the force envelopes are distributed and integrated only in a small domain de- fined as In (Y ) = {x|x ∈ R3 , |x − Y | < a × n} for a positive real constant n. Lastly, the grand-mobility matrix of LC-FCM is inverted by using a preconditioned conjugate gradient (PCG) solver. Hence, the tolerance level of PCG also affects the accuracy of the solution. In this section, we test the effects of these numerical parameters on the convergence of the solution. 2.6.1 Periodic domain We test LC-FCM in a tri-periodic domain. The Stokes equations are solved by using the Fourier spectral method. For a particle-pair interaction problem, PCG converges to the tolerance < O(10−10 ) within one or two iterations. So, we choose a three-particle interaction in a shear flow for a test problem. The three particles are seeded as a equilateral triangle in 62 0 10 -2 10 -4 10 10-6 -8 10 ψ -10 10 -12 10 -14 10 10-16 -18 10 0 -1 -2 -3 -4 -5 -6 -7 -8 10 10 10 10 10 10 10 10 10 δ Figure 2.7: Effects of the tolerance level of the preconditioned conjugate gradient solver to the accuracy. the shear plan as shown in figure 2.6. The distance between the particles is 2.01a. The ratio of the length of the computational domain to the particle radius is fixed, L/a = 25.133. First, the effects of the tolerance level of PCG δ is shown in figure 2.7. For the reference solution, LC-FCM is solved for the grid size a/∆x = 10.186, the tolerance level δ = 10−8 , and the domain I4 . The error Ψ is defined as à 3 !1/2 X .X 3 Ψ= |V i − Vref i 2 | i 2 |Vref | , (2.202) i=1 i=1 in which Vref is the reference solution. For 10−4 ≤ δ ≤ 10−1 , Ψ decreases faster than ∼ δ, indicating that the tolerance level is an effective control for the error. Then, Ψ quickly drops to the machine accuracy for δ < 10−4 . For δ > 10−4 , the error Ψ is at least one order of magnitude smaller than δ. In the simulations of concentrated suspensions, we usually choose δ = 10−2 ∼ 10−3 . Figure 2.8 shows show the effects of In on the accuracy of the solution. The reference solution is obtained by setting a/∆x = 10.186, δ = 10−6 , and the domain size n = 5. It is shown that Ψ exhibits an exponential decay. The exponential decay of Ψ is expected as 63 0 10 -2 10 -4 10 -6 10 -8 ψ 10 10-10 -12 10 -14 10 -16 10 1 2 3 4 n Figure 2.8: Effects of the integration width to the accuracy. the regularized multipoles are Gaussian functions. Based on this result, the domain size is chosen between n = 2.5 ∼ 3.0 in most of the computations in this thesis. Finally, we show the effects of the grid resolution on the numerical accuracy. Here, the reference solution is obtained for a/∆x = 10.186, δ = 10−6 , and the domain size n = 4. Figure 2.9 shows the spectral convergence of LC-FCM, which is expected as a Fourier spectral solver is employed. This result indicates that the numerical error is negligible in LC-FCM, as long as the particle radius is bigger than 3∆x. 2.6.2 Channel flow The angular velocity of a sphere moving parallel to a wall under the influence of an external force is given as aΩ 3YB = , (2.203) V 4YC in which Y B and Y C are resistance functions for the particle-wall hydrodynamic interaction. The resistance functions are given in Dance & Maxey [50] for a very small gap a² between the sphere and the wall. Here, we compare the numerical results obtained for a single sphere 64 0 10 -2 10 10-4 -6 ψ 10 -8 10 -10 10 -12 10 2 4 6 8 10 a/∆x Figure 2.9: Convergence of LC-FCM as a function of the grid resolution. -0.12 -0.125 aΩ / V -0.13 -0.135 4 6 8 10 12 14 P Figure 2.10: The ratio of the angular velocity to the translational velocity as a function of the maximum order of the polynomial in each elements. The dash line is the analytical solution from Dance & Maxey [50]. 65 -1 10 h-refinement -7 p-refinement -3 ~N 10 -5 10 Ψ -7 10 10-1 -3 10 -5 10 -9 10 10-7 -9 10 -11 10 100 200 300 400 -11 10 100 200 300 400 500 Ndof Figure 2.11: Convergence of h−type and p−type refinement. Inset shows the convergence of p−type refinement in a log-linear plot. moving in a channel to the analytical solution given in [50]. The particle is located very close to the lower wall, Y2 /a = 1.01. To reduce the effect of the periodicity and the upper wall, the computational domain is chosen as Hx × Hy × Hz = 40a × 60a × 40a. The length of the element in the vertical direction is fixed, Le = 2a, and the maximum order of the polynomial P is changed to test the convergence. The numbers of Fourier modes in the horizontal directions are fixed, Nx × Nx = 256 × 256. In figure 2.10, it is shown that the numerical solution converges as P increases. The difference between the analytical solution and the numerical solution for P = 14 is about 1.6 × 10−4 . It is worth noting that the analytical solutions are obtained by using an asymptotic expansion in ² and the solutions are supplemented by O(1) corrections, which are obtained by fitting to numerical data. As the resistance functions in the literature have O(²) errors and O(1)-correction terms are shown only up to four significant digits. Finally, so as to check the convergence of the spectral element solver, LC-FCM solutions for a particle pair in a Couette flow are obtained by changing the number of degree of freedom Ndof in the wall-normal direction. In this numerical test, two particles are located 66 at the same wall-normal location Y2 /a = 1.01 and separated in the x−direction by 2.01a. The upper wall is moving with the velocity Vupp = γH ˙ y for the shear rate γ˙ = 1. The dimensions of the computational domain and the resolution in the horizontal directions are the same as the one-sphere simulation shown above. The reference solution is obtained for Le = 6a with P = 60. In p−type refinement, Le is fixed and P is changed from 11 to 50, while h−type refinement is performed by fixing P = 7 and changing Le from 1 to 4. The −7 h−type refinement exhibits an algebraic convergence ∼ Ndof and the p−type refinement shows a much faster convergence close to the spectral convergence. The exponential decrease of Ψ is clearly shown in the inset of figure 2.11 for Ndof < 300. 2.7 Validation 2.7.1 Results I: particles in an infinite domain To verify the lubrication correction method developed in section 2.3, several numerical simulations of spheres in an infinite domain are performed. The FCM mobility matrix is constructed by using the FCM Oseen tensors derived in Lomholt & Maxey [135]. 2.7.1.1 Chain of particles settling under gravity First, consider two horizontally separated spheres settling under gravity (see inset of figure 2.12). The hydrodynamic interaction induces counter-rotation of the particle pair. Near contact, the counter-rotation is frozen by the lubrication forces. The angular velocity cal- culated from the present lubrication correction (FCM-LUB) and FCM with just monopole and dipole terms (FCM-MD) are shown in figure 2.12. It is shown that the present lu- brication correction method can reproduce the angular velocity with good accuracy when r/a < 2.25. At r = 2.1, the angular velocity obtained by Ganatos et al. [74] is 0.137 and the present method gives 0.139. For the resistance functions that have a leading-order log ² singularity (Y A , Y B , Y C , Y G , and Y H ), the near-field form is used only when ² < 0.05 and the far-field form is used for 0.05 < ² < 0.8. The exact resistance functions can be found in [50, 101, 110]. In a second example, the settling of a horizontal chain of seven spheres is calculated using FCM-LUB. The configuration of the particles is illustrated in figure 2.13 with the relative gap between the particles ² = 0.005. The drag coefficient λ and angular velocity Ω for 67 0.16 0.14 0.12 Ω 0.1 0.08 0.06 2 2.2 2.4 2.6 2.8 3 r/a Figure 2.12: Comparison of angular velocity for a pair of equal spheres with horizontal separation. ◦, Ganatos et al. (1978); N, FCM-LUB; –, FCM-MD. 3 2 1 0 1 2 3 aε F Figure 2.13: Illustration of the horizontal chain of 7 spheres. 68 0.3 + + 0.5 0.2 Ω λ + + 0.1 + + 0.4 + 0+ 0 1 2 3 0 1 2 3 Sphere number Sphere number (a) (b) Figure 2.14: Comparison of (a) the drag coefficient λ = F/6πµaU and (b) angular velocity. ◦, Ganatos et al. (1978); 4, Durlofsky et al. (1987); +, FCM-LUB; ¤, Dance & Maxey (2003). 69 x2/a x1/a Figure 2.15: Illustration of a particle-pair in a pure straining flow. each particle are compared to those obtained by Ganatos et al. [74] and FTS-SD[63]. The results are shown in figure 2.14. FCM with the present lubrication correction shows similar accuracy as FTS-SD. For the drag coefficient the maximum difference of λ between FCM- LUB and [74] is about 1%. The lubrication model of Dance & Maxey [50] shows relatively larger error, particularly for the particle 3. It was noted that the error at particle 3 may come from the pair-wise addition of the lubrication force in the mobility problem. This is consistent with the argument of Durlofsky et al. [63] that adding lubrication correction terms to the resistance matrix in a pair-wise manner and inverting the resistance matrix to solve the mobility problem better resolves many-body interactions as compared to adding the lubrication forces to the mobility problem. 2.7.1.2 Particles in a pure straining field The interactions of a particle-pair in a linear flow field can be evaluated analytically and there are many solutions available in the literature [15, 14, 110], which are suitable for benchmarking a numerical scheme. In this section, the numerical computations for a pair 70 Table 2.3: Relative velocity of two spheres in a straining flow when E = 1. ² B&G FCM 0.01 0.08195 0.07647 0.005 0.04087 0.03931 0.001 0.008158 0.008071 of force-free and torque-free spheres in a pure straining flow are performed and the results are compared with the analytical solutions. Consider a particle pair in a uniform straining flow (see figure 2.15), ∞ 1 Eij = Eδij11 − E(δij22 + δij33 ), (2.204) 2 ∞ ∞ ui = Eij xj , (2.205) in which δijkl is 1 if all the indices are the same and 0 otherwise. Batchelor & Green (B&G) [14] solved this problem analytically by using bispherical harmonics. When ri = rδi1 and ² is small, the relative velocity and stresslet in B&G are Vr = (1 − A(r))Er, (2.206) µ ¶ 20 4 2 S11 = πµa3 E 1 + K + L + M , (2.207) 3 3 3 in which A(r) = 1 − 4.077² + O(²3/2 ), (2.208) 3 K + 2L + M = 1.366 + O(²). (2.209) 2 The relative velocities from B&G and FCM-LUB are shown in Table 2.3. At ² = 10−2 , the error is about 7%. However, due to the O(²3/2 ) error of A(r) in B&G, it is difficult to assess the error of numerical method at this separation distance. As ² decreases, the difference between B&G and the present simulation becomes smaller. When ² = 10−3 , the difference between B&G and FCM-LUB is about 1%. The stresslet of a sphere is calculated from equation (2.132), S = S tot − RES L E ∞ + RVS L (V − V ∞ ), (2.210) 71 Table 2.4: FCM resistance functions (X M , Y M , Z M ) C0 C1 C2 C3 C4 M X11 −1.4235 0.9133 −1.0670 0.7197 −0.2120 M X12 −0.5911 0.9453 −0.9702 0.6307 −0.1852 M Y11 −1.0787 0.1991 −0.2617 0.1902 −0.0584 M Y12 0.0806 0.0370 −0.1294 0.0990 −0.0283 M Z11 −1.0035 0.0145 −0.0253 0.0221 −0.0075 M Z12 0.0593 −0.1281 0.1400 −0.0855 0.0229 3 100 2 10 -1 y/a ε -2 1 10 -3 10 0 -10 -5 0 5 10 -3 -2 -1 0 1 2 3 x/a x/a (a) (b) Figure 2.16: (a) Relative trajectory of the centers of two equal spheres and (b) separation distances for different initial positions, ∆y I . The length scale of the contact force Rref = 2.001a. From top to bottom: ∆y I = 0.7, 0.6, 0.4, and 0.2 L = −(RL )T . S tot is obtained by solving the equation (2.130). To estimate S, in which RVS EF L needs to be constructed. The FCM resistance functions X M , the fourth-order tensor RES Y M , and Z M are given in table 2.4. The stresslet given in B&G is S11 = 40.017a3 E + O(²). (2.211) Using the lubrication correction, FCM gives S11 = 39.226a3 E and 39.582a3 E when ² = 0.01 and 0.005, respectively. As expected, the results get closer to B&G as ² decreases, and the differences are consistent with an O(²) scaling. 72 Table 2.5: Upstream ∆y I and downstream center-to-center distances ∆y F for Rref = 2.001. ∆y I /a ∆y F /a 0.2 0.631 0.4 0.631 0.6 0.631 0.7 0.700 0.8 0.800 2.7.1.3 Relative motion of a particle pair in a linear shear flow If the surface of spheres are perfectly smooth and the interaction between a particle-pair is from purely hydrodynamics, the trajectories of the particle-pair should be time reversible as a consequence of the linearity of the Stokes equations. Consider a particle-pair in a shear flow, u∞ 1 (y) = γy, ˙ in which γ˙ is a shear rate, with the initial center-to-center distances ∆xI = δx , ∆y I = δy , and ∆z I = δz . Then, at the downstream location ∆xF = −δx , the center-to-center distances in y− and z−directions should be ∆y F = δy and ∆z F = δz . If δy and δz are small, the minimum separation distance during a tumbling motion of the particle pair can be very small. Da Cunha & Hinch [47] observed that for δx /a = 10, δy /a = δz /a = 0.1, the minimum separation is 4.75×10−5 a. In a physical systems, however, the surface roughness of particles breaks the reversibility, resulting in a net drift after the “collision”. Smart & Leighton [200] measured the surface roughness of particles ranging from 43 to 6350 µm in diameter and found these to be the order of 10−2 to 10−3 a. Corresponding numerical simulations of a particle-pair in a linear shear flow are per- formed. δx /a = −10 and δz /a = 0 are used for all simulations. The time advancement was carried out by using a fourth-order Adam-Bashforth method with time step size dt = 10−3 . To prevent any overlap, an elastic contact force proposed by Dance et al. [49] is used. The contact force to the j-th particle by the i-th particle is given by  µ 2 ¶2   Rref −|r|2 r −Fref 2 2 |r| if |r| < Rref ij Rref −4a F = (2.212)   0 otherwise, in which r = Y i − Y j . Although in most numerical simulations a contact force is used to prevent the overlapping of particles as a result of finite dt, physically the use of a contact force is to model the surface roughness of particles [47]. 73 3 2 y/a 1 0 -10 -5 0 5 10 x/a Figure 2.17: Relative trajectory of the centers of two equal spheres for different initial positions, ∆y I . The length scale of the contact force is Rref = 2.0001a. From top to bottom: ∆y I = 0.4 and 0.2 Figure 2.16 shows the relative trajectories and the separation distances of the two equal spheres in the shear flow for ∆y I /a = 0.2, 0.4, 0.6, and 0.7 when Rref = 2.001. When ∆y I /a = 0.7, the minimum separation is ²min ' 0.0015. Since the contact force is not activated, the trajectory shows the fore-after symmetry. For ∆y I /a ≤ 0.6, it is observed that the symmetry is broken and there is a net displacement in the vertical direction. From the table 2.5, it can be deduced that ∆y F will be the same if ∆y I /a < 0.631. Using a traction-corrected boundary element method, Ingber et al. [97] observed that ∆y F /a = 0.71 for ∆y I /a < 0.71, when the surface roughness of a particle is 5 × 10−4 a. In their simulation, the minimum separation was ²min = 1.03 × 10−3 . If ² < ²min , the normal motion of a particle pair is restricted. On the other hand, in the present simulation, the contact force is activated when ² < 10−3 and ²min ' 0.97 × 10−3 , which may contribute to the quantitative difference. The relative trajectories for Rref = 2.0001 is illustrated in figure 2.17. It is observed 74 that ²min = 1.007783 × 10−4 for ∆y I /a = 0.2. Since ²min > Rref , the contact force is not used and the trajectories are symmetric. 2.7.2 Results II: homogeneous suspensions in an periodic domain Here, the numerical simulation of concentrated suspensions in a periodic cell are illus- trated. Three problems are considered; a simple cubic lattice of neutrally buoyant particles, high-frequency dynamic viscosity, and sheared suspensions of non-colloidal particles. The Fourier-spectral method shown in 2.5.1 is used to solve the Stokes equations. The length of the computational domain is kept constant, 2π, in all directions. The number of Fourier modes is determined by the particle radius. In FCM, as long as the force envelope is resolved the numerical result is insensitive to ∆x. The computational resolution is kept a/∆x ' 3. The lubrication correction is used when the distance between particle centers is less than 2.8a. 2.7.2.1 Rheology of particles in a periodic cell Nunan & Keller [166] showed that the bulk deviatoric stress tensor σij is related through the effective viscosity tensor µ∗ijkl to shear rate γ˙ as σij = 2µ∗ijkl γ˙ kl , (2.213) in which γ˙ kl denotes the shear rate. For a cubic lattice of spheres, the effective viscosity tensor is given by [166], 1 2 1 µ∗ijkl = µ(1 + β)(δik δjl + δil δjk − δij δkl ) + µ(α − β)(δijkl − δij δkl ), (2.214) 2 3 3 in which α and β are functions of the volume fraction. In the case of pure-straining flow, the effective viscosity tensor is a function of α only, while in the linear shear flow only β survives. Figure 2.18 shows α and β for the simple cubic lattice of neutrally buoyant particles. FCM results are compared with the low volume fraction asymptotic solution given in [166] and the high volume fraction asymptotic result by Hoffman [90]. At low φ where far-field interaction plays the major role, the distance between particles are larger than the cut-off 75 50 5 40 4 30 3 α β 20 2 10 1 0 0 0 0.2 0.4 0.6 0 0.2 0.4 0.6 φ φ (a) (b) Figure 2.18: The effective viscosity functions (a) α and (b) β in the functions of the volume fraction φ for the simple cubic lattice. The circle indicates FCM results. The dashed and solid lines are, respectively, the high and low concentration asymptotic solutions. 76 Table 2.6: Simulation parameters of the random static simulations φ 0.1 0.2 0.3 0.35 0.4 0.45 0.5 a 0.35 0.35 0.35 0.35 0.38 0.395 0.409 Np 138 276 414 483 432 432 432 distance of the lubrication correction and FCM-MD alone can reproduce the asymptotic solution. In the intermediate region, FCM-LUB predicts slightly higher effective viscosity, which is also observed in SD simulations [196]. However, the difference with the asymptotic results becomes smaller and smaller as φ → φmax . 2.7.2.2 High-frequency dynamic viscosity The high-frequency dynamic viscosity µ∗ is evaluated using the Monte Carlo approach. For each volume fraction φ, the shear viscosity is obtained by averaging over 1000 different random particle configurations. When φ ≤ 0.35, the random configuration can be achieved by using a uniform random number generator. For φ > 0.35, body centered cubes are used initially to locate particles. Then, small random perturbations are introduced to generate the random configurations. The minimum distance between particles is kept larger than 10−5 a. The contact force is not used for the Monte Carlo simulation. The simulation parameters are listed in table 2.6. The number of grid points are 643 for all static simu- lations. In figure 2.19, µ∗ computed by the present FCM is compared with the empirical formula by Krieger & Dougherty [115], analytical results by Batchelor & Green (B&G) [14], and the multipole-moment simulation by Ladd [121], which is used as the reference. The Stokes-Einstein estimate is obtained in the dilute regime and B&G made φ2 correction in the semi-dilute regime. 1. Stokes-Einstein estimate: 5 µ∗ /µ = 1 + φ. (2.215) 2 2. Batchelor & Green (1972): 5 µ∗ /µ = 1 + φ + 5.2φ2 . (2.216) 2 77 10 8 6 4 µ* /µ0 2 0 0.1 0.2 0.3 0.4 0.5 φ Figure 2.19: Shear viscosity in terms of volume fraction. Solid line, Krieger & Dougherty (1959); dashed line, Stokes-Einstein estimate; dash-dot line, Batchelor & Green (1972); M, Ladd (1990); ◦, FCM-LUB; ¤, FCM-MD. 78 Table 2.7: Simulation parameters of the dynamic simulations φ Ng Np a ∆t 0.3 1283 4337 0.16 2 × 10−3 0.4 963 1458 0.2532 2 × 10−3 The empirical formula by Krieger & Dougherty is given by µ ¶ ∗ φ −[η]φm µ /µ = 1 − , (2.217) φm in which φm is the maximum random packing volume fraction and [η] is a rheological fitting 5 parameter. Following Stickel & Powell[203], [η] = 2 and φm = 0.63 are used. It is observed that, the FCM-MD result is close to B&G. Since FCM-MD is a far-field approximation, it can reproduce the semi-dilute solution. At high φ, the deviation from the exact values becomes larger. It is shown that FCM-LUB gives accurate results. Although FCM-LUB predicts slightly higher µ∗ compared to [121] when φ > 0.4, the difference is about 5% at φ = 0.45. 2.7.2.3 Sheared suspensions Dynamic simulations are performed for φ = 0.3 and 0.4. For the comparison with the prior simulation results [198], a conservative, short-range contact force of the exponential form τ e−τ ² Fpij = −Fref d (2.218) 1 − e−τ ² ˙ 2 . Using these parameters, is used. For all dynamic simulations, τ = 100 and Fref τ = 6πµγa it is observed that the minimum values of ² are about 0.001 and 0.0005 at φ = 0.3 and 0.4, respectively. Table 2.7 shows the simulation parameters used to compute the statistics. We define a mean-square residual as   6Np 5Np 1  X X ||R||2M = |rf,i |2 + |rs,i |2  , (2.219) Np i=1 i=1 in which rf and rs are the residuals of F lub and S tot , respectively. The tolerance level δ is chosen as 10−3 ∼ 10−4 to ensure that the error of the particle velocity is O(10−3 ∼ 10−4 ). 79 3 2 ~NplogNp Time (sec.) ~Np 1 0 0 1000 2000 3000 4000 5000 Np Figure 2.20: Computation time of one time step for φ = 0.3. Using the particle velocities in the previous time step as an initial estimate, the system is solved in under 7 ∼ 8 iterations. In the concentrated suspensions, due to the complex multi-body hydrodynamic interactions and the elastic contact forces, the trajectories of particles exhibit chaotic motions, which contributes to the random fluctuation of macro- scopic variables such as the particle stresses. As a result, a small error in the calculation of the particle velocities is indistinguishable from the statistical noise so that setting very low tolerances does not add to the overall accuracy of a simulation. However, if the tolerance level is set too large, overlap of particles can happen as ² is small. The simulations were performed on a 2.6GHz AMD Opteron Linux cluster. The com- putation time for one time step is shown in figure 2.20. All the computations were done using 16 processors and the computation time is the average over 500dt. It is observed that the computation time is scaled as ∼ Np rather than ∼ Np logNp . This Np -scaling is the result of the large operation counts in the numerical integration of the force envelope and the distribution of the force monopole and dipole to the computational mesh. For example, at φ = 0.3 (table 2.7), the number of operations of the Stokes solver using a Fourier-spectral 80 4 3 〈 σ12〉/µ0 H 2 1 0 10 20 30 Time Figure 2.21: Time history of the deviatoric stress tensor. The top and bottom curves correspond, respectively, to φ = 0.4 and 0.3. method is O(107 ) while the numerical integrations to calculate V and E take O(108 ) op- erations. This gather and scatter processes take about 60% of the total computation time while the Stokes solver is about 15%. About 15% of the total computation time is spent on the communications between processors. In the present implementation, MPI collective communications are used in the FFT and gather-scatter processes. By changing the collec- tive communications to non-blocking communications and overlapping with computations, further speed-up can be achieved. Figure 2.21 shows the time history of the deviatoric stress tensor by hydrodynamic interactions; Np hσ H i = hSi, (2.220) V in which V is the volume of the sampling domain, in this case the computational domain, and h·i denotes ensemble average. Time is normalized by the shear rate, γ. ˙ Initial configurations 81 5 y/a2.5 0 -2.5 -5 -5 -2.5 0 2.5 5 x/a Figure 2.22: Projection of the pair-distribution function onto the x − y plane. The darker the contour, the higher the probability is. H drops were generated from the body centered cubes with small random perturbations. σ12 sharply at first and starts increasing after γt ˙ ' 1 as the microstructure develops in response to the shear flow. To remove the effect of this initial transition, only data for γt ˙ > 20 are used to estimate the statistics. Figure 2.22 shows the pair-distribution function projected onto the x − y plane for φ = 0.4, in which x and y denote the flow and the velocity gradient directions, respectively. In Stokes flow with perfectly smooth hard sphere suspensions, the pair-distribution function should be symmetric due to the reversibility of Stokes equation. However, as shown in figure 2.16, the net displacement caused by the contact force breaks the fore-after symmetry and the probability of finding another particle in the extensive strain direction becomes smaller than the compressible strain direction. The result for the pair-distribution function is consistent with that of Sierou & Brady [197]. 82 1 0.8 0.6 0.4 0.2 0 φ = 0.4 φ = 0.3 -0.2 0 2 4 6 t (a) 1 0.8 0.6 0.4 0.2 0 φ = 0.4 φ = 0.3 -0.2 0 2 4 6 t (b) Figure 2.23: Velocity auto-correlation functions: (a) ρV2 (t) and (b) ρV3 (t) for φ = 0.3 and 0.4. 83 1 10 ~t 0 10 -1 10 -2 ~t 2 10 -3 10 -1 0 1 2 10 10 10 10 Time (a) 1 10 ~t 0 10 -1 10 2 ~t -2 10 -3 10 -1 0 1 2 10 10 10 10 Time (b) Figure 2.24: Mean squared displacement normalized by a2 in the velocity gradient (solid line) and the vorticity directions (dashed line): (a) φ = 0.3 and (b) φ = 0.4. A key element for the shear-induced random dispersion of particles in a suspension is the velocity auto-correlation, which is defined as hVi (t)Vi (t + τ )i ρVi (τ ) = . (2.221) hVi (t)2 i Figure 2.23 shows the velocity auto-correlation for two volume fractions, φ = 0.3 and 0.4. At the larger volume fraction, the velocity auto-correlation decays faster and the magnitude of the negative peak decreases, which is consistent with the earlier results of [58]. 84 Figure 2.24 shows the mean squared displacement normalized by a2 , h(Yi (t) − Yi (0))2 i σXi (t) = . (2.222) a2 It is well known that at short times in which the particle motion is strongly correlated with the initial configuration σX increases as ∼ t2 , while at longer times a steady behavior is established and σX shows linear growth with time, σXi (t) ∼ hVi2 it2 , t ¿ TLi , (2.223) σXi (t) ∼ 2hVi2 iTLi t, t À TLi , (2.224) in which TLi is the integral timescale in i-direction, Z ∞ TLi = ρVi (s)ds. (2.225) 0 The diffusivity Di is given as Di = hVi2 iTLi . (2.226) The diffusivity can be estimated by two methods. One is from the integral of the velocity autocorrelation function, as shown in equation (2.226) and the other is by estimating the long-term slope of σX , which should yield the same results. Table 2.8 shows the diffusion coefficients evaluated from equation (2.226). The agreement with the previous simulations [198] is quite good. There is a large scatter in the diffusion coefficients in the literature due to the variation in experimental conditions. In [198], it is found that the diffusion coefficients from SD simulations are almost half of the experimental results. In the experiments, several poorly quantified features, such as surface roughness, surfactant or steric forces, and residual Brow- nian forces, strongly affect the motion close to contact. In the numerical simulations, the contact force is used to model these effects. Using the contact force (2.212) with large cut-off distance (Rref = 2.2), Abbas et al. [2] could obtain the diffusion coefficients similar to the experimental results. On the other hand, when using the lubrication model by Dance & Maxey [50], the diffusion coefficients were only about 2/3 of those with the large con- tact forces. Although there are many studies on the role of contact forces on suspensions 85 Table 2.8: Diffusion coefficients in y and z directions φ Dy Dz 0.3 0.030 0.016 0.4 0.059 0.037 x2 = Ly c a V2 b x2 = 0 Figure 2.25: Geometry for a sphere placed between two walls. [2, 47, 58, 97, 152], the choice of a contact force to represent the underlying physics correctly is still an open question. 2.7.3 Results III: single particle in a channel We first illustrate the methods described in the previous sections by applying them to the motion of a single particle in a channel bounded by two parallel planar walls. The first example is of a spherical particle moving perpendicular to the walls under the action of an external force but in the absence of any other flow. The configuration of the channel geometry is shown in figure 2.25. Numerical results for the non-dimensional resistance 86 30 25 20 15 λt 10 5 0 0.1 0.2 0.3 0.4 0.5 s=b/(b+c) Figure 2.26: Drag coefficient λt from °, Ganatos et al. (1980); 4, FCM with wall lubrica- tion; ¤, FCM with monopole and dipole. 87 coefficient λt , defined by F2 λt = , (2.227) 6πµaV2 have been obtained previously by [75] using a series solution and boundary collocation scheme for a channel with infinite planar walls. The corresponding FCM simulations are performed for a periodic channel, where Lx /a = Lz /a = 60. The height of the channel Ly is varied according to the parameter s = b/(b + c) used by [75]. The gap between the particle and the wall is 0.1a so that b/a = 1.1. A comparison of the results for λt from the two different approaches is shown in fig- ure 2.26. Also shown are the values from FCM without the lubrication correction, which are roughly half the total for λt . This configuration is the most challenging to calculate with FCM, the gap being larger than at which near-field lubrication forces would dominate, and tests the matching procedure for the near-field and far-field conditions. In general the agreement of the results is very good, with differences of 2% or less for the most part. For the larger values of s the discrepancies are larger. For s = 0.5, the channel height is only 2.2a and the calibration procedures used to obtain the FCM wall resistance functions, which were based on widely separated channel walls are no longer appropriate. The second example is for the translational velocity of a force-free and torque-free sphere in a Couette flow. The computational domain here is Lx = 30, Lz = 20 and Ly = 10 and the particle radius a = 1. The upper wall is moving in the x1 direction with the velocity Vupp = −1.0, while the lower wall velocity is Vlow = 1.0. Figure 2.27 shows the translational velocity of a sphere centered at various y locations, close to the upper wall at y = 10. The FCM results are compared with the boundary collocation results of [76] and the SD simulation by [199]. The results in [76] are used as the reference. In the FCM simulations, the lubrication corrections are used when y/a > 8.8, that is when the gap between the particle and the wall is less than 0.2a. In the region where the far field solution plays the dominant role, the FCM results match those of [76] without any corrections. In the near- wall region, FCM gives more accurate results than the SD results reported in [199]. In the latter, a rigid wall was represented as a layer of fixed spherical “wall particles” rather than a smooth wall with no-slip boundary conditions. Further, the domain size was smaller, with periodic dimensions Lx /a = Lz /a = 14. 88 10 9 8 y 7 6 5 -0.2 -0.4 -0.6 -0.8 -1 V Figure 2.27: Translational velocity of a sphere V in a Couette flow: Solid line, Ganatos et al. (1982); ¥, Singh & Nott (2000); °, present FCM. 89 2 1 ∆y/a 0 -1 -2 -10 -5 0 5 10 ∆x/a Figure 2.28: Relative trajectories in the x − y plane for a particle pair in a linear shear flow at Rep = 1.0. The initial vertical separation distances are ∆y/a = 0.5 (dash-dot), 0.6 (dashed), and 0.7 (solid). 2.7.4 Results IV: a particle pair under a finite fluid inertia In the limit of vanishing Reynolds number, the relative trajectory of a pair of equal-size particles in a linear shear flow can have only two types of trajectories, open or closed trajec- tory, depending on the initial relative position of the particles [15]. Kulkarni & Morris [118] have shown that, at finite Reynolds numbers, the topology of the relative trajectory changes from that of the Stokes flow. Using the lattice-Boltzmann simulations, they found reversing and spiralling trajectories in addition to the open trajectory of Stokes flow suspensions. Here, we used the force-coupling method to reproduce the lattice-Boltzmann simulations of Kulkarni & Morris [118]. Figure 2.28 shows the relative trajectory of a particle pair in the shear plane, i.e. the separation distance in the vorticity direction ∆z/a = 0. The particle trajectories are com- ˙ 2 ργa puted for the particle-scale Reynolds number Rep = µ = 1.0. The initial separation 90 distance in the velocity direction is fixed, ∆x/a = 10, while the vertical separation dis- tances are changed from ∆y/a = 0.5 to 0.7. It is found that the topology changes from the reversing to open trajectories as ∆y/a changes from 0.6 to 0.7. In [118], they found that the bifurcation point is about 0.6 for Rep = 1.0. Considering the difference in the numeri- cal set-up, LBM simulation of a wall-bounded channel in [118] versus FCM in tri-periodic domain in the present study, the agreement in the bifurcation point is satisfactory. For the second example, we show the spiralling trajectory. The initial separation distance is fixed as (∆x/a, ∆y/a, ∆z/a) = (−0.01, 0.005, 2.8) and the Reynolds numbers are changed from Rep = 0.1 to 0.5. Figure 2.29 shows the relative trajectories in the x − z direction. It is seen that the particle moves towards the reference particle at the origin showing an oscillatory motion in the x−z plane. The spiralling trajectory observed in the present study is qualitatively consistent with those in Kulkarni & Morris [118]. Note that the spiralling trajectory is caused by the secondary flow and, hence, accurate representation of the no- slip boundary on the particle surface is important in resolving the dynamics. In the force- coupling method, a particle is represented by a “smooth” Guassian profile, instead of having a sharp no-slip boundary, which results in an underestimation of the non-linear interaction. As a result, the timescale of the spiralling trajectory in FCM is about O(103 ) in contrast to O(102 ) of the lattice-Boltzmann simulation [118]. In the previous lattice-Boltzmann simulations, the particle surface is represented by a staggered non-smooth surface. As the timescale of the multi-particle interaction is usually O(1) or smaller in concentrated suspensions, the detailed mechanism of this spiralling trajectory is not expected to play an important role. A more detailed discussion of a secondary flow induced by the non-linear interaction is given in Chapter 5. 2.8 Conclusions In this chapter, the lubrication-corrected FCM (LC-FCM) has been derived for the inclusion of lubrication forces between particles in the simulation of viscous suspensions using the force-coupling method. These forces are evaluated on the basis of pairwise additions to the resistance formulation, the inverse of the natural mobility formulation provided by FCM. This provides a more reliable estimate of lubrication interactions for concentrated suspensions than using pairwise addition to the mobility problem. Efficient and robust 91 (a) 1 ∆z/a 2 Re = 0.1 -4 -2 0 2 4 ∆x/a (b) 1 ∆z/a 2 Re = 0.3 -4 -2 0 2 4 ∆x/a (c) 1 ∆z/a 2 Re = 0.5 -4 -2 0 2 4 ∆x/a Figure 2.29: Relative trajectories in the x − z plane for a particle pair in a linear shear flow at Rep = 0.1 (a), 0.3 (b), and 0.5 (c). 92 iterative methods are described for solving for the particle stresslets and for lubrication effects in which the problems are written in terms of a symmetric positive definite system. The fully coupled lubrication and far-field interactions are then obtained efficiently through a suitably chosen preconditioned conjugate gradient method, in which the inverse of the ill- conditioned resistance matrix is calculated by a recursive formula. Solving the full system requires typically less than 7 ∼ 8 iterations while ensuring that the errors in the particle velocity are O(10−3 ) or less. The force-coupling method can be easily implemented with any existing numerical Stokes or Navier-Stokes solver. As such, the simulation of inhomogeneous and wall-bounded sus- pensions are possible with FCM. The no-slip boundary condition at a fixed rigid boundary is then naturally incorporated in the numerical simulation. Lubrication forces must be spec- ified for the interaction of particles with a rigid wall following the same general procedures. There is a minor complication in that the force envelopes (2.4)-(2.5) are not compact and may extend outside of the flow domain even though the particles are full contained within the domain. The standard procedure has been to truncate the envelopes at the domain boundaries [135]. Here, a new FCM envelope near a wall is proposed. To accelerate the convergence of the system for dynamic simulations, a fictitious inertia is added to the equation of motion of Stokes flows. Using the fictitious inertia, the computation time can be reduced to almost 1/2 ∼ 2/3, while an additional temporal error of O(δt2 ∼ δt3 ) is introduced. The solution procedure of FCM with the lubrication correction for Navier-Stokes flows is presented. Here, a first-order scheme for concentrated finite-inertia suspensions is provided and tested. A second order scheme is also proposed, which has not been implemented yet. LC-FCM is tested for the various particle-pair interactions and in concentrated sus- pensions. For the computation of dynamically evolving, sheared suspensions of neutrally buoyant particles, numerical simulations with O(1000) particles are performed. It is shown that the results are consistent with the previous theories and numerical simulations. It is also shown that the computational cost for homogeneous suspensions in a periodic cell, us- ing a Fourier spectral method, is close to O(Np ). The results also illustrate the importance of near-contact repulsion forces in non-Brownian suspensions and their contribution to the irreversible interaction of particles. LC-FCM may be readily extended to bidisperse or polydisperse systems of particles. 93 For each combination of particles, the corresponding lubrication forces between the particles must be specified as in (2.134) and as given by [110]. Similarly, the FCM resistance functions for near-field interaction of each combination must be determined as in (2.135) and a table constructed similar to table 2.1. The basic procedures follow as before. The scale coefficient κ in the preconditioner (2.137) should be adjusted as in equation (2.124). Chapter 3 Stokes flow: Concentrated suspensions in Couette flow 3.1 Introduction Suspension of neutrally-buoyant non-colloidal particles in a linear shear flow is one of the most fundamental systems to study dynamics and rheology of suspension flows. Yet, hydro- dynamic interaction between particles becomes so complex as the volume fraction increases that theoretical analysis of the system has been limited to the dilute to semi-dilute regimes (see [159, 203] for review). Most of our current understanding of non-colloidal suspen- sions comes from experimental and numerical observations. In providing detailed informa- tion about the micro-structure as well as the rheological properties, numerical simulations greatly contribute to our understanding on concentrated suspensions in ways not always available in experiments. The most widely used numerical method to study concentrated suspensions is the Stokesian Dynamics (SD) method [24, 23]. In the case of homogeneous sheared suspensions in which periodic boundary conditions are used for all three-directions, modeling an infinite domain, Stokesian Dynamics has been used for a wide range of volume fractions, for example, to investigate the rheology and micro-structure [197], shear-induced diffusion [141, 198], and chaotic behavior [59, 172]. Although the dynamics of concentrated suspensions in the unbounded domain is now well understood, much less is known about the behavior of suspensions in the wall-bounded domain, where the suspension field is highly inhomogeneous. There have been several ob- 94 95 servations on the distinctive mechanisms in the wall bounded suspensions, such as apparent wall-slip [99], particle structuring [113], and swapping of particle trajectories [244]. There have been a few numerical studies on the wall-bounded suspensions using the Stokesian Dynamics, where a wall is replaced with a chain of fixed spheres [164, 199]. Due to the high computational cost, dynamic simulations in the wall bounded flows have been limited to a monolayer simulation with O(10) particles. Bossis et al. [22] and Swan & Brady [207] have extended Stokesian Dynamics for particle-wall interactions using the image method [18]. However, dynamic simulations of concentrated suspensions using these modification have not been reported yet. There are only a few three-dimensional simulations of concentrated suspensions in Cou- ette flow. Most of the simulations are performed using the lattice-Boltzmann (LB) method as developed by Nguyen & Ladd [162]. Kromkamp et al. [116] performed both two- and three-dimensional LB simulations of concentrated suspensions in a Couette flow. They showed that the wall structuring in two-dimensional simulation differs from the three- dimensional results. They observed dependency of some results on the computational res- olution and ascribed the dependency to the ragged particle surface in the LB method. Kulkarni & Morris [119] performed three-dimensional LB simulations for the volume frac- tion up to 0.3 in a Couette flow to investigate the finite-Reynolds-number effects on the suspension rheology. The main goal of this chapter is to investigate the effects of the confinement on the dynamics of suspension flows. The present simulations are among the first computational results on the dynamics of concentrated Stokes suspensions in a Couette flow with a full representation of no-slip conditions at a rigid wall. In section 3.2, the rheology and mi- crostructure of Couette flow suspensions are investigated for wide ranges of volume fraction and channel height. The diffusive behavior and lengthscales of suspension flows are dis- cussed in section 3.3. The effects of confinement and external torques on the disorder-order transitions of non-Brownian suspensions are shown in sections 3.4 and 3.5, respectively. The conclusions are given in section 3.6. 96 3.2 Wall effects on the rheology of concentrated suspensions One of clear presentations about the wall effects on suspension rheology is given in Zarraga et al. [241], where they measured the effective viscosity and normal stress differences in various experimental devices. In a parallel plate geometry, they found that the rheological parameters are functions of the gap width when the gap width is smaller than 40a, in which a is the particle radius. However, detailed mechanism responsible for the behavior is still not understood clearly. The main focus of this section is to investigate the changes of suspension dynamics near the wall. It is shown that particle layers are formed near the wall due to the strong particle-wall lubrication interactions. As a result of the particle layering, the rheological properties such as the relative viscosity and the normal stresses are found to be functions of Hy /a in which Hy is the channel height. At high volume fractions the suspension can be divided into three regions depending on the micro-structure; the wall region where particle layering is dominant, the core region in which suspension behaves similarly to homogeneous suspensions, and the buffer region in which the micro-structure shows the characteristics of both the shear structure and the particle layer. 3.2.1 Simulation parameters FCM simulations are performed for three different volume fractions (φ = 0.2, 0.3, 0.4). The computational domain in the horizontal directions is fixed (Hx /a = 30, Hz /a = 20), while the channel height is varied to investigate the effect of the wall (Hy /a = 10, 20, 30). The number of Fourier modes in x− and z−directions are 96 and 64, respectively. The simulation parameters are shown in table 3.1. The radius of a sphere (a) and the shear rate (γ) ˙ are fixed at reference values: a = 1, γ˙ = 1. The upper wall is moving with the velocity ˙ y /2, 0, 0)T and the lower wall is moving in the opposite direction with the same Vwall = (γH speed. When the separation distance between two particles becomes very small (less than 10−2 a), non-hydrodynamic effects, such as surface roughness, residual Brownian forces, and electrostatic forces, may play an important role. These effects will break the fore-after sym- metry of particle interactions in Stokes flow. For example, Smart & Leighton [200] showed that the roughness of particles ranging from 43 − 6350µm in diameter is on the order of 10−2 a ∼ 10−3 a. Brady & Morris [25] showed the importance of these non-hydrodynamic 97 Table 3.1: Simulation parameters. Ny is the number of elements in the vertical direction. Every elements has the same length EL = 2.5 and quadrature points Qy = 9. φ Hy Ny Np Rref Fref dt C2S 0.2 10 4 285 2.01 100 2 × 10−3 C3S 0.3 10 4 430 2.01 100 2 × 10−3 C4S 0.4 10 4 567 2.01 100 2 × 10−3 C2L 0.2 20 8 570 2.01 100 2 × 10−3 C3L 0.3 20 8 860 2.01 100 2 × 10−3 C4La 0.39 20 8 1122 2.01 100 2 × 10−3 C4Lb 0.39 20 8 1122 2.004 100 2 × 10−3 C4Lc 0.39 20 8 1122 2.001 1000 1 × 10−3 C4H 0.4 30 12 1718 2.01 100 2 × 10−3 forces on the non-Newtonian aspect of suspension flows. To model the non-hydrodynamic interactions, and specifically the effects of surface roughness of non-Brownian particles, a short-range contact force FPij is used in the present simulations when the distance between particle i and j is less than a barrier cut-off distance Rref . The form of the contact force is chosen to be similar to Dance et al. [49],  µ 2 ¶6   2 Rref −|r|2 r −6πµγa ˙ Fref 2 2 |r| if |r| < Rref Rref −4a FPij = (3.1)   0 otherwise, in which r = Y i − Y j and Fref is a constant. It has been discussed that, as long as the contact force is sufficiently short ranged, the detailed expression of the contact force does not change the simulation results significantly [57, 141]. The effects of the roughness model or the contact force on the suspension rheology has been of interest and can be found in the literature [2, 25, 47, 97, 152]. Since the system is chaotic and ergodic, a time average is used to obtain statistics once the suspension flow reaches a statistically stationary state. The initial configurations are obtained by two different approaches; body-centered cubes with small random perturbations and a molecular dynamics simulation. It is observed that, once the suspension reaches a statistically stationary state, the statistics does not depend on the initial configuration. Simulations are performed for γt ˙ ' 50 to reach the stationary state. Then, the time average is performed over the period T = 100γt ˙ except for C4Lc for which the simulation is performed until T = 40γt. ˙ 98 0.8 c(y)0.6 0.4 0.2 0 0 1 2 3 4 5 y/a Figure 3.1: Concentration profiles for C2S (solid), C3S (dashed), and C4S (dash-dot). 3.2.2 Wall effects on particle concentration In wall-bounded suspensions, once a particle moves close to a wall, the particle tends to stay near the wall for a long time due to the strong particle-wall lubrication force, which then results in the formation of stable particle layers near the wall. Using LB simulations, both Kromkamp et al. [116] and Kulkarni & Morris [119] observed a pronounced local concen- tration peak near the wall at high volume fractions, which was not reported in the earlier simulations of coplanar particles in Couette flow [199]. The concentration profiles and the two-dimensional pair-distribution functions are presented to characterize the particle layering and the near-wall effects on suspensions. The local concentration is defined as ZZ 1 hc(x2 )i = h χ(x)dxdzi, (3.2) Hx × Hz in which χ(x) is an indicator function that is only non-zero if x is inside of a particle and h·i denotes the ensemble (time) average. 99 Figure 3.1 shows the concentration profiles in the small domain (Hy /a = 10). Consid- ering the symmetry, only half of the channel is drawn. The concentration profiles for C2S and C3S have a local peak in the wall region and become relatively flat away from the wall. On the other hand, the plateau is not observed in C4S, indicating that the wall effect is dominant across the whole channel. As the layering in the wall region becomes stronger, it is seen that the ratio of the first local peak concentration near the wall to the concentration in the center is an increasing function of φ. For C4S, the peak concentration in the particle layer is almost 1.5 times larger than that in the center region. Consistent with Kromkamp et al. [116], the location of the peak concentration moves toward the wall as φ increases. The local peaks are observed around 1.2, 1.1 and 1.08 for C2S, C3S, and C4S, respectively. Due to the finite size of the particles, there is a particle depletion layer around y/a ' 2.1 even for the lowest volume fraction in the present simulation (C2S). Further insight about the relative position of particles comes from the particle pair- distribution. The pair-distribution functions projected onto the velocity/velocity-gradient plane, g(∆x, ∆y), and velocity-gradient/vorticity plane, g(∆y, ∆z), are shown in figure 3.2 for C4S. The pair-distribution functions are obtained for the reference particles centered in 4.5 ≤ y/a ≤ 5.5. Two distinct streaks of high probability region are observed around ∆y/a ' ±2 and ±4, which correspond to the parallel particle layers at y/a ' 1 and 3. Due to the strong layering, there is a depletion layer between two particle layers (∆y/a ' ±3 ∼ 3.5) in which the probability of finding another particle is low. In the core region −2.5 < ∆y/a < 2.5 or 2.5 < y/a < 7.5, g(∆x, ∆y) shows two shells of high probability near the compressional axis of the linear shear flow, similar to the results for sheared suspensions in an unbounded domain [197]. Figure 3.3 shows the concentration profiles in a large channel (Hy /a = 20). The concen- tration profiles for C2L and C3L are similar to those in the smaller channel (Hy /a = 10). For φ = 0.4, the plateau of the concentration is observed when y/a > 7. In addition, to show the effect of the contact force or particle roughness, the concentra- tion profiles for C4La and C4Lc are compared in figure 3.3. In C4Lc, the contact force is activated when the distance between two particles is a² < 0.001. A very large Fref is used to make the contact force behave similarly as the hard sphere potential. The minimum separation distances for C4La and C4Lc are ²min ' 3 × 10−3 and 5 × 10−4 , respectively. Comparison of the profiles for C4La and C4Lc shows that indeed the specific details of the 100 4 2 ∆y/a 0 -2 -4 -4 -2 0 2 4 ∆x/a (a) 4 2 ∆y/a 0 -2 -4 -4 -2 0 2 4 ∆z/a (b) Figure 3.2: The pair-distribution function projected onto (a) x − y and (b) y − z planes for C4S. Light contours represent high probability. 101 0.8 0.6 c(y) 0.4 0.2 0 0 2 4 6 8 10 y/a Figure 3.3: Concentration profiles for C2L (solid), C3L (dashed), C4La (dash-dot), and C4Lc (dotted). 102 contact force has only a slight effect on the concentration profile. The pair-distribution function g(∆x, ∆y) for C4La is obtained for two different regions for the reference particles; 5.5 < y/a < 6.5 (figure 3.4 a), a middle zone between the wall and the centerline, and 8 < y/a < 12 (figure 3.4 b), close to the centerline. In figure 3.4 (a), three particle layers are observed in the lower half around ∆y/a ' −1.5, −3, and −5, which correspond to the local concentration peaks near the wall. When ∆y > 0, the particle layering is weak and the suspension is essentially homogeneous. Similar to suspensions in an unbounded domain, two shells of high probability region are observed oriented with the compressional axis of the mean rate of strain. Observation of the instantaneous suspension field reveals that the second and third particle layers are formed and then broken repeatedly with some period. In other words, unlike the first particle layer, the second and third particle layers exist on average not at each time instance. The pair-distribution function obtained in the core region (figure 3.4 b) is similar to the previous results for an unbounded domain [197, 233], implying the rheological properties in the core region may be explained by the results in the unbounded domain. The concentration profiles for different Hy /a at φ = 0.4 is compared in figure 3.5. It is shown that, as Hy /a increases, the peak concentration at the location of the first particle layer decreases. It is interesting to see that the concentration profiles for different Hy /a are almost similar except for the value of the first peak, indicating the width of particle layers is independent of Hy /a, if Hy /a is large enough. At φ = 0.4, the suspension field is homogeneous if y/a > 8. This observation is consistent with the experimental result in Zarraga et al. [241] that the rheological properties are functions of the channel width when the channel is not wide enough. As Hy increases, the ratio of the homogeneous region to the particle layers becomes bigger, which makes the volume-averaged rheological properties independent of Hy /a in a wide channel. Similar to the concentration, the average particle flux is defined as Np Z Z 1 X cv(x2 ) = H(a − |x − Y i |)V n dxdz, (3.3) Hx × Hz i=1 where H(x) is the Heaviside step function. Then, the averaged particle-phase velocity is 103 4 2 ∆y/a 0 -2 -4 -4 -2 0 2 4 ∆x/a (a) 4 2 ∆y/a 0 -2 -4 -4 -2 0 2 4 ∆x/a (b) Figure 3.4: The pair-distribution functions for C4La projected onto x − y plane for particles in (a) 5.5 < y/a < 6.5 and (b) 8 < y/a < 12. Light contours represent high probability. 104 0.8 0.6 c(y) 0.4 0.2 0 0 5 10 15 y/a Figure 3.5: Concentration profiles for C4S (solid), C4La (dashed), and C4H (dash-dot). 105 10 8 6 y/a 4 2 0 -10 -8 -6 -4 -2 0 v Figure 3.6: Average particle velocities for C3L (solid) and C4La (dash-dot). The dashed line is the linear velocity profile of Couette flow without suspensions. 106 10 8 6 + µr + 4 + + 2 + + 0 0.2 0.4 φ Figure 3.7: Relative viscosity µr : solid line, [115]; dashed line, Eilers fit; ¤, C2S, C3S, and C4S; M, C2L, C3L, and C4La; O, C4Lb; ◦, C4Lc; ¦, C4H; +, [233]. obtained from hcv(x2 )i hv(x2 )i = . (3.4) hc(x2 )i The particle-phase velocity profiles for C3L and C4La are shown in figure 3.6. Consistent with the previous results of Singh & Nott [199] and Kromkamp et al. [116], the particle- phase velocity profile is almost linear in the core region (5 < y/a < 15). Near the wall (y/a < 0.7), the particle-phase velocity is faster than the linear profile, which is responsible for the apparent wall slip observed experimentally in concentrated suspension flows [99]. 3.2.3 Relative viscosity The particle contribution to the bulk stresses is given by P Ns ¡ e ¢ hσij i= D hSij i + hSij i , (3.5) Q 107 in which QD is the volume of the sampling domain over which the contribution is summed and Ns is the number of particles in the sampling domain. The hydrodynamic Sij and e to the total stresses are calculated by contact-force contributions Sij Ns 1 X n hSij i = Sij , (3.6) Ns n=1 XNs Ns X e 1 1 hSij i = − (rink FP,j nk + rjnk FP,i nk ). (3.7) Ns 2 n=1 k=n+1 The relative viscosity of the suspension µr in a linear shear flow is 1 P µr = 1 + hσ i. (3.8) µγ˙ 12 In figure 3.7, µr for QD = |ΩD | is compared with two viscosity models, 1. [115], µ ¶ φ −[η]φm µr = 1 − , (3.9) φm 2. Eilers fit [203], à !2 1 2 [η]φ µr = 1+ , (3.10) 1 − φ/φm in which φm is the maximum packing fraction and [η] is a fitting parameter. Stockel & Powell [203] showed that the relevant experimental data are well represented by Eilers’ fit with [η] = 2.5 and φm = 0.65. The high-frequency dynamic viscosity, µ∞ , in Yeo & Maxey [233] is obtained by the ensemble average of 1000 random configurations at each φ. It is shown that µr in the present simulation is in the range of previous empirical results. As micro-structures form in sheared suspensions, µr is typically larger than µ∞ . The difference between µr and µ∞ , called the excess viscosity ∆µ = µr − µ∞ , is positive and an increasing function of φ [197]. Of particular note, Zarraga et al. [241] found in their parallel plate experiments for φ = 0.45 that µr is an increasing function of Hy /a when Hy /a < 40 and then reaches a plateau if Hy /a > 40. It was suggested that the wall slip is a major mechanism responsible for the 108 P i estimated in the core region ΩC and the wall region Table 3.2: The deviatoric stress hσ12 ΩW . ΩC ΩW C4La 4.18 3.65 C4H 4.58 3.86 [197] 4.49 ± 0.037 decrease in the relative viscosity [5, 241]. It appears instead that the Hy /a dependency of µr is largely due to the organized micro-structure in the near-wall region (particle layering). In homogeneous concentrated suspensions, the dominant contribution to the relative viscosity comes from the normal lubrication interactions between particles, which is proportional to ∼ 1/² for a gap width a² [73]. On the other hand, the lubrication interactions between two particle layers is mainly through tangential motions between particles in each layers for which the singularity is ∼ log ². Hence, if the particle layering is dominant, the viscosity may be smaller than that in a corresponding homogeneous suspension. These observations may be compared with the simulation results for µr at different φ and Hy /a shown in figure 3.7. Since the particle layering is weak at low volume fractions, µr is less sensitive to Hy /a when φ ≤ 0.3. On the other hand, µr is a non-decreasing function of Hy /a at φ = 0.4. The comparison may be refined by dividing the channel into zones and evaluating the average particle stresses in each zone as opposed to the whole domain ΩD of the channel. Table 3.2 shows the deviatoric particle stress estimated from the particles in the core region ΩC and wall region ΩW . ΩC and ΩW are chosen as ΩW = (0, Hx ) × {(0, Hw ) ∪ (Hy − Hw , Hy )} × (0, Hz ), (3.11) ΩC = ΩD \ ΩW , (3.12) in which Hw is the width of the wall region. Here, Hw = 5a is used at φ = 0.4. For the comparison, the result for a homogeneous suspension from Sierou & Brady [197] is included. P in ΩC is close to the results obtained in an unbounded domain, and It may be seen that σ12 P in ΩC is consistently larger than that in ΩW . The ratios of the deviatoric further that σ12 stress in ΩC to that in ΩW are about 1.15 and 1.19 for C4La and C4H, respectively. 109 Table 3.3: The relative viscosity for different Rref at φ = 0.4. Rref Fref ²min µr C4La 2.01 100 3 × 10−3 4.92 C4Lb 2.004 100 1 × 10−3 5.31 C4Lc 2.001 1000 5 × 10−4 5.70 Typically, µr obtained in the numerical simulations is somewhat lower than the exper- imental results. Sierou & Brady [197] argued that the discrepancy may come from the uncertainty in the contact force model. To show the effects of the contact force on rheologi- cal parameters, the relative viscosity µr at φ = 0.4 is computed for three different values of Rref ; Rref = 2.01 (C4La), 2.004 (C4Lb), and 2.001 (C4Lc). Together with the values of µr , the minimum separation distances ²min in each case are shown in table 3.3. It is apparent that µr is dependent on the choice of the contact force. Since the major contribution to µr from the viscous lubrication interaction is proportional to 1/², µr increases as the minimum separation decreases. Comparing C4Lc and C4La, there is about 16% difference in µr . 3.2.4 Normal stresses In sheared suspensions of non-colloidal particles, the irreversible effects originating from any non-hydrodynamic forces will result in an anisotropic micro-structure, which in turn lead to the development of the non-Newtonian stresses. Among these, the normal stresses in suspensions are of interest because of their importance in the prediction of the particle migration in inhomogeneous flows [160, 155, 164]. The first (N1 ) and second (N2 ) normal stress differences are defined as P P N1 = σ11 − σ22 , (3.13) P P N2 = σ22 − σ33 . (3.14) Zarraga et al. [241] measured linear combinations of the normal stresses in suspensions of non-colloidal particles. They showed that, in Couette flow, both N1 and N2 are negative and |N2 | > |N1 | for φ = 0.3 − 0.5. Brady & Morris [25] showed theoretically that all three normal stress components should be compressive in a uniform shear flow. Using SD simulation, Sierou & Brady [197] calculated the normal stress differences for a wide range of φ (0.1 ≤ φ ≤ 0.5) for a uniform shear flow. Their results show qualitatively consistent 110 behaviour with the experimental results in Zarraga et al. [241]. However, it was observed that |N2 | ' |N1 | in contrast to the experimental fit |N2 | ' 3.6|N1 | found by Zarraga et al. [241]. Origin of this difference between the numerical simulations and experiments is not clear. Possible influences include the effects of the particle-particle contact force or the shear-induced migration of particles in the circular Couette flow device, which introduces curvilinear flow effects, as opposed to the simple homogeneous shear of the SD simulations. Additionally, there is the influence of the walls and the layered structures in the experiments. Figure 3.8 shows the normal stress differences from the present simulations normalized by the shear stress τ = µr µγ. ˙ For comparison, the previous numerical results from SD simulations [197] are also presented. Since there is an appreciable wall effect even in the larger channels for the present simulations, a quantitative match with the previous SD simulations in an unbounded domain is not expected. Nevertheless, N1 and N2 in the larger domains (Hy /a = 20, 30) correspond well with the SD results. At φ = 0.4, reducing Rref , the differences in N1 and N2 between the present simulations and Sierou & Brady [197] decrease. The second normal stress difference (N2 ) calculated in the small channel (Hy /a = 10) shows a similar trend with the results in the larger domain but is larger in magnitude. On the other hand, there is a significant difference in N1 . Other SD simulations of unbounded sheared suspensions of non-colloidal particles have shown that the normal stresses are all P | > |σ P | > |σ P | [240]. However, in the small channel (C4S), it is negative and that |σ11 22 33 P | ≈ |σ P | > |σ P | with all negative values. It seems that the large N /N in found that |σ11 22 33 2 1 C4S is related with the particle layering, which spans the whole channel. The ratio of the normal stress differences, N2 /N1 , estimated in the present simulations are shown in figure 3.9 along with the SD data of Sierou & Brady [197]. It is shown that N2 /N1 > 1 for all the simulations while, from SD, N2 /N1 < 1. Since the normal stress is sensitive to the micro-structure, the particle layers may be responsible for the difference. For example, in C4S where the wall effect is the strongest, N1 ' 0. To identify the effects of the walls, N2 /N1 in the core ΩC and the wall ΩW regions are shown in figure 3.9 (b), where Hw = 2a and 3a are used for φ = 0.2 and 0.3, respectively, in (3.11). N2 /N1 in ΩC quantitatively agrees well with Sierou & Brady [197]. On the other hand, N2 /N1 W C estimated in ΩW is larger than 3. In C2L, C3L, and C4La, 1 < N2Ω /N2Ω < 1.6 while 0.2 < N1 ΩW /N1 ΩC < 0.3. This results suggest that the normal stresses measured in the 111 10-1 ∗ - N1/τ 10-2 10-3 10-4 0.2 0.3 0.4 0.5 φ (a) 0 10 ∗ 10-1 - N2/τ 10-2 10-3 0.2 0.3 0.4 0.5 φ (b) Figure 3.8: Normalized normal stress differences (a) −N1 /τ and (b) −N2 /τ . ♦, C2S, C3S and C4S; ¤, C2L, C3L and C4La; O, C4Lc; ∗, C4H; ◦, [197]. 112 1 10 ∗ N2 / N1 100 10-1 0.2 0.3 0.4 0.5 φ (a) 1 10 N2 / N1 100 10-1 0.2 0.3 0.4 0.5 φ (b) Figure 3.9: (a) Ratio of N2 to N1 in ΩD ; ♦, C2S, C3S and C4S; ¤, C2L, C3L and C4La; O, C4Lc; ∗, C4H. (b) Ratio of N2 to N1 in ΩC (¤) and ΩW (M) for C2L, C3L and C4La. ◦ denotes [197]. 113 0.8 0.5 0.4 0.7 λ2 0.3 λ3 0.2 0.6 0.1 0.5 0 0.2 0.3 0.4 0.2 0.3 0.4 φ φ (a) (b) Figure 3.10: Anisotropy parameters (a) λ2 and (b) λ3 . ♦, C2L, C3L and C4La; ◦, C4H. wall bounded domain will be different from those obtained the numerical simulation in the unbounded flows. If Hy /a is large enough such that |ΩC |/|ΩW | >> 1, it is expected that the rheological properties would be similar to those observed in an unbounded domain. However, in figure 3.9 (a), it is observed that the difference between C4La and C4H is negligible, in which |ΩC |/|ΩW | = 1 and 2 for C4La and C4H, respectively. It seems that, in the largest channel used in the present simulation (Hy /a = 30), the presence of the wall has a significant effect on the suspension rheologies. As a result, it is not straightforward how to extrapolate the rheological properties for large |ΩC |/|ΩW | from the present simulation results. 3.2.5 Normal stresses and continuum models While numerical simulations based on the individual particle motion provide invaluable insight into the detailed dynamics, most of the simulation techniques are computationally too intensive to apply for engineering problems with complex geometries. To predict the behaviour of the suspensions in such a flow condition, there have been several attempts to develop a continuum model [126, 160, 155, 164, 227]. One of the successful approaches is proposed by Morris & Boulay [160], in which the constitutive law for the particle stress for shear flows is given by σ P = −µµn (φ)γQ ˙ + 2µµr (φ)e. (3.15) 114 In this formulation, µn is the normal stress viscosity defined as µ ¶2 φ/φm µn (φ) = Kn , (3.16) 1 − φ/φm where Kn is a rheological fitting parameter. The material property tensor Q is   1 0 0     Q = 0 λ2 0  , (3.17)   0 0 λ3 P /σ P and λ = σ P /σ P . in which the anisotropy parameters are λ2 = σ22 11 3 33 11 Morris & Boulay [160] suggested that λ2 ≈ 0.8 and λ3 ≈ 0.5 provide a good approxima- tion to the suspension rheology and migration. Figure 3.10 shows the anisotropy parame- ters λ2 and λ3 as a function of φ. It is shown that both λ2 and λ3 approach the suggested value at high φ. In the experiments by Zarraga et al. [241], a measure of the anisotropy (N2 − N1 )/σ33 is an increasing function of φ for φ < 0.4 and then reaches a plateau. Hence, there is a possibility that both λ2 and λ3 become constants at higher φ. The values of µn calculated from FCM and equation (3.16) are shown in figure 3.11. Kn = 0.75 and the maximum packing φm = 0.63 are used. The qualitative agreement of the model equation with the FCM results are satisfactory. 3.2.6 Micro-structure Here, we investigate further the details of the micro-structure using the pair distribution function g(r, θ, ψ) for C4La. Here, θ denotes the azimuthal angle measured from the flow direction (positive x) and ψ is the polar angle about the x-axis, measured from the vorticity direction (positive z). Since most contributions to the rheological properties come from the particles near contact, the pair distribution function is shown only for 2 < r/a < 2.1. Figure 3.12 shows the pair-distribution function g(θ) in the x − y plane, the plane of shear, obtained by averaging over π/2 − δ < ψ < π/2 + δ, in which δ = π/60, and over the reference particles whose centers are located in the core region. Considering the mirror symmetry, only 0 < θ < π is shown. It is shown that the probability to encounter another particle around the compressive axis is much higher than that in the extensive axis, which is consistent with g(∆x, ∆y) (figure 3.4 b). The asymmetry, which is responsible for the 115 4 3 µn 2 1 0 0 0.1 0.2 0.3 0.4 φ Figure 3.11: The normal stress viscosity in terms of φ. ¤, C2L, C3L and C4La; ♦, C4Lb; M, C4Lc; ◦, C4H; [160]. 116 15 g(2 < r < 2.1) 10 5 0 0 0.5 1 1 - θ/π Figure 3.12: The pair distribution function for particles of which center is located in 5 ≤ y/a ≤ 15. 117 15 g(2 < r < 2.1) 10 5 0 0 0.5 1 1 - θ/π, θ/π - 1 Figure 3.13: The pair distribution function for particles of which center is located in 2 ≤ y/a ≤ 5. Solid line, 1 − θ/π; dashed line, θ/π − 1. non-Newtonian behaviour, has been observed both in experiments [169] and in numerical simulations [119, 197]. The corresponding pair-distribution function g(θ) for the reference particles centered in 2 < y/a < 5 is shown in figure 3.13. Because the particles are in a buffer region between the strongly structured particle layer around y/a ' 1 and the free shear region (core region), the pair-distribution function is no longer mirror-symmetric. In other words, g(θ) in the upper hemisphere 0 < θ/π < 1 is different from the lower hemisphere 1 < θ/π < 2. It is observed that g(θ) in the upper hemisphere resembles that in the core region qualitatively. However, g(θ) in the lower hemisphere is distinguished by the increased probability near θ/π ' 3/2, which comes from the interaction with the particle layer below. Figure 3.14 shows g(θ) for the reference particles close to the wall, 1 ≤ y/a ≤ 2. Distinctive peaks are observed near θ/π ' 0 and 1, indicating that it is much more probable for a particle near the wall to experience the lubrication interaction with another particle in the same particle layer than particles in another region. A plateau is observed near 118 40 g(2 < r < 2.1) 20 0 0 0.5 1 1- θ/π Figure 3.14: The pair distribution function for particles of which center is located in 1 ≤ y/a ≤ 2. 119 θ/π = 0.5 due to the interaction with the second particle layer. The location of this plateau is consistent with g(θ) for the lower hemisphere shown in figure 3.13 for particles centered further away from the wall. Although g(θ) is not symmetric, the asymmetry is much weaker compared to the core region (figure 3.12), which is responsible for the small |N1 | in the wall region (figure 3.8, 3.9). 3.3 Effects of confinement on the mobility of the particles The interaction between three or more particles in Stokes flow is chaotic [100] and, thus, even small non-hydrodynamic effects, which come from, for example, surface roughness or short-range interparticle forces, make the system unpredictable and irreversible [58, 172]. As a result, non-colloidal particles in sheared suspensions, where the dynamics is deter- mined mainly by hydrodynamic interactions, exhibit a diffusive behavior similar to col- loidal suspensions, where Brownian motion is the driving mechanism [64, 126, 26]. Due to a substantial number of studies over the past decade (see [203, 159] for review), now our knowledge on the shear-induced diffusion process of concentrated non-colloidal suspensions in an open domain, i.e. far from a solid boundary, is much more advanced. However, much less is known about the diffusion process in wall-bounded suspensions. Most studies of the diffusion processes in confined suspensions have been focused on hard-sphere fluids or colloidal suspensions in thermodynamic equilibrium, i.e. in the absence of flow. Mittal et al. [156] calculated the position-dependent diffusivity of confined hard- sphere fluids using discontinuous molecular dynamics simulations. Michailidou et al. [153] have studied the short-time self-diffusion of colloidal suspensions near the wall from both experimental and numerical results. Nugent et al. [165] and Sarangapani & Zhu [188] studied the diffusion of colloidal suspensions bounded by two parallel walls for a wide range of the gap width between two walls to investigate the dynamic lengthscale of structural reorganization. Eral et al. [66] investigated the effects of smooth and rough walls on the diffusion of concentrated colloidal suspensions. They showed that even in the center of the channel, the particles exhibit subdiffusive behavior. In flowing suspensions, Zurita- Gotor et al. [244] estimated the local self-diffusivity in the dilute limit from binary particle interactions. Asmolov [7] has computed the diffusion of non-colloidal particles in dilute wall-bounded suspensions (volume fraction φ ≤ 0.02) under a steady shear using a constant 120 dipole model. Still, there is only a limited number of studies on the shear-induced diffusion in wall-bounded concentrated suspensions. In this section, we study the confinement effects on suspension dynamics focusing on the crossflow diffusion of the particles as a function of the distance from the wall on the intermediate timescale t∗ ∼ O(100), in which t∗ is the time normalized by the shear rate ˙ t∗ = γt. γ, ˙ The main results are presented for concentrated suspensions in a wall-bounded Couette flow, where the bulk volume fraction is φ = 0.40. The channel is divided into four zones, considering the suspension microstructure [230], and the behavior of particles in each zone is investigated. Particularly, we are interested in the variance of particle displacement, which is a measure of the spreading of the suspended particles relative to the mean position. It is found that the variance of the wall-normal displacement shows anomalous diffusive behavior, σy2 ∼ tν , in which ν 6= 1. Depending on the initial location, particles exhibit super- (ν > 1) or subdiffusion (ν < 1) in the velocity-gradient (wall-normal) direction. Anomalous diffusion has been observed in transport processes in porous media [60] or passive tracer dispersion in flows with coherent structures [167, 201, 239]. In this study, we report on anomalous diffusion induced not by a coherent structure in flow but by complex multibody hydrodynamic interactions. The anomalous diffusion near the wall is linked to strong intermittency of particle motion. Unexpectedly, even particles in the core of the channel show a subdiffusive behavior, which may be related with the restriction on the available lengthscale by the confinement. The diffusive behaviors of particles in the core of the channel for four different volume fractions (φ = 0.25, 0.30, 0.35, and 0.40) are compared. The computational domain size is Hx × Hy × Hz = 30a × 30a × 20a, in which a is the particle radius and x, y, and z denote the velocity (streamwise), velocity-gradient (wall- normal), and vorticity (spanwise) directions, respectively. The periodic boundary conditions are used in the horizontal (x − z) directions. The no-slip boundary condition is used on the wall. Unless otherwise stated, all results are obtained at the volume fraction φ = 0.4 with 1718 neutrally-buoyant spheres. The smooth repulsive potential is applied for particle pairs whose center-to-center distance is smaller than 2.01a. The magnitude of the near-contact force is determined to keep the minimum separation distance between particle surfaces a²min ' 0.002. Figure 3.15 (a) shows the area fraction φA as a function of the distance from the wall. 121 (a)0.6 0.4 φA 0.2 I II III IV 0 0 5 10 15 y/a (b)10 (c) 20 y/a y/a 5 10 00 200 400 600 00 200 400 600 t* t * (d)30 (e) 10 20 y/a y/a 5 10 00 200 * 400 600 0 0 5 10 15 t z/a Figure 3.15: (a) Area fraction profile for the bulk volume fraction φ = 0.4. Considering the symmetry, φA is shown only for the lower half of the channel. The wall-normal displacements as a function of time are shown for particles which are initially in (b) Zone I, (c) Zone II + III, and (d) Zone IV. (e) shows a representative trajectory in y − z plane for a particle whose initial location is in Zone I. 122 RR φA (y) is defined by φA (y) = χ(x)dxdz/(Hx × Hz ), where χ(x) is a particle-phase in- dicator function. A significant fluctuation of φA is observed near the wall. Due to the strong particle-wall lubrication interaction, a well structured particle layer develops near the wall (Zone I). Above the particle layer, there is a region in which the behavior of the suspension flow is still strongly affected by the wall (Zone II). In the core region (Zone IV), the rheology of suspension flow behaves similarly to a homogeneous shear flow. There is a buffer region (Zone III) between the discrete (Zone I + II) and the continuous (Zone IV) regimes. Detailed analysis of the rheology and suspension microstructure in each zone is given in 3.2. Figure 3.15 (b–d) illustrate the trajectories of particles whose initial locations are in (b) Zone I, (c) Zone II + III, and (d) Zone IV. Most particles in Zone I stay in the first particle layer for a long period of time t∗ ' O(100) before moving to Zone II. In Zone II, the particles are again trapped in the second particle layer (y/a ' 3). The existence of another particle layer is observed around y/a ' 5, in which the wall-normal displacement is largely restricted to 4 < y/a < 6 for a long period of time. In figure 3.15 (c), it is shown that some particles in Zone II move toward the core of the channel while some particles are trapped in the second or third particle layers exhibiting oscillatory motions around y/a ' 3 or 5. In contrast, the particles in Zone IV exhibit more standard diffusive behavior (figure 3.15 d). However, some of the particles in Zone IV migrate towards the wall relatively quickly (t∗ ' 200) and they are trapped in the particle layers. Figure 3.15 (e) shows a representative trajectory of a particle initially located in Zone I plotted in y − z plane. The wall-normal displacement of the particles in the particle layers (y/a < 7) is strongly suppressed by cages formed by the surrounding particles, while the particles are more mobile in the horizontal direction (z). A similar behavior has been observed for confined hard sphere fluids in equilibrium, where the diffusivity in the horizontal direction is larger than that in the wall-normal direction [156]. Near the wall, the travel time of a particle between each particle layers is much shorter than the residence time inside of the particle layers. The particle trajectories suggest that there is a strong spatial coherence induced by the wall in concentrated suspensions. Moving from the first particle layer (Zone I) to the second (Zone II) and the third particle layers (Zone III), the residence time of a particle in each particle layer decreases and, at the same time, the wall-normal fluctuation around the center of a particle layer becomes stronger. The variance of each component of the particle displacement is defined as σi2 (t) = 123 (a) 101 ~t 0 10 -1 10 σ2z σ* 2 10-2 10-2 σ2y ν = 1.17 σ* 2 / t* ν 2 ~t -3 -3 10 10 ν = 1.69 -4 10-40 100 200 300 10 -1 0 1 2 10 10 *10 10 t 0 (b) 10 P(y*,t*|Y2(0)∈ZI ) -1 10 -2 10 * t = 200 * 10 -3 t = 250 * t = 300 * t = 350 * t = 400 -4 10 0 5 10 y* Figure 3.16: (a) Variances of the wall-normal and spanwise displacements of the particles in Zone I at t∗ = 0. The inset shows the variances divided by tν . (b) The probability density functions (PDF) of the standardized wall-normal displacements at different time instances. The dashed line is ∼ exp(−0.8y ∗ ). 124 hY˜i2 (t)i − hY˜i (t)i2 , in which Y (t) is the location of a particle at time t and Y˜ (t) = Y (t) − Y (0). Figure 3.16 (a) shows the normalized variance (σ ∗ 2 = σ 2 /a2 ) for the particles in Zone I at t∗ = 0. σ 2 in both y and z directions show a ballistic behavior ∼ t2 for short time intervals while the particle velocity remains strongly correlated. For t∗ ∼ O(100), both σy2 and σz2 grow faster than ∼ t, indicating superdiffusion. In the inset of figure 3.16 (a), it is clearly seen that σy2 ∼ t1.69 and σz2 ∼ t1.17 . In homogeneous suspensions, the variances are highly anisotropic and σy2 (t)/σz2 (t) > 1 [58]. However, in Zone I, σy2 is much smaller than σz2 . Figure 3.16 (b) shows the probability density functions (PDF) for the normalized wall- normal displacement. The PDFs are computed for the particles in Zone I at t∗ = 0. Here, the normalized wall-normal displacement is defined as y ∗ = (Y˜2 (t) − hY˜2 (t)i)/σy (t). The normalized PDFs show a remarkable self-similarity. The positive tails of the PDFs almost collapse onto one curve after the superdiffusion is observed (t∗ > 200). The tail of the PDF shows an exponential decay ∼ exp(−αy ∗ ) with α = 0.8 (inset of figure 3.16 b). As a measure of the intermittency, we computed the flatness factor, which is the fourth moment of the normalized PDF. The flatness factor of the PDF is about 18, compared to 3 of a Gaussian distribution, indicating that there is a large fraction of the particles, which travel in a much faster rate compared to the mean spreading rate. The intermittent events, jumps to Zone II and III shown in figure 3.15 (b, e), seem to be responsible for the slow decay of the PDF. Figure 3.17 shows the changes of the PDF of the wall-normal particle velocity (V2 ) in time for the particles initially located in Zone I. The particle velocity is normalized as V2∗ = (V2 (t) − hV2 (t)i)/σV (t), in which σV is the standard deviation of V2 . The third moments of the normalized PDFs in figure 3.17 (a) are about 0.1, suggesting that the wall- normal velocity PDFs P (V2∗ ) are nearly symmetric. Similar to the wall-normal displacement PDF, P (V2∗ ) has a much wider tail compared to the Gaussian distribution. The flatness factor of P (V2∗ ) is a decreasing function of time. The flatness factors are 18, 13, and 10 at t∗ = 200, 300, and 400, respectively. This decrease of the flatness factor in time is related with the reduced frequency of the intermittent events, as the particles migrate into Zone III and IV. To investigate the changes of P (V2∗ ) in time in more detail, the positive tail of P (V2∗ ) is shown in a log-log plot in figure 3.17 (b). Interestingly, P (V2∗ ) consists of three distinctive 125 (a) 1 10 P(V2,t |Y2(0)∈Z1) * t = 200 0 * 10 t = 300 * t = 400 -1 10 * * -2 10 -3 10 -5 0 5 * (b) 1 V 2 10 P(V2,t |Y2(0)∈Z1) 10 0 ~V -2 -1 10 ~V -1/2 * * 10-2 10-3 10-1 100 101 V*2 Figure 3.17: (a, b) PDFs of the wall-normal velocity V2 of the particles initially located in Zone I. The velocity PDFs are shown for t∗ = 200, 300, and 400. The dashed line in (a) is the Gaussian distribution. In (b), B is the velocity PDF for t∗ = 10. 126 (a) (b) Zone II Zone III 10 1 Zone IV ~t 10 1 ~t 100 100 σ*z 2 σ*y 2 -1 -1 10 ~t 2 10 ~t 2 0.3 ν = 0.79 σ 2 / tν 0.2 -2 -2 10 ν = 0.8 10 0.1 ν=1 00 100 200 300 -3 -3 10 -1 0 1 2 10 -1 0 1 2 10 10 10 10 10 10 10 10 t* t* Figure 3.18: Variances of the (a) wall-normal and (b) spanwise displacements of particles in Zones II – IV. ranges. In the central range of small V ∗ , the velocity PDF shows an algebraic decay ∼ V2∗ β with β ' −2. The V ∗ −2 behavior in this core seems to be due to the particles remaining in Zone I. For comparison, P (V2∗ ) at t∗ = 10, when most of the particles are still in Zone I, is drawn in the same plot. It is shown that P (V2∗ ) at t∗ = 10 is well approximated by V ∗ −2 . Around V2∗ ' 0.3 ∼ 0.4, the exponent β changes from −2 in the core to −1/2 in an intermediate range. The crossover point gradually moves towards the origin as the particles leave Zone I and migrate to the channel center. A mechanism responsible for the slow decay ∼ V ∗ −1/2 is not clear. The behavior seems to be related with the particles in Zone II or III. However, the V ∗ −1/2 behavior is somewhat surprising, because the velocity PDF for the particles in other Zones, even that calculated in Zone II, is essentially Gaussian. For large V2∗ , P (V2∗ ) shows an exponential decay. σy2 for Zones II – IV are shown in figure 3.18 (a). As the diffusive behavior changes from superdiffusion in Zone I to subdiffusion in Zones III, Zone II shows normal diffusion (σ 2 ∼ t) with Dy∗ = 2.1 × 10−2 , in which D∗ is the diffusivity normalized by γa ˙ 2 . In figure 3.15 (a), it is shown that φA reaches a plateau in Zone IV, implying that the suspension may behave similarly to a homogeneous suspension. Indeed, in 3.2, it is shown that rheology and 127 suspension microstructure in Zone IV are comparable to those in homogeneous suspensions. Therefore, it is natural to assume that σ 2 would show normal diffusion as in homogeneous suspensions. Unexpectedly, the particles in Zone IV exhibit subdiffusion for t∗ > 150. The inset shows the variances divided by tν . It is shown that σy2 ∼ t0.8 for the particles in Zones III and σy2 ∼ t0.79 for Zone IV. Figure 3.18 (b) shows σz2 for Zones II – IV. Unlike σy2 , normal diffusion is observed in the vorticity direction. In the wall-normal displacement, the variance for Zone IV is significantly larger than that for Zone III; σ ∗ 2y = 17.5 and 26.6 at t∗ = 300 for Zone III and IV, respectively. However, σ ∗ 2z for Zone III and IV are almost the same; 21.4 and 22.3 for Zone III and IV at t∗ = 300. In figure 3.15 (e), it is observed the wall-normal movement is strongly suppressed near the wall while the spanwise displacement is less restricted. As a result, σz is larger than σy for Zones I – III. In Zone IV, σy becomes larger than σz similar to the results in homogeneous suspensions. The diffusivities in the vorticity direction computed from σz2 are Dz∗ = 3.1×10−2 , 3.6×10−2 , and 3.7×10−2 for Zone II, III, and IV, respectively. Dz∗ in Zone III and IV are similar to that in homogeneous shear flow. The changes of area fraction profiles in time for different initial groups of particles are shown in figure 3.19. As the outward flux from Zone I should be equal to the inward flux from Zone II to Zone I, the particles in Zone II have much higher probability of migrating towards the core (Zone III + IV). In figure 3.19 (a), it is shown that only a small fraction of particles move into Zone I while most of them migrate towards the outer region, which may explain the apparent normal diffusion in Zone II (figure 3.18 a). On the other hand, for the particles in Zone III, a large fraction of particles move to Zone I+II, in which they are trapped in particle layers (figure 3.19 b). For y/a > 6, the spreading of φA to Zone IV is almost Gaussian. However, for y/a < 6, φA shows a complex behavior as particles migrate into Zone II and subsequently into Zone I. φA in Zone I is an increasing function of t∗ , while φA in the second particle layer (y/a ' 3) stays almost constant for t∗ = 100 ∼ 400. Subdiffusion may occur if there are stagnation or recirculating regions in the fluid flow, in which particles are arrested [239]. The subdiffusion in Zone III can be understood in the same context. Some particles in Zone III quickly move to the near wall region and they are trapped in the particle layers hovering around the same wall-normal position for a long time, while others migrate to the channel core. However, for Zone IV, only a small fraction of the particles reach Zones I and II and most of the particles remain in Zones III and IV 128 (a) φA (y,t | Y2(0)∈ZII ) 0.4 t* = 0 * t = 100 t* = 200 * t = 300 0.2 00 10 20 30 y/a (b) φA (y,t | Y2(0)∈ZIII ) 0.4 0.2 00 10 20 30 y/a (c) φA (y,t | Y2(0)∈ZIV ) 0.4 0.2 00 10 20 30 y/a Figure 3.19: The area fraction profiles at different time instances for the particles whose initial locations are in (a) Zone II, (b) Zone III, and (c) Zone IV. 129 in the time interval considered (figure 3.19 c). The subdiffusion in Zone IV can not be explained solely by the particle entrapment in the near-wall region. To investigate the diffusive behavior of the particles in the core of the channel more clearly, σy2 is computed for the particles only near the center of the channel, 12 ≤ y/a ≤ 18, for 0.25 ≤ φ ≤ 0.40 (figure 3.20 a). At t∗ = 200, σy∗ = 2.87, 3.58, 4.18, and 4.54 for φ = 0.25, 0.30, 0.35, and 0.40, respectively, indicating that most of the particles have not reached the near-wall region (Zone I+II) yet. The variances in the wall-normal direction divided by t∗ are shown in figure 3.20 (b). For φ = 0.25, normal diffusion is observed for t∗ > 50 with Dy∗ = 2.1 × 10−2 . However, for φ ≥ 0.30, subdiffusive behaviors are observed for t∗ > 100. Considering that σy∗ at t∗ = 100 are 2.59, 3.02, and 3.32 for φ = 0.30, 0.35, and 0.40, respectively, the origin of the subdiffusivity is not the particle entrapment as in the case of Zone III. For φ = 0.25, it is interesting to observe that the linear region (∼ t) is developed for t∗ > 50, while such a linear behavior has been observed at smaller t∗ in homogeneous suspensions (t∗ > 20) [58]. One possible explanation of this subdiffusive behavior is the restriction on the length- scale of suspension flows by the size of confinement. In previous studies of confinement effects on the dynamics of equilibrium colloidal suspensions, it has been shown that, as the lengthscale of structural reorganization is limited by Hy /a, the mobility of particles and relaxation processes are significantly slowed [165, 188, 66]. In sheared suspensions, the dynamics is closely related to the formation of hydroclusters, which have a long correlation length [151]. However, in a confined suspensions, the lengthscale available for the formation of hydroclusters is limited by the confinement size and the emergence of a wall-induced microstructure, namely, a particle layer. For example, for φ = 0.40 and Hy /a = 30, the suspension field is uniform only in 10 < y/a < 20, indicating that the available lengthscale for a shear-induced suspension structure to develop is only L ∼ 10a. This restriction on lengthscale may disable some of large-scale dynamic modes. The disturbance of these large- scale collective motions leads to the subdiffusion on the intermediate timescale t∗ ∼ O(100), over which normal diffusion has been observed in homogeneous suspensions. To support the argument, σy2 for φ = 0.40 and Hy /a = 40 is shown in figure 3.20 (b), where σy2 is computed from the particles in the core of the channel 17 ≤ y/a ≤ 23. Normal diffusion is observed from t∗ > 50 with Dy∗ = 5.7 × 10−2 , which is comparable to the result in homogeneous suspensions [198], suggesting that Hy /a = 40 channel is large enough to 130 (a) 0.6 0.4 φA 0.2 0 0 5 10 15 y/a (b) 0.15 σ*y 2 / t * 0.1 0.05 0 0 50 100 150 200 * t Figure 3.20: The area fraction profiles (a) and variances of the wall-normal displacement (b) for φ = 0.25 (solid), 0.3 (dashed), 0.35 (dash-dot), and 0.40 (dash-dot-dot), Hy /a = 30. In (b), the circles are for φ = 0.40 and Hy /a = 40. 131 accommodate a large-scale suspension structure. This observation is consistent with the experiments of [241], where they found that the suspension rheology is independent of Hy , when Hy /a > 40. Comparing φ = 0.25 to 0.30, the diffusive behavior changes from normal diffusion at φ = 0.25 to subdiffusion at φ = 0.30, though the available lengthscales esti- mated from the φA profiles are similar, suggesting that the lengthscale of the hydroclusters increases with the volume fraction. However, to confirm the changes of the lengthscale with φ, extensive studies for a wide range of φA and Hy /a are still required. We have shown that concentrated non-colloidal suspensions in a wall-bounded Couette flow exhibit anomalous diffusion on the intermediate timescale, on which normal diffusion is observed for homogeneous suspensions. The channel is divided into four zones and the behavior of particles in each zone is investigated. For the diffusion perpendicular to the wall, particles exhibit anomalous diffusion due to a strong spatial coherency induced by the wall effects. Particularly, the particles in a well-structured particle layer located next to the wall (Zone I) show superdiffusive behavior. The probability density functions both for the displacement and the wall-normal velocity show wider tails compared to the Gaussian distribution, indicating the stochastic processes involved in the motion of the particles are highly intermittent. Subdiffusion is observed for the particles in Zone III and IV. As a buffer zone between superdiffusion in Zone I and subdiffusion in Zone III, normal diffusion is observed in Zone II. The subdiffusion in Zone III is related with the trapping of the particles in the particles layers near the wall, Zone I + II. In confined suspensions, the available lengthscale for the suspension dynamics is restricted by the size of the confinement, which will disable some of the large-scale dynamic modes [66]. The subdiffusion in Zone IV seems to be related with a disturbance in large-scale collective motions, such as the formation of hydroclusters, by the size of the confinement. As the volume fraction increases from φ = 0.25 to 0.40, the diffusive behavior of the particles in the channel center changes from normal to subdiffusion, implying the largest lengthscale related with suspension dynamics increases with the volume fraction. It is worthwhile to note that, while most modeling approaches to suspension flows are based on the results in homogeneous suspensions, the present study indicates that the models may fail to predict the diffusive behavior not only near the wall (Zone I+II) but also in the core (Zone IV). 132 3.4 Ordering transitions Colloidal suspensions undergo an intriguing phase behavior when subjected to a shear flow. In the absence of flow, a colloidal suspension develops a crystalline structure above a freezing volume fraction, φf = 0.494 for hard-sphere colloids [179, 184], which can be melted by applying a strong shear to the system [54]. On the other hand, if a shear is applied to disordered colloidal suspensions for the volume fraction φ > φf , crystallization takes place rapidly at low shear rate yet it melts into a fluid at higher shear rate [222]. The shear ˙ 3 /kB T is of O(1). Here, µ is melting is observed when the P´eclet number P e = 6πµγa the viscosity of the solvent, γ˙ is the shear rate, a is the particle radius, and kB T is the thermal energy. Interestingly, another disorder-order transition has been observed at much higher shear rate P e À 1 [4]. Sierou & Brady [197] have demonstrated the existence of shear-induced ordering in homogeneous non-Brownian suspensions (P e → ∞) using the accelerated Stokesian Dynamics simulations. Subsequently, Kulkarni & Morris [120] performed numerical simulations for a wide range of P e, 1 ≤ P e ≤ 104 , and showed the similar phase behavior for P e ≥ 103 . Non-equilibrium phase transitions, in general, are of interest both in theoretical studies [43, 44] and in engineering applications [71]. As the dynamics are determined by a balance between driving forces, the process can be controlled by altering external fields. For example, in colloidal suspensions under an oscillatory shear, different types of crystalline structure can be obtained by changing the frequency and magnitude of the oscillation [150]. When a concentrated suspension is confined by solid boundaries, the dynamics of the suspension becomes dramatically different from the bulk properties [66, 153, 165, 230]. In the absence of flow (P e = 0), phase behaviors of confined molecular or colloidal systems have been extensively studied over the last two decades (see [182] for review). Courtemanche & Swol [45] showed that crystallization of hard-sphere (HS) fluid occurs at a smooth boundary earlier than in the bulk fluid, i.e. below liquid-crystal coexistence, which is later known as ‘wall-induced ordering’. Schmidt & L¨owen [191, 192] calculated the phase diagram of HS fluids confined by two parallel walls for 2 < H/a < 4, where H is the separation distance between walls and a denotes the particle radius. Varying H/a for a fixed volume fraction φ, they observed strong discontinuous phase transitions between different crystal structures, e.g. layered, buckled, and rhombic crystals. Recently, Fortini & Dijkstra [72] performed 133 extensive Monte Carlo simulations and calculated the equilibrium phase diagram of HS fluids for 2 < H/a < 10. However, considering the relevance of highly confined suspensions to many industrial processes such as surface coating, lubricants, and microfluidic devices [80, 178, 195], surprisingly little is known for the effects of confinement on the dynamics of concentrated suspensions under shear flow. Sheared suspensions are different from the aforementioned equilibrium HS fluids in that the dynamics are determined by both long- range multi-body hydrodynamic interactions and short-range lubrication and interparticle forces. In colloidal suspensions under oscillatory shear flows, Haw et. al. [88] observed that crystal structures near a wall are more ordered than those in the center. However, they did not show any quantitative results. Cohen et al. [40] found that a new crystalline structure emerges in a strongly confined system in colloidal suspensions for φ = 0.61 ± 0.02 under large oscillatory shear. In this section, we study the ordering transition of concentrated suspensions confined by two parallel walls under steady shear in the limit of infinite P e, where dynamics are solely determined by hydrodynamic interactions and short-range interparticle forces. It is found that the particles near the walls start forming hexagonally organized strings in the plane normal to the flow at a volume fraction as low as φ ' 0.48, while the center of the channel remains disordered. The ordered state depends not only on the volume fraction but also on the ratio of the channel height to the particle radius Hy /a. The effect of the channel height on the order structure is investigated for 8 ≤ Hy /a ≤ 21 at φ = 0.52. 3.4.1 Simulation parameters The computational domain in x- and z-directions are fixed, Hx /a = 30 and Hz /a = 20, and Hy is varied; 8 ≤ Hy /a ≤ 30. Periodic boundary conditions are used in x− and z−directions. The number of particles ranges from 688 for φ = 0.48 and Hy /a = 10 to 2235 for φ = 0.52 and Hy /a = 30. To model non-hydrodynamic effects, we employ a contact force model. The contact force on particle i from particle j is given by  µ 2 ¶6   Rref −|r|2 r ˙ 2 Fref −6πµγa 2 2 Rref −4a |r| if |r| < Rref FCij = (3.18)   0 otherwise, 134 Figure 3.21: (a) Transient behavior of the relative viscosity normalized by the relative viscosity in the stationary state. The channel height is fixed; Hy /a = 20. Snapshots (end view) for φ = 0.52 and Hy /a = 20 at γt˙ = 1 (b) and 50 (c). For visualization, the particle radius is reduced to 1/2 of the actual size. in which r = Y i − Y j , Fref is a constant, and Rref is a cut-off distance. In the present study, the contact force is activated if the shortest distance between two particle surfaces (²) is less than 0.01a, i.e. Rref /a = 2.01. Fref is chosen to keep the minimum separation distance ²min ' 0.002a. In this study, Fref = 200 is used for φ = 0.46 ∼ 0.54 and Fref = 600 for φ = 0.60. Once FC is computed for all the neighboring particles, it is added to F P . To generate initial configurations, small particles are seeded randomly in the compu- tational domain. Then, a molecular dynamics simulation with a repulsive potential is performed, while slowly increasing the particle radius until the desired volume fraction is reached. Figure 3.21 shows the development of the relative viscosity µr in time for Hy /a = 20. Once shear is applied, the suspension exhibits a disordered fluid state at first 135 Figure 3.22: 2-dimensional pair distributions in the velocity-gradient–vorticity (y − z) plane for Hy /a = 20 obtained in (a,b) ΩW and (c,d) ΩC . (figure 3.21 b), which accompanies a sharp increase of µr . The peak µr is observed to be in between the high-frequency shear viscosity and the dynamic shear viscosity. For φ = 0.60, the peak µr is around 70. As order develops (figure 3.21 c), µr drops slowly. In most cases, a stationary state is reached after γt ˙ ' 40 ∼ 50, which is much faster than for the homogeneous shear results in Kulkarni & Morris [120] (γt ˙ ' 150). 3.4.2 Wall-induced ordering In wall-bounded suspensions, the behavior of suspensions near the wall is radically different from the center of the channel. Depending on the micro-structures, Yeo & Maxey [230] showed that the wall-bounded suspensions of non-Brownian particles can be divided into three regions, the wall, buffer, and core regions. The wall region is distinguished by a strong particle layering. In the core region, the suspension is similar to that in a homogeneous shear 136 flow. In the buffer region, the suspension micro-structure is no longer reflexional symmetric due to the interactions with particle layers below (wall region) and the shear structure above (core region). To show the different levels of ordering near the wall and in the channel center, we investigate the micro-structures in the wall ΩW and the core ΩC regions; ΩW = (0, Hx ) × {(0, 1.5a) ∪ (Hy − 1.5a, Hy )} × (0, Hz ) and ΩC = (0, Hx ) × (5a, Hy − 5a) × (0, Hz ). For Hy /a = 10, ΩC is defined as ΩC = (0, Hx ) × (4a, 6a) × (0, Hz ). First, the ordering transition is investigated by varying φ for Hy /a = 20. The pair distribution function in spherical polar coordinates g(r, θ, ψ) is calculated for particles in ΩW and ΩC , in which r is the radial distance, θ is the azimuthal angle measured from the velocity direction, and ψ denotes the polar angle measured from the vorticity direction. Figure 3.22 shows g(r, θ, ψ) in the velocity-gradient–vorticity (y − z) plane, i.e. θ = π/2. At φ = 0.46, the dominant structure in ΩW is the particle layering and a weak hexagonal order is observed, while g(r, ψ) in ΩC shows an isotropic ring-like structure indicating the suspension is homogeneous. At φ = 0.48, a hexagonal order begins to be developed in ΩW , while the suspension in ΩC is still in disordered state. At φ = 0.52, the whole channel is almost completely ordered (figure 3.22 c). As the suspension in ΩC is in the disorder-order coexistence state, g(r, ψ) for φ = 0.50 shows a mixture of the hexagonal structure (figure 3.22 d) and the ring-like structure (figure 3.22 c). Figure 3.23 shows the area fraction φA as a function of y for Hy /a = 20. φA is calculated RR by φA = χ(x)dxdz/(Hx × Hz ). Here, χ(x) is an indicator function which is non-zero if x is inside of particles. For φ ≤ 0.46, φA is higher near the wall. The values of the first peaks near the walls are insensitive to φ. It seems that the high volume fraction near the wall leads to the earlier transition in ΩW than in ΩC . At φ = 0.46, the value of the local peaks is a decreasing function of distance from the wall. Similar to φ = 0.40, if Hy /a is large enough, φA may become uniform around the core of the channel. However, once the ordered structure is present across the entire channel (φ = 0.52), φA is no longer a decreasing function of distance from the wall. The peaks of φA are almost constant across the channel. The area fraction profiles suggest that the particles form a layered structure at high volume fractions in confined suspensions, in which the suspension dynamics are dependent on the interaction between discrete particle layers, and, thus, the previous continuum model approaches, which rely on the rheological functions based on well-mixed suspensions, may not be suitable to apply for suspensions near the ordering transitions. 137 0.8 0.6 φA(y) 0.4 0.2 φ = 0.40 φ = 0.46 φ = 0.52 0 0 5 10 15 20 y Figure 3.23: Area fraction profiles for Hy /a = 20. (a) 1 (b) 1 C6 C6 0.5 0.5 Hy /a = 10 Hy /a = 15 Hy /a = 20 Hy /a = 30 0 0 0.45 0.5 0.55 0.6 0.45 0.5 0.55 0.6 φ φ Figure 3.24: Order parameter C6 in (a) ΩW and (b) ΩC . 138 Figure 3.25: Snapshots (end view) for φ = 0.52; (a) Hy /a = 20 and (b) Hy /a = 30. For visualization, the particle radius is reduced to 1/2 of the actual size. Figure 3.24 shows the hexagonal order parameter C6 estimated in ΩW and ΩC . Similar to the bond-orientational order parameter, C6 is defined as [120] R 2π 0 g(ψ) cos(6ψ)dψ C6 = R 2π , (3.19) 0 g(ψ)dψ in which g(ψ) is the pair-distribution function in the y−z plane averaged over 2a < r < 2.1a. The values of C6 lie in the range 0 ≤ C6 ≤ 1, i.e. C6 = 1 for a perfect hexagonal structure and C6 = 0 for an isotropic micro-structure. In general, C6 in ΩW is larger than that in ΩC consistent with the wall-induced ordering. For Hy /a = 15, the hexagonal ordering in ΩW is not evident when φ ≤ 0.48. On the other hand, for Hy /a = 20, C6 ' 0.8 at φ = 0.48, indicating that the particles in ΩW begin to be ordered into hexagonal strings at φ = 0.48. When φ ≥ 0.52 and Hy /a = 20, C6 in ΩC and ΩW are almost the same, implying the presence of the hexagonal order in the whole channel. The decrease of C6 in ΩW at high volume fraction is mainly due to the small sampling volume. Since the sampling volume of C6 for ΩW is small, even if there are only a few defects in the sampling volume, C6 can be significantly reduced by these defects. If C6 is computed for the whole domain, it is a non-decreasing function of φ for a given Hy /a. 139 25 Hy /a = 10 Hy /a = 15 20 Hy /a = 20 Hy /a = 30 15 µr 10 5 0.45 0.5 0.55 0.6 φ Figure 3.26: The relative viscosity for various φ and Hy /a. In a larger channel (Hy /a = 30), the wall effects become weaker in the channel center, which in turn weakens the ordered structure in ΩC . At φ = 0.50, C6 in ΩC are 0.65 and 0.5 for Hy /a = 20 and 30, respectively. On the other hand, a better order is observed near a wall. Figure 3.25 shows snapshots for different Hy at φ = 0.52. For Hy /a = 20, most particles are assembled into nearly linear strings which are organized as a hexagonal array with a few defects. However, for Hy /a = 30, a disordered fluid region emerges in the channel core. Typically, most experiments of non-colloidal suspensions are performed in a wide channel, for example Hy /a > 40 [241]. Therefore, it is likely that the suspension is in the disorder-order coexistence state, for which the bulk behavior resembles that of the homogeneous suspension. This may be one reason that ordering transitions in the non-colloidal suspension has not been investigated so far in experiments. The relative viscosity µr is shown in figure 3.26. For Hy /a = 15 and 20, µr begins to decrease at φ = 0.48 as the hexagonal order develops. Once the hexagonal order is dominant across the entire channel, µr increases again (φ > 0.52). The value of µr for Hy /a = 30 is larger than those in the smaller channels due to the emergence of disordered region in ΩC , 140 Figure 3.27: Snapshots (end view) for φ = 0.52; (a) Hy /a = 9, (b) Hy /a = 10, (c) Hy /a = 11. For visualization, the particle radius is reduced to 1/2 of the actual size. in which µr is higher. It is expected that, in a large channel Hy /a > 40, the disordered region is much larger than the ordered region so that the bulk properties would resemble those without the ordered structures. Unlike the results in the three wider channels, µr for Hy /a = 10 is an increasing function of φ. 3.4.3 Effects of the channel height on the order structures In the equilibrium phase transition (P e = 0) of strongly confined colloidal suspensions, a sequence of crystal layers is observed depending on the commensurability of the crystal structures with the channel height; n¤ → n4 → (n + 1)¤ → · · · . Here, n¤ and n4, respectively, denote n crystal layers of square and hexagonal lattice symmetries in the hor- izontal (x − z) plane. Depending on the commensurability, first-order melting and freezing transitions are observed between different crystal structures [192]. At higher volume frac- 141 Table 3.4: Channel height Hy /a, characteristic gap width η, number of particle layers N , and order parameter C6 for φ = 0.52. Hy /a 8 9 10 11 12.5 15 17.5 19 19.5 20 21 η 4.46 5.04 5.62 6.20 7.06 8.51 9.95 10.82 11.10 11.39 11.97 N - 5 - 6 7 8 10 11 11 11 12 C6 0.59 0.88 0.50 0.82 0.88 0.67 0.89 0.94 0.86 0.78 0.86 tions, the alternating sequence of square and hexagonal crystal layers is disturbed by the emergence of other crystal structures, such as rhombic, buckling, or prism phases [72]. In flowing suspensions, the particle layers slide over each other and ordered structures must accommodate this. Hence, it is not expected that the ordered structures and the commensu- rability will exactly follow what has been observed in the thermodynamic phase transitions of HS fluids or colloidal suspensions. To investigate the effects of Hy /a on the ordering transition in flowing suspensions, the numerical simulations are performed for varying Hy /a from 8 to 20 for φ = 0.52 and 0.54. Figure 3.27 shows snapshots for φ = 0.5 at different channel heights Hy /a. Increasing Hy /a from 9 to 11, it is shown that the suspension exhibits phase transitions from the hexagonal order (figure 3.27 a) to a mixed state (figure 3.27 b) and, then, again to the ordered state (figure 3.27 c). Figure 3.28 shows the order parameter C6 as a function of Hy /a for φ = 0.52 and 0.54. It is shown that the order structure is very sensitive to Hy when Hy /a ≤ 11. For φ = 0.52, the suspensions are in an ordered state for Hy /a = 9 and 11 and in a disordered state for Hy /a = 8 and 10. The similar behavior is observed for φ = 0.54. The oscillation in C6 becomes smaller for larger Hy . When Hy /a is changed from 10 to 11, C6 decreases from 0.88 to 0.50, while, increasing Hy /a = 19 to 20, it changes from 0.94 to 0.78. The change in the order state is due to the commensurability of the order structures with the available space between two walls. As the most distinguishable structure is the simple hexagonal lattice in the y − z plane, a characteristic gap width can be defined as Hy − 2a η= √ + 1. (3.20) 3a The characteristics gap width are shown in table 3.4 together with the number of particle layers N and the order parameter C6 . The number of particle layer is well approximated 142 25 φ = 0.52 φ = 0.54 20 Hy / a 15 10 5 0.4 0.6 0.8 1 C6 Figure 3.28: The order parameter as a function of Hy /a for φ = 0.52 and 0.54. 143 0 10 Hy = 8 Hy = 9 ~t Hy = 10 Hy = 12.5 -1 Hy = 19 10 Hy = 20 〈∆y2〉 / a2 -2 10 -3 10 10-1 100 101 102 γt Figure 3.29: The normalized mean-square vertical displacements h[Y2 (t) − Y2 (0)]2 i/a2 for φ = 0.52. by η. As expected, the suspension is in more ordered state (large C6 ) when η is close to an integer, or small |N − η|. When η is increased or decreased from an integer, the distance between particle layers increases making particles more mobile. As a result, the order structure becomes unstable and the suspension becomes disordered. The mean-square displacement (MSD) in the vertical direction h[Y2 (t)−Y2 (0)]2 i normal- ized by a2 is shown in figure 3.29. After a short ballistic regime (γt ˙ ∼ O(10−1 )), MSDs for Hy /a = 8 and 10 show a sub-diffusive behavior, h[Y2 (t) − Y2 (0)]2 i ∼ tν with the exponent ν = 0.86 on the intermediate time scale of the present simulation. Previously, Yeo & Maxey [229] showed that confined non-Brownian suspensions for φ = 0.40 exhibit sub-diffusion on the time scale of γt ˙ ∼ O(100). In the long term, the vertical particle displacement is confined by the size of the channel height. When an ordered structure is developed, the vertical displacement of particles is restricted by the cage formed by neighboring particles. For Hy /a = 19 and 20 , in which the suspensions are in a mixed state, the particles in the core of the channel behaves similarly to the disordered state, while mobility of the parti- 144 (a) C6 (b) C6 0.4 0.6 0.8 1 0.4 0.6 0.8 1 25 25 20 20 Hy / a Hy / a 15 15 10 10 5 5 0 5 10 0 5 10 15 Π Π Figure 3.30: The particle pressure Π (¨) and the order parameter C6 (•) as a function of Hy /a for (a) φ = 0.52 and (b) 0.54. cles near the walls is significantly reduced due to the ordered structure. Hence, MSDs for Hy /a = 19 and 20 are significantly lower than those in a disordered state (Hy /a = 8 and 10). For Hy /a = 9, the entire channel is in an ordered state and the vertical displacement of particles is observed only near defects. Hence, there is a large increase in MSDs between ordered (Hy /a = 9) and mixed states (Hy /a = 12.5, 19, and 20). The bulk particle pressure Π normalized by µγ˙ is shown in figure 3.30. The definition of Π is 1 Π = − (σ11 + σ22 + σ33 ). (3.21) 3 Here, σij denotes the particle stress computed from the stresslet and the interparticle po- tential. In a strongly confined system (Hy /a < 12), the oscillation of Π is almost exactly opposite to C6 . However, in larger channels, the correlation between Π and C6 becomes less clear. Because the wall effect is strong across the entire channel in small channels, the order structure and, thus, the rheological parameters are mainly determined by the com- mensurability. On the other hand, in a larger channel, there is a competition between the 145 confinement effect and the shear around the core of the channel. Therefore, rheology does not exactly follow the commensurability of the channel. Figure 3.31 (a) shows the order structure in the horizontal (x − z) plane for φ = 0.52 and Hy /a = 9. A rhombic phase is observed for the horizontal structure. A stable rhombic phase is also observed in the equilibrium phase transitions of confined HS fluids as an interpolating structure between square and hexagonal symmetric structures [192]. Fortini & Dijkstra [72] found that the rhombic phase is stable between n¤ and n4 for n ≤ 5, i.e. Hy /a < 10. However, in flowing suspensions, a rhombic phase is observed for a wide range of Hy /a. When φ ≤ 0.54, only rhombic phases are observed for the range of parameters in the present study of non-Brownian (infinite P e) suspensions. In flowing suspensions, the disturbance flow induced by a particle decays slowly, which leads to a long-range correlation. The long-range hydrodynamic interactions result in the earlier order transition in sheared non-Brownian suspensions than that seen in HS fluids. However, near the lower boundary of order transitions, the volume fraction is still too low to develop a fully three-dimensional crystal structure. As the hydrodynamic force induced by the shear flow is anisotropic, an ordered structure is formed in the y − z plane first. It seems that the rhombic phase in the x − z plane in figure 3.31 (a) is a transitional structure that supports a hexagonal structure in the y − z plane at the given volume fraction. In contrast, suspensions at higher volume fractions may show a phase behavior similar to HS fluids. At φ = 0.60, 5 layers of the hexagonal symmetric structure are observed for Hy /a = 9 (figure 3.31 b), which is followed by 6 layers of nearly rectangular structure for Hy /a = 10 (figure 3.31 c). 3.5 Effects of external torques on rheology and ordering tran- sitions Manipulating rheological properties of suspension flows by imposing electric or magnetic fields is of current interest due to its relevance to a wide range of technological applications [71, 86, 111]. In electrorheological (ER) or magnetorheological (MR) fluids, a dramatic in- crease of the apparent viscosity is observed, when particles are assembled into well-organized large-scale structures, such as chains or stripes, by applied fields [46, 170]. It is also known that the apparent viscosity can be reduced by applying an external torque on the particles in ferrofluids [10]. In ER fluids, such a decrease of apparent viscosity can be achieved by a 146 Figure 3.31: Order structures in the horizontal plane for (a) φ = 0.52 and Hy /a = 9, (b) φ = 0.60 and Hy /a = 9, and (c) φ = 0.60 and Hy /a = 10. Green (lighter) particles are in the lower layer and red (darker) particles are in the upper layer. 147 DC electro-rotation of suspended particles (Quincke rotation) [102]. If a DC electric field is applied in the velocity-gradient direction on suspensions under a steady shear, the particles experience an electric torque in the vorticity direction, which makes the particles act as a colloidal motor [127, 134]. Lemaire et al. [127] have suggested a simple model to predict the apparent viscosity of sheared suspensions under Quincke rotation. However, the detailed micro-structure and rheological behavior of sheared suspensions under an external torque are not fully understood. Rheological properties of suspension flows, such as the apparent viscosity and the nor- mal stress differences, are closely related to the micro-structure of the suspension. For example, in suspensions of non-Brownian particles under a steady shear, irreversible effects introduced by small roughness elements on the particle surface, residual Brownian force, and/or surface charge result in an anisotropic micro-structure, which is responsible for the non-Newtonian rheology [25]. In recent studies of the colloidal [120] and non-Brownian sus- pensions [197, 231] under a steady shear, an ordering transition is observed at high volume fractions (φ ' 0.50). This non-equilibrium phase transition of non-Brownian particles is different from the thermodynamic phase transition of colloidal particles in that the process is driven mostly by the shear-induced hydrodynamic interactions between particles. As the suspension undergoes the ordering transition, the apparent viscosity decreases significantly. Particularly, Yeo & Maxey [231] showed that the ordering transition and, hence, the changes in the apparent viscosity are complex functions of the ratio of the channel height to the particle radius and the volume fraction. In this section, we show a possibility of manipulating the ordering transition of non- Brownian suspensions in a Couette flow by applying external torques in the vorticity di- rection to the particles. At high volume fractions φ ≥ 0.48, where the ordering transition occurs, the suspension can be more ordered or disordered depending on the sign of the external torque. Applying negative torque can hinder the ordering transition, which is then accompanied by an increase in the shear viscosity. As a consequence, contrary to previous results at low volume fractions (φ ≤ 0.20) [127], the shear stress of the suspension is not reduced dramatically by negative torques. On the other hand, a positive torque has a fa- vorable effect on the hexagonal order. However, above a certain threshold, the order begins to be weakened by the positive torque. At moderate volume fractions φ ≤ 0.4, the shear and vortex viscosities are neither sensitive to the sign nor the magnitude of torque. 148 3.5.1 Simulation parameters We focus on the volume fractions around which an ordering transition of non-Brownian suspensions occurs, φ = 0.48 ∼ 0.52 [231]. The computational domain is Hx × Hy × Hz = 30a × 20a × 20a, in which a is the particle radius and Hx , Hy , and Hz denote the lengths of the domain in the velocity (x), velocity-gradient (y), and vorticity (z) directions, respectively. Periodic boundary conditions are used in the horizontal directions (x and z). The computational domain is bounded by two parallel walls located at y = 0 and Hy . The lower wall is fixed and the upper wall is moving in the x-direction with velocity Vupp = γH ˙ y, where γ˙ is the nominal shear rate. The external torque is varied between −3 ≤ T ∗ ≤ 3, in which T ∗ is the torque normalized by the fluid viscosity µ0 and γ, ˙ T ∗ = T /8πµ0 γa ˙ 3 . In Couette flow, the suspended particles rotate in the clockwise (negative) direction. When a negative torque is applied, the particles rotate faster and a positive torque retards the rotation. To model irreversible forces, an elastic contact force is used when the shortest distance between two particle surfaces (a²) is smaller than 0.01a [230]. The magnitude of the contact force is set to keep the minimum separation distance a²min ' 0.002 in all simulations. Initial random configurations for the simulations are generated by a molecular dynamics procedure. First, the simulations with zero torque are allowed to evolve until the suspension reaches a stationary state. Then, an external torque is applied to the suspension. It usually takes about tγ˙ ' 100 to reach a new stationary state after a torque is applied. 3.5.2 Results and discussions Figure 3.32 shows illustrative examples of the effects of an external torque on ordering transitions. For Hy /a = 20, an ordering transition begins around φ = 0.48 [231]. At φ = 0.48, the suspension is in a mixed disordered-ordered state whereby a hexagonal order exists near the wall with a disordered state in the core of the channel (figure 3.32 a). The suspension has a fully hexagonal order across the whole channel when φ ≥ 0.52 (figure 3.32 c). When the positive torque T ∗ = 2 is applied to the suspension in a mixed state (φ = 0.48), the hexagonal order near the wall is more pronounced (figure 3.32 b). On the other hand, at φ = 0.52, the hexagonal order in the core of the channel is disturbed after the particles are subjected to the negative torque T ∗ = −2 (figure 3.32 d). 149 Figure 3.32: Snapshots (end view) for φ = 0.48 (a,b) and φ = 0.52) (c,d). For visualization, the particle radius is reduced to 1/2 of the actual size. 150 1 0.8 0.6 C6 0.4 φ = 0.48 0.2 φ = 0.50 φ = 0.52 0 -3 -2 -1 0 1 2 3 * T Figure 3.33: The order parameter C6 as a function of the non-dimensional torque T ∗ . 151 To investigate the hexagonal order quantitatively, Kulkarni & Morris [120] suggested a hexagonal order parameter C6 , R 2π 0 g(ψ) cos(6ψ)dψ C6 = R 2π , (3.22) 0 g(ψ)dψ in which ψ denotes the azimuthal angle measured from the positive vorticity (z) direction and g(ψ) is the pair-distribution function in y − z plane averaged over the radial interval 2a < r < 2.1a. C6 = 1 for a perfect hexagonal order in y − z plane and C6 = 0 if the suspension micro-structure is isotropic. C6 for different φ is shown in figure 3.33 as a function of T ∗ . It is clearly seen that the hexagonal order is always weakened by the negative torque. On the other hand, the effect of positive torque on the suspension is not straightforward. When a positive torque is applied, C6 increases at first and then starts ∗ . T ∗ seems to depend on the order state at T ∗ = 0. decreasing slowly after a threshold Tcrit crit ∗ Both for φ = 0.50 and 0.52, Tcrit is observed around T ∗ ' 1, while Tcrit ∗ for φ = 0.48 is found at larger T ∗ , T ∗ ' 2. The shear stress of suspension flows at low Reynolds number in the presence of particle torques is [13, 230] τ ∗ = 1 + hσxy ∗ i + 3φT ∗ , (3.23) in which h·i denotes an average over the whole suspension, and τ ∗ and σxy ∗ are, respectively, the shear stress and x − y component of the symmetric part of the particle stress tensor normalized by µ0 γ. ˙ The sum of the first two terms on the right hand side of (3.23) corre- ∗ i. In suspensions under steady shear, sponds to the effective shear viscosity; µs /µ0 = 1+hσxy normal relative motions between particles, which may occur during tumbling of a particle doublet by the shear flow, generate stresslets proportional to the inverse of the gap between particles (∼ 1/²), which is a major contributor to µs [73]. However, once the hexagonal order is developed, most particle interactions are tangential relative motions between par- ticle strings, whose contribution to µs (∼ log ²) is much smaller than that from the normal motion. As a consequence, in the previous studies on suspensions under a steady shear, the decrease of µs is observed if a hexagonal order is present in suspensions [120, 197, 231]. The shear viscosity is shown in figure 3.34 (a). In general, µs is a decreasing function 152 (a) 12 10 µs / µ0 8 6 4 φ = 0.40 φ = 0.48 2 φ = 0.50 φ = 0.52 0 -3 -2 -1 0 1 2 3 * (b) T 12 10 8 µapp / µ0 6 4 φ = 0.40 φ = 0.48 2 φ = 0.50 φ = 0.52 0 -3 -2 -1 0 1 2 3 * T Figure 3.34: The shear (a) and apparent viscosities (b) as functions of the non-dimensional torque. 153 of T ∗ . At φ = 0.40, where the suspension is in a disordered state, µs is much less sensitive to the external torque. There is only less than 10% difference between µs for T ∗ = −2 and 2. At φ = 0.48, in which the ordered region is confined near the wall, the changes in µs by negative T ∗ is not significant similar to φ = 0.40. A dramatic increase of µs is observed at φ ≥ 0.50 when negative T ∗ is applied. As the hexagonal order is disturbed by negative T ∗ , a disordered region emerges in the channel core, which in turn results in the increase of µs . On contrary, the decrease of µs by positive T ∗ is most pronounced for φ = 0.48; µs is decreased about 15% by changing T ∗ from 0 to 2. However, when a positive torque is applied on the already ordered suspensions (φ ≥ 0.52), the changes in µs are not noticeable. Local minima of µs are observed for φ = 0.50 and 0.52, which correspond to the maxima of C6 in figure 3.33. Although the shear viscosity is an important parameter in studying suspension rheology, it is not easy to obtain µs directly in the experiments of sheared suspensions under an external torque. Here, we show the apparent viscosity, which can be obtained by measuring the shear stress on the top wall. The apparent viscosity of suspension µapp is defined by ˙ or µapp = µs + 3µ0 φT ∗ . The definition of µapp a simple constitutive equation, τ = µapp γ, implies that, if suspension micro-structure, or µs , is not altered by an external torque, µapp can be increased or decreased by changing T ∗ and it may be possible to observe even a “negative viscosity” [127]. The theoretical prediction of the “negative ER effect” by [127] assumes that µs is a function of φ only, i.e. the micro-structural change is negligible. The result for φ = 0.40 indicates that the assumption in [127] is indeed valid at lower volume fractions (φ ≤ 0.40). However, once a ordering transition occurs, µs becomes a function of both φ and T ∗ and the previous analysis is no longer applicable in this regime. If suspensions are in a ordered state, the magnitude of negative T ∗ needed to observe the “negative ER effect” would be much higher than that predicted by the analytical model [127]. The apparent viscosity is shown in figure 3.34 (b). As µs is only weakly dependent on T ∗ at φ = 0.40, µapp is seen to be a linear function of T ∗ . A notable change in µapp is observed when φ ≥ 0.50. For φ ≥ 0.50, the contribution from negative T ∗ to µapp is somehow balanced by the increase of µs . As a result, µapp remains almost unchanged regardless of the magnitude of negative T ∗ in the range of the present study. For φ = 0.52, it is observed that, after a threshold T ∗ ' −2, µapp begins to decrease. One of the important rheological parameters in ER or MR fluids is the vortex viscosity 154 3 Marchioro et al. [19] Monte Carlo T* = -2 T* = -1 T* = 1 µv / µ0 2 T* = 2 1 0 0.2 0.3 0.4 0.5 φ Figure 3.35: Variations in the vortex viscosity µv as a function of volume fraction for different values of the non-dimensional torque in a confined shear flow. Also shown are results of Monte Carlo simulations for randomly seeded suspensions in a homogeneous uniform shear flow, compared to prior results [143]. µv , which is a measure of the hydrodynamic resistance to an applied external torque. The vortex viscosity µv is defined as [27, 68] µv 3φT ∗ = , (3.24) µ0 2hΩz i/γ˙ + 1 in which hΩz i is the averaged angular velocity of the particles in the vorticity direction. In an infinite domain, Marchioro et al. [143] have calculated µv by a Monte Carlo procedure for a wide range of φ. For a comparison, µv computed by FCM in a tri-periodic domain is shown in figure 3.35 together with the numerical fitting curve by Marchioro et al. [143]. µv in the present simulation is estimated from an ensemble average of 100 random configurations for each φ. The agreement with [143] is excellent. Due to the absence of any microstructure, µv obtained by the Monte Carlo approach does not depend on the sign or magnitude of T ∗ . In figure 3.35, the vortex viscosity estimated by dynamic simulations of the confined 155 0.5 * 0 T =2 Ωz T* = 1 -0.5 * T =0 * T = -1 -1 T* = -2 -1.5 0 5 10 15 20 y/a Figure 3.36: Effects of the torque on the angular velocity in the vorticity direction for φ = 0.52 and Hy /a = 20. steady shear flow is presented. At φ = 0.4, it is observed that µv obtained in the dynamic simulations is larger than that by the Monte Carlo approach in an infinite domain. In the previous boundary element computation of confined sheared suspensions, the vortex viscosity of confined suspensions was slightly lower than that of the infinite domain [68]. Hence, the increase of the vortex viscosity seen in the present dynamic simulations may be due to the formation of hydroclusters [151]. At φ = 0.40, µv is not sensitive to the signs and magnitudes of T ∗ . On the other hand, a bifurcation of µv is observed when φ ≥ 0.48, where the hexagonal order is present in the suspension. In general, µv for negative T ∗ is always larger than that for positive T ∗ . At φ = 0.48, µv for positive T ∗ is a decreasing function of T ∗ for T ∗ ≤ 2, as the suspension becomes more ordered, while the effect of the negative torque, disturbing the order structure, is not noticeable. It seems that, because the suspension is already in a disordered state except near the wall, µv is less sensitive to the magnitude of torque for negative T ∗ . On the contrary, φ ≥ 0.50, µv does not change significantly when T ∗ > 0. The changes in the ordered state correspond to the changes in the angular velocities of the suspended particles. In a homogeneous suspension, the additional hydrodynamic force from neighboring particles due to the external torques tend to be canceled out. In a wall- bounded sheared suspensions, however, the symmetry in the suspension microstructure is 156 broken and the response of the suspension to an external torque now depends on the sign of the torque. Figure 3.36 shows the mean angular velocity in the vorticity direction for φ = 0.52 and Hy /a = 20. Similar to the mean particle velocity (3.4), the mean angular velocity is defined as Np Z Z 1 X φA Ωz (y) = H(a − |x − Y i |)Ωnz dxdz, (3.25) Hx × Hz i=1 h(φA Ωz )(y)i hΩz (y)i = . (3.26) hφA (y)i The particle-wall resistance relation for the hydrodynamic torque in a shear flow is given by [22, 230], Y C ΩD ˙ = T ∗ + 3Y A (Vx − Vx∞ )/4aγ˙ − Y H = T ∗ + TG∗ , z /γ (3.27) where ΩD D ˙ Y C , Y A , and Y H are the z is the retarded angular velocity Ωz = Ωz + 0.5γ, resistance functions, which depend on the distance from the wall, Vx is the translational velocity in the velocity direction, and Vx∞ is the background velocity by the imposed shear rate. When a particle is close to the wall, both Y C and TG∗ are positive, which indicates that the suspended particles near the wall rotate more slowly than the bulk when T ∗ = 0. Similarly, for negative T ∗ , |T ∗ + TG∗ | < |T ∗ | and the particles near the wall rotates slower than those in the core of the channel. On the other hand, if a positive torque is applied, |T ∗ + TG∗ | > |T ∗ |, indicating |ΩD | for the particles near the wall becomes larger than that in the core. The asymmetric response to the sign of T ∗ may be explained by these wall-induced hydrodynamic interactions. As the dynamics of the wall layer is decoupled from that in the core before an ordering transition occurs, the wall effects are localized near the wall and the bulk dynamics does not change noticeably. However, near the ordering transition, a long-range correlation develops and the wall-induced hydrodynamic interactions may alter the overall suspension dynamics. Here, we have not specified a way to impose an external torque on the particles. The external torque may be achieved by applying magnetic or electric fields as seen in ferro- or ER fluids. However, in concentrated suspensions, electric or magnetic coupling between particles may be important, which is not considered in the present study. The effect of electric or magnetic coupling between particles is a subject of further investigation. 157 3.6 Conclusions In this chapter, we have reported on the fully three-dimensional numerical simulations of concentrated suspensions of non-colloidal particles in Couette flow using the force coupling method. To investigate the wall effects on the suspension dynamics, wide ranges of volume fraction and channel height are considered; φ = 0.20 ∼ 0.60 and Hy /a = 10 ∼ 40, in which φ is volume fraction and Hy and a denote the channel height and the particle radius. In the Couette flow suspensions, there is a critical volume fraction around which the suspension dynamics are completely changed. In an analogy to hard-sphere liquids, the suspension can be categorized by disordered fluid and ordered crystal regimes below and above the critical point, respectively. It is found that the critical volume fraction depends on the channel height. We have investigated the dynamics in the disordered fluid regime in the first two sections (3.2 and 3.3) and in the ordered crystal regime in the last two sections (3.4 and 3.5). In concentrated suspensions, the strong particle-wall lubrication inhibits particles near the wall from being resuspended into the core region. As a result, one or more particle layers are formed near the wall depending on the volume fraction. Although the present simulations are performed only for the plane Couette flow, it seems that the particle layering at high volume fraction is common in most wall-bounded flows. Even in the Poiseuille flows where most particles migrate toward the center of the channel, there is evidence of the particle layering [84]. In the disordered fluid regime, since the micro-structure inside of the particle layer is significantly different from that in homogeneous suspensions, it is not straightforward to predict the rheological properties in wall-bounded flows from the results obtained in unbounded domain. It is found that the width of the particle layer is not sensitive to Hy /a once the channel is wide enough. At φ = 0.4, it is observed that the layering of particles is dominant for 0 ≤ y/a ≤ 7 and beyond this the suspension is quasi- homogeneous similar to unbounded suspensions. Based on the observation, the rheological properties are calculated separately in the core and wall regions. The rheological properties, such as the relative viscosity and normal stress differences, in the core region agree well with the results in an unbounded domain. In the wall region, the formation of particle layers leads to rheology of the suspension distinct from unbounded suspensions. As Hy /a increases, the ratio of the core region to wall region grows larger so 158 that the bulk rheological properties should resemble more those in unbounded suspensions. Interestingly, there is no difference in the normal stress differences between Hy /a = 20 and Hy /a = 30 at φ = 0.4 in the present simulations, even though the ratio of the core to wall region increases from 1 to 2. It should be noted that even at Hy /a = 30 it seems that suspensions are still in a discrete regime. Hence, a simple extrapolation to higher Hy /a from the present results may not be appropriate. The micro-structure clearly shows the different dynamics depending on the distance from the wall. At φ = 0.4, the micro-structure in the core region resembles that in the homogeneous suspensions. The shear structure (higher probability near the compressive axis) is missing in the pair-distribution function g(θ) for particles in the particle layer (y/a < 2). In the particle layer, most particle-particle interactions arise from the normal motion between the particles in the same particle layer (θ = 0 or π) or from the tangential interaction with particles above (θ = π/2). In the buffer layer (2 < y/a < 5), the pair- distribution function g(θ) shows the characteristics of both the homogeneous suspension and the particle layer. It is shown that concentrated Couette flow suspensions exhibit anomalous diffusion on the intermediate timescale, on which normal diffusion is observed for homogeneous suspen- sions. Based on the results in section 3.2, the channel is divided into four zones and the behavior of particles in each zone is investigated. For the diffusion perpendicular to the wall, particles exhibit anomalous diffusion due to a strong spatial coherency induced by the wall effects. Particularly, the particles in a well-structured particle layer located next to the wall (Zone I) show superdiffusive behavior. The probability density functions both for the position and the wall-normal velocity show wider tails compared to the Gaussian distribu- tion, indicating the stochastic processes involved in the motion of the particles are highly intermittent, which may be related with the superdiffusivity of the particles. Subdiffusion is observed for the particles in Zone III and IV. As the diffusive behavior changes from superdiffusion in Zone I to subdiffusion in Zone III, normal diffusion is observed in Zone II. The subdiffusion in Zone III is related with the trapping of the particles in the particles layers near the wall, Zone I + II. In confined suspensions, the available lengthscale for the suspension dynamics is restricted by the size of the confinement, which will disable some of the large-scale dynamic modes [66]. It seems that subdiffusion in Zone IV is related with a disturbance in large-scale collective motions, such as the formation of hydroclusters, by 159 the size of the confinement. As the volume fraction increases from φ = 0.25 to 0.40, the diffusive behavior of the particles in the channel center changes from normal to subdiffusion, implying the largest lengthscale related with suspension dynamics increases with the vol- ume fraction. It is worthwhile to note that, while most modeling approaches to suspension flows are based on the results in homogeneous suspensions, the present study indicates that the models may fail to predict the diffusive behavior not only near the wall (Zone I+II) but also in the core (Zone IV). As the bulk volume fraction approaches φ ' 0.50, non-colloidal Couette flow suspensions undergo a phase transition. This non-equilibrium phase transition is of interest in that the suspension will cease to flow due to the divergence of the effective viscosity as the volume fraction approaches the random close packing limit if the ordering transition does not occur. When a Couette flow suspension is in an ordered state, the effective viscosity is significantly lower than that in the high frequency limit. One of the first observations of the shear- induced ordering of non-colloidal suspensions is done by Abbott et al. in 1991 [3] in the experiments of concentrated suspensions of bimodal particles in the circular Couette flow. However, the ordering transitions of non-colloidal suspensions had been virtually unexplored until the first numerical observation by Sierou & Brady in 2002 [197] in an infinite domain. Here, it is shown that the shear-induced crystallization of non-Brownian suspensions under a strong confinement is dramatically different from the previous results in homoge- neous suspensions [120, 197]. As the volume fraction is higher near the wall than the bulk, an ordering transition occurs earlier in the wall region. At φ = 0.48 and Hy /a ≥ 15, a hexagonal structure (y − z plane) of particle strings (x-direction) is observed near the wall, while the suspension in the core of the channel is still in disordered state. For a strongly confined system Hy /a ≤ 11, the order state and rheology depend on the commensurability. However, due to the competition between the wall effects and shear-induced hydrodynamic forces, the relation is not so clear for larger channels Hy /a ≥ 20. At φ = 0.60, it is ob- served that the order structure in horizontal plane exhibits a transition from triangular to rectangular structures, similar to the equilibrium phase transitions in hard-sphere fluids. It is shown that due to the complex phase behavior, the rheological parameters, such as the relative viscosity and the particle pressure, are nonlinear functions of both the channel height and the volume fraction. We have shown a possibility of manipulating the ordering transition of non-Brownian 160 suspensions in a Couette flow by applying external torques in the vorticity direction to the particles. At high volume fractions φ ≥ 0.48, where the ordering transition occurs, the suspension can be more ordered or disordered depending on the sign of the external torque. Applying negative torque can hinder the ordering transition, which is then accompanied by an increase in the shear viscosity. As a consequence, contrary to previous results at low volume fractions (φ ≤ 0.20) [127], the shear stress of the suspension is not reduced dramatically by negative torques. On the other hand, a positive torque has a favorable effect on the hexagonal order. However, above a certain threshold, the order begins to be weakened by the positive torque. At moderate volume fractions φ ≤ 0.4, the shear and vortex viscosities are neither sensitive to the sign nor the magnitude of torque. Chapter 4 Stokes flow: Concentrated suspensions in Poiseuille flow 4.1 Introduction The study on suspensions of solid particles in Poiseuille flows is essential in understanding many biological and engineering flows [178, 203]. Since the pioneering works by Batchelor [13, 14, 15], homogeneous suspensions, e.g. suspensions in a linear shear flow far from a solid boundary, have been extensively studied and, at least, the bulk rheological behaviour of a homogeneous suspension is well understood [159, 197]. In Poiseuille flow, however, the shear rate varies in the wall-normal direction and, as a consequence, a migration of the suspended particles towards the core region of the channel has been observed, which in turn makes the suspension field inhomogeneous. One of the first observations of the inhomogeneities in the suspension field dates back to Karnis et al. [105]. Nevertheless, the theoretical development on Poiseuille-flow suspensions has been slow due to the complex hydrodynamic interactions and lack of detailed experimental as well as numerical data about the suspension dynamics. Karnis et al. [105] found, in the experiments in a pressure-driven tube flow, that a plug flow is developed at high volume fractions in the core region of the channel, in which the translational particle velocity is almost uniform. Since then, several experimental studies on concentrated Poiseuille-flow suspensions have been performed both in a channel [98, 98, 112, 139, 194] or in a tube flow [84]. The experimental studies reveal that the suspended particles migrate towards the core region of the channel, resulting in a gradient in the local 161 162 volume fraction, which is highest in the core of the channel and decreases closer to the wall, and a blunting of the velocity profile in the core region. Leighton & Acrivos [126] proposed a phenomenological model based on the gradient of volume fraction and the local shear rate to predict the shear-induced migration. Nott & Brady [164] suggested a suspension balance model, which relates the shear-induced migration to the particle stresses, based on the phase-averaged mass and momentum conservation equations [61]. Morris & Boulay [160] and, subsequently, Miller & Morris [155] proposed modifications of the suspension balance model, which has now become a standard predictive model. A more systematic theoretical analysis of slightly inhomogeneous suspensions has been performed by [143, 242]. Through the theoretical studies, it is now generally understood that the imbalance of the normal stresses in the wall-normal direction induces the particle migration. However, still there is no direct measurement of the local suspension properties across the channel, supporting the theoretical studies. Measuring suspension properties, such as particle stresses, is a challenging task. For example, it is only recently that the particle pressure of Couette-flow suspensions has been successfully measured [52]. Numerical simulations, based on Stokesian dynamics, have pro- vided invaluable information in understanding suspension dynamics [23]. However, due to a lack of suitable numerical techniques, numerical simulations of concentrated suspensions have largely been limited to homogeneous suspensions [197] or, for inhomogeneous suspen- sions, to two-dimensional simulations [164, 199]. Since Nguyen & Ladd [162] developed a numerical method to incorporate lubrication interactions into the lattice-Boltzmann method (LBM), LBM becomes a popular tool to investigate the three-dimensional wall-bounded suspensions [116, 119, 215]. More recently, Yeo & Maxey [233] modified the force-coupling method to account for the near-field interactions and for the fast three-dimensional simula- tions of concentrated suspensions. This scheme, which incorporates low-order force multi- pole representations and viscous lubrication corrections, has been successfully employed to study wall-bounded Couette flows of concentrated suspensions [230, 231, 232]. Here, we report on three-dimensional numerical simulations of concentrated suspensions of O(1000) monodisperse non-colloidal particles in plane Poiseuille flows using the force- coupling method. The main goals of this study are to investigate the dynamics of Poiseuille- flow suspensions via phase-averaged suspension quantities, examining these as functions of the distance from the wall and both the bulk and local volume fractions, and to provide 163 statistics essential in developing and verifying the suspension models. The remainder of this chapter is organized as follows: the simulation parameters are shown in section 4.2; the volume fraction profiles are compared with the previous experiments in section 4.3.1; section 4.3.2 describes the velocity statistics, such as mean and fluctuating particle velocities. In section 4.4, the particle stresses are investigated as functions of the distance from the wall and compared with the suspension model by Morris & Boulay [160]. 4.2 Simulation parameters We use the force-coupling method (FCM) developed in Chapter 2 to compute the particle- particle and particle-wall hydrodynamic interactions for Poiseuille flow in a channel. Here, the interparticle potential force is computed as  µ 2 ¶6   Rref −|r|2 r −6πµaVref 2 2 |r| if |r| < Rref Rref −4a FijP = (4.1)   0 otherwise, in which Vref is a constant, r is a relative position vector Y j − Y i , and Rref is a cut-off distance. To minimize the effects of the contact force on the suspension dynamics, the contact force is activated only when the separation distance between particles ² < 0.002a, i.e. Rref /a = 2.002. Vref is chosen to keep the minimum separation distance ²min ' 0.001a. To verify the numerical method, we compare the translational velocity of a single par- ticle parallel to the wall calculated by FCM to the numerical result by Ganatos et al. [76] (GWP82). In GWP82, a boundary collocation technique was used to compute the trans- lational velocity of a sphere, which is almost exact for Y /a ≥ 1.1. Lomholt & Maxey [135] have shown that FCM results agree well with GWP82 for Y2 /a ≥ 1.1. Here, we compare FCM simulations with GWP82 for Y2 /a < 1.1, where the lubrication interaction becomes important. The FCM simulations are performed for the computational domain Hx × Hy × Hz = 30 × 20 × 20 and the particle radius a = 1. Table 4.1 shows the translational velocity normalized by the centerline velocity V /uc for a particle near the bottom wall. The agreement with GWP82 is very good. The difference in the velocity is at most O(10−3 ). For the particle very close to the wall, Y /a ≤ 1.01, there is about 7% difference between FCM and GWP82. There is some uncertainty in the simulation 164 Table 4.1: The translational velocity of a particle at different vertical locations Y . The channel height is Hy /a = 20. The particle velocity is normalized by the centerline velocity uc . Y /a FCM GWP82 1.1 0.156 0.155 1.04 0.128 0.128 1.01 0.102 0.097 1.001 0.078 0.073 Table 4.2: Simulation parameters. Φ is the bulk volume fraction and Np is the number of particles. Nx and Nz are the number of Fourier modes in the streamwise and spanwise directions, respectively. Ny is the number of spectral elements in the wall-normal direction. The length of each spectral elements are the same, lE = Hy /Ny . The same numbers of quadrature points QE and the spectral modes PE are used for every elements; QE = 10 and PE = 8. Φ Hx /a Hy /a Hz /a Nx Ny Nz Np P3S 0.3 30 18 20 96 9 64 773 P4S 0.4 30 24 20 96 12 64 1,376 P2L 0.2 40 40 30 128 20 96 2,292 P3L 0.3 40 40 30 128 20 96 3,438 P4L 0.4 40 40 30 128 20 96 4,584 results by GWP82 at these small separation distances. Due to the computational limitations at that time, they were able to compute a fully converged solution only for Y /a ≥ 1.1 and, for locations below Y /a = 1.1, some functions were estimated by an extrapolation from the value at Y /a = 1.1. Still, the difference between the FCM results and GWP82 in the absolute value is small ∼ O(10−3 ). In order to resolve the hydrodynamic forces exactly for the parabolic velocity profile of a Poiseuille flow, it would be necessary to include the force quadrupole and degenerate sextupole. If a channel is much larger than the particle radius, however, the quadratic variation of the fluid velocity at the location of a particle can be effectively approximated by a linear function, which is resolved by the stresslet. It is worth noting that, using lower order, regularized force multipoles, FCM approximately resolves higher-order multipoles such as the degenerate force quadrupole and octapole, which come from the Fax´en corrections. A numerical simulation of Poiseuille flow can be performed either fixing the driving force (mean pressure gradient) or keeping the mass flux constant. In the previous Stokesian Dynamics simulations, constant mass flux simulations were performed and the pressure 165 (a) 0.6 0.4 φ 0.2 0 0 0.5 1 1.5 2 y/h (b) 0.8 0.6 φ 0.4 0.2 0 0 0.5 1 1.5 2 y/h Figure 4.1: The area fraction profiles for (a) P3S and (b) P4S. The solid lines are the present simulation results and the circles are from Lyon & Leal [139]. gradient was estimated a posteriori, which is a natural choice for a resistance formulation [158, 164]. On the other hand, in the force-coupling method, which is written as a mobility problem, it is easier to control the pressure gradient. Hence, we use a constant pressure gradient condition for the present simulations and the mass flux is estimated in a post- processing stage. The simulation parameters are shown in table 4.2. 4.3 Volume fraction and velocity statistics 4.3.1 Volume fraction profile In Poiseuille flows, the shear rate varies across the channel, which results in an inhomo- geneous stress field. As a result, the suspended particles tend to migrate into the center region of the channel. It is well known that, due to the migration, the local volume fraction becomes an increasing function of the distance from the wall. 166 Here, we define the averaged area fraction, φ(y), as ¿Z Z À 1 φ(y) = χp (x)dxdz , (4.2) Hx × Hz in which χp (x) is an indicator function for the particle phase and h·i is an ensemble average. The bulk volume fraction is given by the average of φ(y) across the channel, i.e. Φ = R φ(y)dy/Hy . In figure 4.1, the area fraction profiles obtained by the present simulations are compared with the experiments 562 (Φ ' 0.3) and 482 (Φ ' 0.4) in Lyon & Leal [139]. Both the simulation and the experimental results show a local peak at the core of the channel. The φ profiles of the present simulations agree well with [139] near the core of the channel. However, near the wall, the volume fraction profiles of the experiments quickly drop to zero, while the FCM simulation predicts the existence of a particle layer and that φ decreases more slowly in the intermediate region, y/h < 0.5. Here, h is the half channel gap h = Hy /2 The existence of a wall layer has been reported in several numerical simulations of wall- bounded Couette-flow suspensions [116, 119, 215, 230]. As there is no net hydrodynamic lift force acting on a particle in Stokes flows, the upward (or downward) vertical displacement of a particle in a shear flow is usually understood by the interaction with another particle below (or above) the center of the reference particle. However, once a particle is located next to the wall, the particle interacts only with particles above and, moreover, a strong particle-wall lubrication interaction makes it harder for the particle to move away from the wall. In experiments, however, usually there are some uncertainties in the experimental conditions, such as size and shape variations, roughness elements on the surface of the particle or the wall, and interaction forces due to electrostatic charge or polymer coating. There is a possibility that such imperfections may disrupt the particle layer observed in the numerical simulations. It is also important to note that, in concentrated suspensions, it is challenging to measure the volume fraction accurately, particularly, near the wall. For example, in figure 4.1, the local volume fractions in [139] are much smaller than the simulation results near the wall, while similar in the core. As a result, if the bulk volume fraction is estimated by an average of the data over the channel height, it will be lower than the actual volume fraction. Roughly, the bulk volume fractions estimated from the experimental data are Φe ' 0.23 and 0.32 for Φ = 0.3 and 0.4, respectively. 167 (a) 0.6 0.4 φ 0.2 0 0 5 y/a (b) 0.8 0.6 φ 0.4 0.2 0 0 5 10 y/a Figure 4.2: The area fraction profiles near the wall for (a) P3S and (b) P4S. The solid lines are the present simulation results and the circles are from Hampton et al. [84]. 168 0.6 0.4 φ 0.2 0 0 0.2 0.4 0.6 0.8 1 y/h Figure 4.3: The area fraction profiles for P2L (solid), P3L (dashed), and P4L (dash-dot). The symbols are the experiments by Gilchrist [79]: ◦, Φ = 0.298; 4, Φ = 0.411. Hampton et al. [84] have measured the volume fraction profiles of concentrated sus- pensions in a pressure-driven tube flow using nuclear magnetic resonance imaging. In their experiments of non-colloidal particles of diameter 2a = 3175µm, they observed the particle layering at high volume fractions Φ ≥ 0.30. Figure 4.2 shows the volume fraction profiles near the wall. Because of the difference in the geometry, straight channel in the simulations versus circular tube in the experiment, it is not straightforward to compare the volume fraction profiles over the whole flow. Still, there is a good agreement in the volume fraction profiles near the wall. Unlike the results in [139], which showed a rapid decrease of φ profile, the volume fraction profiles of [84] agree well above the one particle diameter y/a > 2 both for Φ = 0.30 and 0.40. At Φ = 0.30, the volume fraction profile of [84] shows a weak local peak near y/a ' 1, suggesting the existence of the particle layer. However, the local volume fraction is somewhat lower than the simulation results. For Φ = 0.40, the quantitative agreement is quite satisfactory even in the particle layer. It is worthwhile to note that the volume fraction in [84] is obtained by an average over a small bin, while φ in the simulation is computed by a direct integration of the indicator function. Further discussions about the volume fraction profiles are provided in section 4.5. One of the most recent measurements of concentrated suspensions in Poiseuille flows were performed by Gilchrist’s group [78, 77] in a wide channel, Hy /a = 80. In figure 4.3, 169 (a) 0.6 0.6 0.4 0.4 φL 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 y/h (b) 0.6 0.6 0.4 0.4 φL 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 y/h Figure 4.4: The local volume fraction profiles for (a) Φ = 0.3 and (b) Φ = 0.4; •, P3L and P4L; 4, Gilchrist [79]; , Yapici et al. [227]. The dashed line in (a) is the area fraction (φ) profile of P3L. the volume fraction profiles for Hy /a = 40 are compared with the experimental data [79]. The wall-normal distance is normalized by the channel half gap h. The particle radius in the experiments is about 0.5µm, suggesting that there may be some limited effects from Brownian motion. Nevertheless, it is shown that the simulation and experimental data are consistent. Again, it is observed that the bulk volume fractions estimated by an average over the channel height in the experiments are lower than the actual volume fractions. Near the wall, the local volume fractions in the experiments are lower than the simulations. The first data points of the experiments are at y/a ' 2a, where the φ profiles show local minima. For Φ = 0.20, the particle layering from the simulations is less pronounced, consistent with the results of [84]. A local volume fraction can be computed from φ as Z y+w 1 φL (y) = φ(s)ds, (4.3) 2w y−w in which 2w is the width of the averaging interval. The definition is similar to the procedure 170 used in the experimental measurements and we set w = a. In contrast to the φ profiles, the near-wall layer and the particle depletion layer near y/a ' 2 are smoothed out in the φL profiles, as shown in figure 4.4. Elsewhere φL and φ are nearly indistinguishable. For comparison, we also show the volume fraction profiles from the continuum model of Yapici et al. [227]. While the volume fraction profiles computed by the continuum model are qualitatively similar with the experiments and the present simulations, it is shown that the continuum model predicts higher volume fraction around the core of the channel and the volume fraction decreases more rapidly with the distance from the center. 4.3.2 Velocity statistics As the suspension field is homogeneous in a plane parallel to the wall, we define a phasic average of a variable g as Drew [61], * Np Z Z + 1 X hgχp i(y) = g i χip (x)dxdz , (4.4) Hx × Hz i=1 in which g i and χip are the value of g and the indicator function of the i-th particle. Then, an average of g for the particle phase can be defined as, hgχp i(y) hgi(y) = . (4.5) φ(y) The particle-phase velocities normalized by the centerline velocity uc for P3S and P4S are compared with the experiments by Lyon & Leal [139] in figure 4.5. Following [139], the centerline velocity is estimated from a parabolic velocity profile, which has the same bulk velocity as that of the particle phase Qp ; Z 1 Qp = hVx idy. (4.6) Hy Here, Vx is the particle velocity in the streamwise (x) direction. The velocity profiles show an excellent agreement except very near the wall y/h < 0.2. In the FCM simulations, due to the particle-wall lubrication interaction, there is an almost step-function increase in the particle velocity around y/h ' 0.2. Figure 4.6 shows the average particle-phase velocity profiles for different Φ in a larger 171 (a) 1 〈V〉/uc 0.5 0 0 0.5 1 1.5 2 y/h (b) 1 〈V〉/uc 0.5 0 0 0.5 1 1.5 2 y/h Figure 4.5: The averaged particle velocity for (a) P3S and (b) P4S. The solid lines are the present simulation results and the symbols are from [139]. 1 〈V〉/uc 0.5 0 0 5 10 15 20 y/a Figure 4.6: The average particle-phase velocity profiles for Hy /a = 40. Dotted line, P2L; dashed line, P3L; solid line, P4L. 172 5 4 3 µapp / µ 2 1 0 0 0.1 0.2 0.3 0.4 0.5 Φ Figure 4.7: Apparent viscosity estimated from the mean flux: ¤ P3S and P4S; ◦ P2L, P3L, and P4L; ∗ Nott & Brady [164]. channel (Hy /a = 40). It is shown that the velocity profile becomes more blunted as Φ increases. The blunting of the velocity profile is more pronounced as Φ increases from 0.3 to 0.4. The maximum φ at the center of the channel for P4L is φ(h) ' 0.58, while that for P3L is about 0.50. It has been observed that a sheared suspension may undergo a phase transition around Φ ' 0.52 in wall-bounded suspensions [231] and Φ ' 0.54 in homogeneous suspensions [120, 197]. In the experiments of concentrated suspensions in a pressure-driven channel flows for Φ = 0.41, Gao et al. [77] found a crystal structure in the core region of the channel and noted that the suspension in the core of a pressure-driven channel flow behaves similarly to a confined fluid. The drop in the maximum velocity at the center of the channel at Φ = 0.40 may be related to the high local volume fraction. In a standard Poiseuille flow in a straight channel, the relation between the bulk velocity Q∞ , mean pressure gradient, and fluid viscosity is H2 D Q∞ = f . (4.7) 12µ 173 Similarly, the apparent viscosity of Poiseuille-flow suspensions is defined by Cokelet [41] as H2 D µapp = f . (4.8) 12Qp Hence, the apparent viscosity in the present simulation is simply µapp Q∞ = p. (4.9) µ Q Figure 4.7 shows the apparent viscosity estimated in the present simulations. Also shown are results from the Stokesian Dynamics simulations (SD) of Nott & Brady [164]. These SD simulations were performed for a constant suspension velocity, so µapp is estimated from their mean pressure gradient data. The SD data are for (Φ = 0.20, Hy = 40.40a) and (Φ = 0.30, Hy = 18.32a). As the SD results of [164] are from the two-dimensional simulations of co-planar particles, the bulk volume fraction is estimated from the bulk area fraction as Φ = 23 φbA . For Φ = 0.20, the agreement between P2L and SD is good, while µapp for P3S is about 10% larger than that of SD obtained in the same Hy . However, considering the differences in the simulation conditions, the overall agreement is reasonable. It has been observed, in Couette-flow suspensions, that the apparent viscosity is an increasing function of the channel height Hy /a for Hy /a < 40 [230, 241]. By contrast, in Poiseuille- flow suspensions, we observe that µapp decreases in a larger channel. A similar behaviour has been observed by [164] for φbA = 0.4. However, their simulations for φbA = 0.30 predicted the opposite; an increase of µapp in wider channels. The particle-phase velocity fluctuation is computed as " µ ¶2 #1/2 hVi2 χp i hVi χp i vi = − . (4.10) φ φ Figure 4.8 shows the velocity fluctuations normalized by the mean velocity Qp in a larger channel Hy /a = 40. For all three components, the maxima of the velocity fluctuations are observed at y/a ' 2, where φ becomes minimum. Due to the wall slip, vx and vz have a finite value at y ' ²wall , in which ²wall is the minimum separation distance between the particle and the wall. In the present study, ²wall ' 0.001a. The vertical velocity fluctuation is a smooth function of y near the wall, which is a natural consequence of the impermeability of the wall. It is shown that the normalized velocity fluctuations increase with Φ. Near the 174 (a) 0.1 vx 0.05 0 0 10 20 30 40 y/a (b) 0.05 vy 0 0 10 20 30 40 y/a (c) 0.05 vz 0 0 10 20 30 40 y/a Figure 4.8: The particle-phase velocity fluctuations normalized by the mean particle-phase velocity Qp for (a) streamwise, (b) wall-normal, and (c) spanwise components: Solid line, P2L; dashed line, P3L; dash-dot line P4L. 175 Table 4.3: The coefficients of the fitting curves for velocity fluctuations. vx vy vz C0 C1 C2 C0 C1 C0 C1 P2L 0.0863 -0.117 0.0359 0.0439 -0.0402 0.0243 - 0.0212 P3L 0.0928 -0.126 0.0411 0.0598 -0.0607 0.0363 - 0.0326 P4L 0.103 -0.146 0.0516 0.0756 -0.0850 0.0476 - 0.0449 core of the channel (16 ≤ y/a ≤ 24), vy is not sensitive to the changes in Φ unlike the other two components, vx and vz . As a consequence, in the core of the channel, vx > vy > vz at Φ = 0.20, which becomes vx > vz > vy at Φ = 0.40. The streamwise and spanwise velocity profiles become more blunted near the core as the volume fraction increases. At Φ = 0.40, vx and vz are almost constant over 16 ≤ y/a ≤ 24. Near the wall and the core of a channel, the finite size of the suspended particles plays an important role in the suspension dynamics and, hence, it is not expected that a continuum analysis would apply. If a channel is wide enough, there exists an intermediate region, in which the spatial variation of the ensemble averaged suspension field is quasi-homogeneous on a particle scale. The suspension dynamics may be well approximated by a continuum theory in this intermediate region. Studying ensemble averaged suspension quantities in such a “well-mixed” region is important in developing a continuum model to predict the behaviour of suspension flows [61, 160, 164]. Here, we find fitting expressions for each velocity fluctuations normalized by Qp . The fitting curves are defined as MP X ³ y ´p P (vi ) = Cp , (4.11) h p=0 in which MP is the maximum order of the polynomial and Cp ’s are fitting coefficients. The coefficients Cp are shown in table 4.3. Different MP are tested and it is found that vx is well represented by a quadratic polynomial of y/h, while vy and vz are linear in y/h. For vy and vz , the slopes of the curves are linearly proportional to Φ. That is, vy ∼ 2κ(y/h)Φ and vz ∼ κ(y/h)Φ in the intermediate region, in which κ is a constant, κ ' 0.1. Figure 4.9 shows the velocity fluctuations normalized by the fitting curves. The velocity fluctuations are well represented by the fitting expressions above y/h ' 0.3 for 0.2 ≤ Φ ≤ 0.4. On the other hand, near the core of the channel, the velocity fluctuations start to 176 (a) 1.5 v / P(v) 1 0.5 0 0.2 0.4 0.6 0.8 1 y/h (b) 1.5 v / P(v) 1 0.5 0 0.2 0.4 0.6 0.8 1 y/h (c) 1.5 v / P(v) 1 0.5 0 0.2 0.4 0.6 0.8 1 y/h Figure 4.9: The particle-phase velocity fluctuations normalized by the fitting curves for (a) P2L, (b) P3L, and (c) P4L: solid line, vx ; dashed line, vy ; dash-dot line, vz . 177 1.5 1 -Ωz / γc 0.5 0 0 0.2 0.4 0.6 0.8 1 y/h Figure 4.10: The average angular velocity normalized by γ˙ c = uc /h. Solid line, P2L; dashed line, P3L; dash-dot line, P4L. Long-dash line is the rate-of-rotational of clear flow. deviate from the fitting expressions around y/h ' 0.75, 0.7, and 0.65 for Φ = 0.2, 0.3, and 0.4, respectively. These changes in the functional form of the velocity fluctuations indicate that the width of the core region is an increasing function of Φ, while that of the wall layer is not so sensitive to the bulk volume fraction. Next, we consider the mean angular velocity of the particles. The anti-symmetric part of the velocity gradient tensor (rate-of-rotation) of the clear flow, i.e. without suspended particles, is 1 du/dy y A12 = − = −1 + , (4.12) 2 γ˙ c h in which γ˙ c is the shear-rate based on uc , γ˙ c = uc /h. Using this scaling, we show the average angular velocity in the spanwise (vorticity) direction hΩz i of the particle phase normalized by γ˙ c in figure 4.10. Similar to vx and vz , the angular velocity has a finite value very near the wall y ' ²wall . Note that, in the present study, the suspended particles never touch the wall due to the lubrication and potential forces. There is a thin liquid film, whose thickness ²wall is determined mainly by the potential force and the effects of the wall on the angular velocity of the suspended particles are through the particle-wall lubrication interaction. In experiments, ²wall may be determined by the surface asperities and, hence, the angular velocity of the particles may be affected not only by the particle-wall lubrication but also by 178 a friction force between the particles and the wall. Due to the lack of any experimental data, it is difficult to quantify the effects of the friction at present. However, it is not expected that the effect of the particle-wall contact is significant in the suspension dynamics outside of the first particle layer. As observed in the velocity fluctuations, the angular velocity profile can be divided into three regions. Near the wall, the rotation of the particles in the particle layer is significantly hindered due to the particle-wall lubrication interaction, which is followed by a sudden increase of Ωz above the first particle layer. In an intermediate region, the angular velocity decreases faster than that of the clear flow and the slope of Ωz is an increasing function of the volume fraction. Consistent with the blunting of hV i, the slope of Ωz near the core of the channel decreases at larger Φ. In figure 4.11, the angular velocity is compared with the wall-normal gradient of the particle-phase velocity. The velocity gradient is multiplied by 1/2 analogous to the rate-of- rotation. Near y/a ' 2, the velocity gradient diverges because the particle-phase velocity profile is not smooth at the depletion layer just above the particle layer (figure 4.6). In an intermediate region, both the angular velocity and the particle velocity gradient can be approximated by linear functions, indicating that in the intermediate region the particle velocity is still a quadratic function of y. 4.4 Particle stresses 4.4.1 Normal stresses Following Drew [61], the ensemble averaged momentum conservation equations for the par- ticle phase in Stokes flows are ∇ · hχp σi − hσ · ∇χp i = 0, (4.13) in which σ is the stress tensor and χp is the particle-phase indicator function. The ensemble averaged equations are exact for every x ∈ ΩD , where ΩD is a domain of interest. However, due to the difficulties in dealing with the singular term, ∇χp , a volume average of (4.13) on a scale much smaller than the spatial variation of the suspension field is sometimes more useful. After integrating over a small volume ΩS and applying the mean value theorem to 179 (a) 1.5 1 -Ωz 0.5 0 0 0.2 0.4 0.6 0.8 1 y/h (b) 1.5 1 -Ωz 0.5 0 0 0.2 0.4 0.6 0.8 1 y/h (c) 1.5 1 -Ωz 0.5 0 0 0.2 0.4 0.6 0.8 1 y/h Figure 4.11: The angular velocity (solid line) and 12 dhV i dy (dashed line) normalized by γ ˙c = uc /h with fitting curves (dotted line) for (a) P2L, (b) P3L, and (c) P4L. 180 the first term, the averaged momentum conservation equations become *Z + ∂ 1 hχp σij i + S σij nj dS = 0, (4.14) ∂xj |Ω | ∂Sp ∩ΩS in which ∂Sp is the surface of the particles in a realization and nj is a unit outward normal vector on the particle surface. The surface integral is performed over the particle surfaces inside of ΩS . The ensemble average and the surface integral do not commute because ∂Sp changes for each realization. Equation (4.14) is accurate to O(a/L), where L is a macroscopic lengthscale. The second term of (4.14) represents the interphase force [142] and is usually modeled as the average hydrodynamic drag force on the particles [160, 164]. A proper treatment of the interphase force in an inhomogeneous suspension is still not clear [128]. In a suspension where the lengthscale of the spatial variation of suspension quantities is much larger than p the particle size, the phasic-averaged stress may be approximated as hχp σij i ' φL hσij i, in p which hσij i is the volume averaged particle stresses in a small volume Ωs ; Z 1 φL = hχp id3 x, (4.15) |Ωs | Ωs Z p 1 hχp σij i 3 hσij i = s d x. (4.16) |Ω | Ωs hχp i The volume average may be taken in |Ωs | ∼ O(a3 ). In microfluidic devices, however, usually the spatial variation of suspension fields is non-negligible even at the particle scale and the volume averaging process has to be performed very carefully. Thus, in the present study, we study the behaviours of the phasic-averaged stresses hχp σi, avoiding the issues with the volume average. To compute the ensemble average, the stress inside of a particle is assumed to be uniform in a statistical sense and evaluated from the sum of the particle stresslet (2.132) and the interparticle force (3.18) contributions, à Nb ! 1 X 1 p σ= S− rF , (4.17) |Ωp | 2 in which |Ωp | is the volume of a particle, Nb is the number of the particles within the cut-off distance, Rref = 2.002a. The normal particle stresses are shown in figure 4.12 as functions of the distance from 181 (a) 0.05 0 χσii / f h D P -0.05 -0.1 0 0.2 0.4 0.6 0.8 1 y/h (b) 0.1 0 χσii / f h D -0.1 P -0.2 -0.3 0 0.2 0.4 0.6 0.8 1 y/h (c) 0 χσii / f h D -0.2 P -0.4 -0.6 0 0.2 0.4 0.6 0.8 1 y/h Figure 4.12: The phasic-average of the particle stresses normalized by f D h for (a) P2L, (b) p p p P3L, and (c) P4L. Solid line, hχp σ11 i; dashed line, hχp σ22 i; dash-dot line, hχp σ33 i. 182 (a) 0.15 0.1 χN1 / f h D 0.05 0 -0.05 0 0.2 0.4 0.6 0.8 1 y/h (b) 0 χN2 / f h D -0.2 -0.4 0 0.2 0.4 0.6 0.8 1 y/h Figure 4.13: The (a) first and (b) second normal stress differences normalized by f D h. Solid line, P2L; dashed line, P3L; dash-dot line, P4L. the wall. The particle stresses are normalized by f D h. These show that the normal stresses are relatively uniform in the intermediate region. Now, it is generally accepted that the migration of the suspended particles in Poiseuille flows is induced by the gradient of the particle normal stresses, ∇ · (φL hσ p i) [159]. The uniform particle normal stresses hχp σii i in the intermediate region indicates that indeed the suspension in the region has reached an equilibrium state in an average sense. Consistent with the velocity statistics, the particle stresses show anomalous behaviour near the wall and in the core of the channel. One of the important rheological parameters in suspension flows are the first and second normal stress differences, which are defined as p p N1 = σ11 − σ22 , (4.18) p p N2 = σ22 − σ33 . (4.19) Figure 4.13 shows the normal stress differences normalized by f D h. For suspensions in a homogeneous linear shear flow, it was shown that all of the normal stress components 183 are negative and |σ11 | ≥ |σ22 | ≥ |σ33 | [240]. Both in experiments [241] and in numerical simulations [197, 230], it has been observed that both N1 and N2 for the suspensions of non- colloidal particles are negative. However, in figure 4.12, it is found that, in the intermediate region, N1 is negative only for Φ = 0.20 and becomes positive for Φ ≥ 0.30. A similar feature of the normal stress differences has been observed in the dissipative particle dynamics simulations of colloidal suspensions in a parabolic flow by Pan et al. [168]. They showed that the first normal stress difference normalized by the local shear rate is positive for some range of y. However, their simulations are for Brownian suspensions in a reverse Poiseuille flow in a periodic domain [9], i.e. without no-slip wall boundaries, and data are presented in terms of a local P´eclet number, so it is difficult to make a direct comparison. It is important to note that the normal stress difference is relatively sensitive to the details of the near- contact force model, in contrast to the effective viscosity or particle pressure. For example, in the experiments of Zarraga et al. [241], the ratio of the normal stress differences is shown to be |N2 |/|N1 | ' 3.6, while |N1 | ' |N2 | in the numerical simulations [197, 230]. Sierou & Brady [197] have shown that introducing a frictional contact force between the suspended particles increases |N2 |. Still, it is not clear how to incorporate these non-hydrodynamic effects into numerical simulations. These issues remain to be explored further. The particle pressure of suspension flows can be defined as 1 P P P Π = − (σ11 + σ22 + σ33 ). (4.20) 3 In the modeling of suspension flows, it is assumed that the particle-phase contribution to the total pressure, φL Π ∼ µf (φL )γ˙ L . Here, φL is the local volume fraction, f (φL ) is an empirical relation and γ˙ L is the local shear rate. In other words, φL Π normalized by the local shear stress µγ˙ L is a function of the local volume fraction only. In figure 4.14 (a), the particle-phase contribution to the total stress normalized by the local shear stress is shown as a function of φ. As shown in figure 4.4, φL is almost the same as φ except near the wall. Hence, hereafter we do not distinguish φ from φL . The local shear rate is estimated by ³ y ´ uc γ˙ L = 2 1 − . (4.21) h h We find that there is a remarkable similarity in the normalized particle pressure profiles. Except for the data near the core of the channel, which diverges as γ˙ L → 0, the normalized 184 (a) 102 1 10 χΠ / µ γL 100 10-1 -2 10 0.2 0.4 0.6 φ (b) 101 χΠ / µn γL 0 10 10-1 0.2 0.4 0.6 φ (c) 10-1 ~ Φ2.38 Ψ 10-2 10-3 0.2 0.4 0.6 0.8 Φ Figure 4.14: The average particle pressure normalized by the local shear rate γ˙ L (a) and the normal viscosity model by Morris & Boulay [160] (b): ¤, P2L; 4, P3L; ◦, P4L. (c) The volume average of the particle pressure scaled by the apparent viscosity as a function of the bulk volume fraction. The dashed line is Ψ = e−0.8 Φ2.38 . 185 pressure profiles in the intermediate region almost collapse onto one curve, suggesting that the particle pressure is indeed a function of the local volume fraction and γ˙ L only. Morris & Boulay [160] proposed an empirical relation for the particle pressure as µ ¶ 1 + λ2 + λ3 φΠ = µn (φ) γ˙ L , (4.22) 3 µ ¶ µ ¶ φ 2 φ −2 µn (φ) = µKn 1− . (4.23) φm φm Here, Kn is an empirical coefficient, φm is the maximum random packing fraction, λ2 = p p p p σ22 /σ11 , and λ2 = σ33 /σ11 . In Miller & Morris [155], they suggested that λ2 ' 0.8, λ3 ' 0.5, Kn = 0.75 and φm = 0.68 would give a good agreement with suspension rheology. In figure 4.14 (b), the particle pressure scaled by the empirical relation of [160] is shown in terms of φ; hχp Πi hχp Πi = ³ ´2 ³ ´−2 . (4.24) µn γ˙ L µ φφm 1 − φφm γ˙ L Since the parameters λ2 and λ3 suggested by [155] are different from the simulation results (figure 4.14 a), here we use a rheological fitting parameter Kp = (1 + λ2 + λ3 )Kn /3, instead of dealing with each normal stress component separately. It is shown that the functional form of their normal stress viscosity model µn shows an excellent agreement with the present simulation results. The model parameter Kp = 0.63 gives a good approximation for the range of 0.25 ≤ φ ≤ 0.55. The volume average of particle pressure over ΩD = Hx ×Hy ×Hz scaled by the apparent viscosity is shown in figure 4.14 (c) as a function of the bulk volume fraction. The scaled particle pressure is defined as µ hΠi Ψ= . (4.25) µapp f Dh Here, the volume averaged particle pressure is Z * Np + 1 3 1 X i hΠi = D hχp Πi d x = Φ Π , (4.26) |Ω | ΩD Np i=1 in which Φ is the bulk volume fraction. The scaled particle pressure is shown in figure 4.14 186 (a) 1 σ12tot = 1 - y/h χσ12 / f h D 0.5 P 0 0 0.2 0.4 0.6 0.8 1 y/h (b) 20 15 10 µeff / µ 5 0.1 0.2 0.3 0.4 0.5 0.6 φ Figure 4.15: (a) The particle shear stress, χp σ12 , normalized by f D h and shown with cor- responding fitting curves. The long-dashed line represents the total shear stress normalized by f D h. From top to bottom, the lines indicate P4L, P3L, and P2L. (b) The effective viscosity as a function of the local volume fraction: ¤, P2L; 4, P3L; ◦, P4L. Solid line is an empirical relation by Krieger & Dogherty [115] and dashed line is Eilers’ fit. Table 4.4: The fitting coefficients for the particle-phase contribution to the total stress hχp σ12 i. P2L P3L P4L C0 0.41 0.63 0.80 C1 -0.37 -0.63 -0.78 (c). This shows that, with the limited data available, the particle pressure scaled by the apparent viscosity is well approximated by Ψ = e−0.8 Φ2.38 . 4.4.2 Shear stresses The shear component of the particle stress tensor hχp σ12 i normalized by f D h is shown in figure 4.15 (a). As with the velocity statistics, there is an intermediate region in which χp σ12 is a linear function of y/h. The fitting coefficients are given in table 4.4. As σ12 is proportional to γ˙ (σ12 ∼ γ), ˙ in a linear shear flow, it is somewhat surprising to observe that p σ12 becomes negative in the core of the channel, where γ˙ L is positive. Recently, Guasto et 187 al. [83] performed experiments of concentrated suspensions under an oscillating pressure gradient for the bulk volume fraction Φ = 0.40. They found that there is a long-range correlated motion for the particles near the core of the channel. A similar long-range correlation has been seen in the present simulations. The magnitude of the negative values p p of σ12 increase with the volume fraction, indicating that such negative values of σ12 are a distinguishing feature of concentrated suspensions. In a Poiseuille flow, the total shear stress normalized by f D h is linear in y/h, ³ y´ D hσ12 i = hχp σ12 i + hχf σ12 i = 1 − f h, (4.27) h in which χf is the fluid-phase indicator function. Analogous to Couette-flow suspensions, we define an effective viscosity as, µef f hσ12 i hχp σ12 i = =1+ . (4.28) µ0 hχf σ12 i (1 − y/h)f D h − hχp σ12 i The effective viscosity is compared with empirical viscosity relations in figure 4.15 (b). The empirical viscosity relations are: 1. Krieger & Dogherty [115], µ ¶ µef f φ −[η]φm = 1− , (4.29) µ0 φm 2. Eilers’ fit [203], à !2 1 µef f 2 [η]φ = 1+ , (4.30) µ0 1 − φ/φm in which φm is the maximum packing fraction and [η] is a fitting parameter. Following Stickel & Powell [203], φm = 0.63 and η = 2.5 are used. It is shown that the effective viscosity evaluated in the intermediate region agrees well with these empirical relations. p As a first step to investigate the origin of the negative σ12 in the core, the particle stress is decomposed into the hydrodynamic stresslet and the interparticle force contributions. As the stresslet is also related with the interparticle force, it is not possible to exactly decouple the interparticle force contribution from the hydrodynamic contribution. Still, 188 0.8 0.4 χσ12 / f h D 0 -0.4 0 0.2 0.4 0.6 0.8 1 y/h Figure 4.16: Decomposition of hχp σ12 i into the contributions from the hydrodynamic stresslet (solid line) and the interparticle force (dashed line) for P4L. All the variables are normalized by f D h. the simple decomposition is helpful to investigate the behaviour qualitatively. Figure 4.16 shows the stresslet and interparticle force contributions for P4L. Here, the interparticle force contribution is uniform except near the wall and in the core of the channel. The interparticle p force contribution is always positive. The negative σ12 is due to the hydrodynamic stresslet. 4.4.3 Pair-distribution function The pair-distribution functions g(x, y) projected onto the velocity-velocity-gradient (x − y) plane are shown in figure 4.17. The pair-distribution functions are obtained at Φ = 0.40 (P4L). The pair-distribution function obtained for 7 ≤ y/a ≤ 13 is very similar to that in a homogeneous linear shear flow [221, 230]. There are two shells of high probability region around the compressional axis of the shear flow at r/a ' 2, which is followed by the second high probability region around r/a ' 3.5. The second high probability shell shows the similar behaviour with the first shell, higher probability around the compressional axis and lower probability near the extensional axis. Since the volume fraction is an increasing function of y in this region, the contour plot shows a background gradation in color in the wall normal direction, indicating the higher probability of a particle encounter at the higher y/a. The pair distribution function near the core of the channel shows a very different be- haviour from that in a linear shear flow. In the upper hemisphere, there are three shells 189 (a) (b) 0.8 0.88 0.96 1.04 1.12 1.2 1.28 1.36 1.44 1.52 1.6 1 1.3 1.6 1.9 2.2 2.5 2.8 3.1 3.4 3.7 4 4 4 2 2 y/a y/a 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 x/a x/a Figure 4.17: The pair-distribution functions for the reference particles in (a) 7 ≤ y/a ≤ 13 and(b) 18 ≤ y/a ≤ 19. The lighter the contour, the higher the probability. of the high probability regions and the probability of a particle encounter in each shells is relatively isotropic. Near θ = 5/4π, in which θ is the azimuthal angle measured from the positive x−axis, the probability in the first shell becomes relatively low, suggesting that, in the lower hemisphere, the micro-structure still resembles that in a linear shear flow. How- ever, such a low probability region is not observed in the first high probability shell in the upper hemisphere. Unlike the pair-distribution in 7 ≤ y/a ≤ 13 , the highest probabilities at r/a ' 2 are observed around θ ' 0, π/2, and 3π/2. A similar behaviour has been observed by Gao et al. [77] in their experiments of colloidal suspensions in a pressure-driven flow. It is interesting that the pair-distribution function qualitatively agrees well, even though Brownian motion is not considered in the present simulations. 4.5 Discussions In the preceding sections, we have summarized the results obtained from the numerical simulations of concentrated suspensions in a plane Poiseuille flow. In this study, we focus on suspensions in narrow channels (Hy /a ≤ 40). These include results for the nonuni- form distribution of particles across the channel, the particle velocity statistics and particle stresses. The volume fraction profiles calculated from the force-coupling simulation at the bulk volume fractions Φ = 0.3 and 0.4 show a quantitatively good agreement with the experi- 190 ments by [139] near the core of the channel under experimental conditions similar to the simulation parameters in the present study. However, near the wall, the volume fraction profiles in [139] show a much faster decrease than the present simulations. On the other hand, the numerically simulated volume fraction profiles near the wall match the experi- ments by Hampton et al. [84] in pressure-driven tube flows. This discrepancy may come from the differences in the experimental setup and measurement techniques. In the laser Doppler velocimetry measurements of Lyon & Leal [139], they first measured the parti- cle velocity and the local volume fraction is then estimated indirectly from the velocity measurement data based on the time interval between particles arriving in the measuring volume. The simulated velocity profiles show a good agreement with [139] across the whole channel, suggesting that there might be some measurement error or statistical bias involved in estimating the local volume fraction. The authors also noted the limitations of the mea- surement technique near the wall. Using nuclear magnetic resonance imaging, Hampton et al. [84] were able to measure directly the local volume fraction. In their experiments with large monodisperse particles (2a ' 3000µm), they also observed the formation of a coherent particle layer near the wall similar to that seen in the present simulations. The results for the velocity fluctuations indicate that the channel can be divided into three regions; near-wall, intermediate and core region. In the near-wall region, the particle- wall lubrication interaction and the finite size effects become important and the behaviour in the region is very different from what is expected from the homogeneous suspensions. In the intermediate region, it is found that the streamwise velocity fluctuation is well approximated by a quadratic function of y/h, while the wall-normal and spanwise velocity fluctuations show nearly linear dependencies on y/h. It is shown that both the spanwise angular velocity and the wall-normal velocity gradient are linear functions of y/h in the intermediate region, which suggest that the wall-normal and spanwise velocity fluctuations are associated with the local shear rate. In homogeneous suspensions, the squared wall-normal and spanwise velocity fluctuations, which are linked to “suspension temperature”, are shown to be ∼ γΦ ˙ and hvy2 i/hvz2 i ' 4 for a wide range of Φ [59]. Similarly, the ratio of the vertical to the spanwise velocity fluctuations in Poiseuille flow suspensions are vy0 /vz0 ' 1.8 for the range of volume fraction Φ = 0.2 ∼ 0.4 in the intermediate region. However, the linear behaviours of wall-normal and spanwise velocity fluctuations (see table 4.3) scale well with the local shear gradient and the bulk volume fraction Φ, not the local volume fraction. As the 191 volume fraction increases, the intermediate region, where the velocity statistics can be approximated by a linear or quadratic function, decreases and the core region becomes wider. In the core region, the velocity fluctuations become almost uniform. While the streamwise and spanwise velocity fluctuations are shown to be increasing functions of the volume fraction, the wall-normal velocity fluctuations remain almost constant in the range of volume fraction considered in the present study. Although the local shear rate is zero at the center of the channel, the velocity fluctuations remain finite, suggesting a non-local structure contributes to the dynamics in the region. The behaviour of the normal particle stresses is very different from what is expected based on the results for homogeneous suspensions in a linear shear flow. The normal stress differences are not only functions of the distance from the wall, but also functions of the bulk volume fraction. At the bulk volume fraction Φ = 0.20, both the first and second nor- mal stresses differences are negative, as in a homogeneous suspensions. However, at higher volume fractions Φ ≥ 0.30, the first normal stress difference becomes positive while the second normal stress difference remains negative. A similar behaviour has been observed in the dissipative particle dynamics simulations by Pan et al. [168]. The contribution of the particle pressure to the total pressure scaled by the local shear rate of the suspension flow is shown to be a function of the local volume fraction, as suggested in some suspension mod- eling approaches. The particle pressure contribution is well represented by the functional form of the normal stress model given by Morris & Boulay [160]. The effective viscosity is estimated by the ratio of the total shear stress to the fluid- phase shear stress. It is shown that the effective viscosity, at least in the intermediate region, shows a good agreement with empirical relations commonly used in Couette-flow suspensions. Unexpectedly, the particle shear stress becomes negative in the core region of the channel, in which the local strain rate is small but still positive. It is shown that the negative shear stress arises from hydrodynamic stresslet. The origin of this anomalous behaviour is a subject of a further investigation. Chapter 5 Finite-Reynolds-number flow: Hydrodynamic interaction between spinning particles 5.1 Introduction Dynamic self-assembly of discrete particles, which occurs by the subtle balance between a driving force and dissipation, has been attracted a great attention due to its potential application for a smart material [71]. This study is motivated by the experiments of dy- namic self-assembly of magnetic particles reported by Grzybowski et al. [81, 82]. In their experiment, millimeter-size disks with a permanent dipole moment coplanar with the disk are placed just beneath the liquid-air interface. As a bar magnet rotates at a constant angular frequency in the plane parallel to the liquid surface, a magnetic torque is applied to the disks, which makes the disks rotate about their centers at the same angular frequency with the bar magnet, and, at the same time, a gradient of the magnetic force results in a centripetal force toward the axis of the rotation of the bar magnet. The balance between the body force/torque on the disks induced by the rotating magnetic field and hydrodynamic interactions due to the vortex-like flow generated by the spinning of the disks drives the formulation of supraparticle aggregates. The goal of this chapter is to investigate the hydrodynamic interactions in a system analogous to the experiments of Grzybowski et al. [81, 82]. In the present study, instead 192 193 of the disks beneath the liquid-air interface, hydrodynamic interaction between spheres completely immersed in a liquid and initially seeded in a coplanar configuration is studied. The similarity in the hydrodynamic interactions between two different situations will be briefly discussed in 5.2. The external body force and torque on the spheres by the rotating magnetic field are, respectively, modeled as a constant body force towards the center of the computational domain and a constant torque in the axis perpendicular to the plane of the spheres (vertical direction). Here, we focus particularly on the hydrodynamic interaction between spinning particles under finite fluid inertia. The force-coupling method for the Navier-Stokes equations described in 2.4 is employed for the numerical simulations. A cubic tri-periodic domain is used and the Navier-Stokes equations are solved by using a Fourier spectral method. The length of the each directions of the computational domain is chosen as 2L = 24a or 48a, in which a is the particle radius. We found that the computational domain is big enough that the effect of the periodicity on hydrodynamics is negligible. 5.2 Results First, we study the flow field induced by the rotation of a sphere at finite Reynolds number. The Reynolds number is based on the particle radius (a) and angular velocity (ω), Reω = ωa2 /ν, in which ν is the kinematic viscosity of the fluid. Figure 5.1 shows the azimuthal velocity uφ as a function of the radial distance (r) in the equatorial plane (θ = π/2) specified in terms of the spherical polar coordinate (r, θ, φ). Comparing the analytical solution and FCM solution in Stokes flow, it is shown that FCM reproduces the analytical solution exactly in the far field r/a > 1.3. Near the particle surface 1 ≤ r/a < 1.3, FCM slightly underestimates uφ , which is a result of the regularized multipole. It is shown that the azimuthal velocity at Reω = 8 is very similar to that in Stokes flow. In this study, the rotating magnetic field is represented by a body torque applied on the suspended particles. From the estimate in the Stokes limit, the body torque is initially set to T ext = 8πµa3 ω0 and µ is gradually adjusted to achieve the desired Reω . Here, ω0 is an arbitrary reference value. The relation between T ext and ω has been studied thoroughly in the literature. Let us define a non-dimensional torque coefficient as M = 2T ext /ρa5 ω 2 . In the Stokes limit, Lamb has shown that M = 16π/Reω [122]. Sawatzki [189] showed in his experiment that the Stokes-limit theory by Lamb is valid for Reω < 3. Dennis et al. 194 1 Exact + FCM - Stokes + + Reω = 8 + 0.5 + + uφ /aω ++ ++++ 0+ + ++ ++++ ++ + + + -0.5 + + -1 + -4 -2 0 2 4 r/a Figure 5.1: The azimuthal velocity in the equatorial plane normalized by the particle radius a and the angular velocity ω. The solid line is the analytical solution at zero Reynolds number. 195 3 10 FCM - T FCM - TS Lamb (1932) 2 10 Dennis (1980) M 1 10 0 10 -1 10 -1 0 1 2 10 10 10 10 Reω Figure 5.2: The non-dimensional torque coefficient M as a function of the Reynolds number Reω . The solid and hollow symbols are, respectively, FCM simulations without and with the stresslet. 196 [53] investigated the same system numerically and confirmed the experiments of Sawatzki. At finite Reω , the M − Reω relation is given by M = 16π(1 + f (Reω ))/Reω , in which f (Reω ) is an empirical correlation. In figure 5.2, the FCM results are shown together with the Lamb’s Stokes-limit theory [122] and the empirical correlation of Dennis et al. [53]. First, the FCM simulations are performed without considering the stresslet (FCM-T), i.e. R without the rigidity constraint Eij ∆D dx = 0. In Stokes flow, the flow field induced by the rotation of a sphere is purely due to the couplet and the stresslet is zero. Hence, at the smallest Reynolds number Reω = 2, M obtained by FCM-T agrees well with the Lamb’s theory. However, for larger Reω , FCM-T overestimates the torque coefficient. Once the stresslet is added (FCM-TS), the torque coefficient obtained by FCM-TS shows a very good agreement for Reω < 10. At Reω = 16, FCM-TS somehow underestimates M compared to the empirical correlation [53]. Rotation of a sphere in an otherwise quiescent flow induces a secondary meridional circulation with an inflow at the poles and outflow on the equator. Figure 5.3 shows the secondary flow induced by the rotation of a sphere at Reω = 2. The flow inside of the sphere is shown to illustrate the internal circulation and the location of the stagnation points. At low Reω , the flow velocity around a sphere can be estimated analytically by using a regular perturbation method. Bickley [17] has found the solution to O(Reω ), ωa3 ³ a ´2 2 ur = − (3 cos θ − 1) 1 − Reω , 8r2 r ωa4 ³ a´ uθ = 1 − sin θ cos θReω , (5.1) 4r3 r ωa3 uφ = sin θ + O(Re2ω ). r2 The analytical solution illustrates that the radial-direction component of the stresslet is non- zero, which explains the error in the prediction of M − Reω relation by FCM-T simulations (figure 5.2). Figure 5.4 shows the flow field induced by two co-rotating coplanar spheres. The spheres are subject to a fixed attraction force which keeps them at a constant separation distance. Here, the center-to-center distance is kept 2R/a = 6. In Stokes flows, a pair of co-rotating coplanar spheres will follow a circular path moving in response to the flow generated by the other particle and there is no net hydrodynamic force in the direction connecting the centers of the spheres. However, at finite Reynolds number, there is a net repulsive hydrodynamic 197 4 2 z/a 0 -2 -4 -4 -2 0 2 4 x/a Figure 5.3: Secondary flow in the x − z plane for a sphere spinning about the z axis at Reω = 2. 198 8 2 0 4 -2 -4 -2 z/a 0 -4 -8 -8 -4 0 4 8 x/a Figure 5.4: Flow field in the x − z plane for a pair of co-rotating spheres at Reω = 2. The spheres are rotating about the z axis. Inset shows the detailed velocity field for the sphere located at x/a = −3. 199 0 10 Reω = 0.25 Reω = 2 + Reω = 8 + 2 + F/ρa ω 4 10-1 + + 10-2 2 4 6 8 10 R/a Figure 5.5: Hydrodynamic repulsion force between two co-rotating spheres as a function of the distance between the particles 2R. force between the spheres due to the outflow near the equator. It is worth noting that, even in the case of counter-rotating spheres, the hydrodynamic interaction between coplanar spheres will be repulsive. The experiments of Grzybowski et al. [81, 82] used the disks with a permanent dipole, not spheres in the present simulations. It is well known that, when a disk rotates under finite fluid inertia, a secondary flow is generated by the nonlinear interaction, which is the celebrated “von K´arm´an viscous pump” problem. Similar to the rotating sphere, an inflow towards the rotating disk in the direction perpendicular to the disk is generated and, near the disk surface, there is an outflow (positive ur ) parallel to the disk surface. Hence, the hydrodynamic repulsion they observed in the experiments also comes from the secondary flow. Although we use spheres instead of disks, the hydrodynamics involved is very similar in essence. 200 (a) 10 0 (b) 10 0 Stokes Re = 0.25 Re = 0.25 Re = 2 -1 -1 + Re = 8 10 10 Ω/ω + Ω/ω + 10-2 10 -2 + 10 -3 -3 + 1 2 3 4 5 10 1 2 3 4 5 R/a R/a Figure 5.6: Precession angular velocity Ω for a pair of co-rotating spheres in terms of the separation distance R: (a) comparison of the FCM results at Reω = 0.25 with the Stokes- limit estimate and (b) FCM results for Reω = 0.25, 2, and 8. The lines in (b) are the fitting curves. The hydrodynamic repulsion force F between the spheres is estimated from the force required to keep R constant. The two spheres are initially seeded at the desired separation distance and the attraction force (G = −F ) is adjusted by using a penalty method; dG = −λ(R − Rref ), (5.2) dt in which λ is a penalty coefficient and Rref is the target separation distance. The hydrody- namic repulsion force is shown in figure 5.5. The repulsion force shows a power-law decay ∼ (R/a)−κ . It is shown that the exponent is a decreasing function of Reω ; κ = 2.64, 2.42, and 2.05 for Reω = 0.25, 2, and 8, respectively. In Stokes flow, the precession angular velocity Ω of the co-rotating particle pair can be predicted analytically by a superposition of the velocity perturbations of each particle. The azimuthal velocity of a particle induced by the rotation of a second particle is u0φ = ωa3 /(2R)2 . Then, the angular velocity of the two particle system is uφ ω ³ a ´3 Ω= = . (5.3) R 4 R In figure 5.6 (a), Ω for Reω = 0.25 are shown together with the theoretical prediction in Stokes flow. It is shown that the simulation results agree well with the theoretical prediction. The precession angular velocity as a function of both R and Reω is shown in figure 5.6 (b). Similarly to F , the precession angular velocity shows a ∼ (R/a)−ξ behavior 201 0.01 Analytical + Re = 0.25 Re = 2 Re = 8 ur / (aω Re) 0.005 + + + + + + + + + + + + + + + + 0 1 3 5 7 9 r/a Figure 5.7: Comparison of the radial velocity ur obtained from FCM-TS with the analytical solution by Bickley [17]. with the exponent different from that for F (κ). Here, the exponents are ξ = 3.21, 3.22, and 3.69 for Reω = 0.25, 2, and 8, respectively. We now consider the quantitative difference between the analytical and FCM solutions. Figure 5.7 shows the radial velocity obtained by FCM and the analytical prediction of Bickley [17]. The FCM solutions for Reω = 0.25 and 2 collapse onto one curve while that of Reω = 8 is somehow smaller. It is shown that the FCM solutions show qualitatively similar behavior with Bickley [17]. However, the magnitude of ur is much smaller than the analytical result. The flow field generated by the rotation of a sphere in Stokes flow is given by u = (ω × x)f (r). (5.4) The nonlinear term for the perturbation solution of order Reω is obtained from the Stokes 202 1.2 1 α2(r/a) 0.8 0.6 1 2 3 4 5 r/a Figure 5.8: Ratio of the magnitude of the non-linear interaction of FCM-TS to that of the analytical Stokes solution. 203 solution as µ ¶ xk xl df (u · ∇)ui = ²ijk ²lpq ωj ωp xq f (r) δkl f (r) + r dr 2 2 = −(|ω| δij − ωi ωj )xj f (r). (5.5) The scalar function f (r) for Stokes flow is f S (r) = (a/r)3 and that of FCM f F CM is " µ ¶ µ ¶1/2 µ ¶# r 2 2 r2 f F CM (r) = f S (r) Erf √ − exp − 2 = f S (r)α(r). (5.6) σD 2 σD π 2σD The ratio of the nonlinear term in the first-order perturbation solution of FCM to Stokes solution is simply α2 (r). The behavior of α2 is shown in figure 5.8. As FCM resolves the far-field solution almost exactly, α for r/a > 2 becomes almost 1. On the other hand, near the particle surface, FCM underestimates the nonlinear term as low as 65% of the exact solution. To show the effect of the difference in the nonlinear term more clearly, FCM for Stokes flow is solved using the nonlinear term from the analytical solution as an external force. That is 1 2 [−(u · ∇)u]Stokes = −∇p + ∇ u + (G · ∇)∆D . (5.7) Re To minimize the effects of the periodicity, we use a larger computation domain; L/a = 48. The radial velocity obtained by solving (5.7) is shown in figure 5.9. It is shown that the agreement between the FCM solution with a nonlinear-correction and the analytical solution is very good. It should be noted that this system is one of the worst cases for FCM to simulate. The secondary flow is very sensitive to the no-slip surface representation. While, due to the use of a regularized multipole, the no-slip particle surface is approximated by a smooth profile, which results in the underestimation of the gradient of the flow velocity near the particle surface. This numerical test suggests that the nonlinear interaction can be computed correctly by adding a correction force. 204 0.01 a/L = 48 Analytical a/L = 48 (Modified) ur / aω Re 0.005 0 10 20 30 40 r/a Figure 5.9: Comparison of the radial velocity ur obtained from FCM-TS with a nonlinear- interaction-correction with the analytical solution by Bickley [17]. 205 5.3 Summary In this chapter, the hydrodynamic interaction between spheres rotating under an external body torque at small but finite Reynolds number. It is shown that a secondary meridional circulation develops around a rotating particle by the nonlinear interaction and, due to the secondary flow, there is a net repulsive hydrodynamic force between a pair of rotating, coplanar sphere. This result suggests that the dynamic self-assembly observed in the ex- periments [81, 82] is a result of the balance between the magnetic attraction force and the hydrodynamic repulsion force. It is found that FCM underestimates the magnitude of the secondary flow, which is related with the smooth particle surface representation. It is shown that, by adding a simple correction force, the secondary flow can be reproduced correctly in FCM. A more general and robust correction scheme is a subject of further investigation. Chapter 6 Finite-Reynolds-number flow: Concentrated suspensions in a linear shear flow 6.1 Introduction Most studies on the suspension flows, particularly for moderate to high volume fractions, have been focused on the Stokes limit, i.e. at zero Reynolds number. However, even under very small fluid inertia, e.g. Reynolds number of O(10−2 ) ∼ O(10−1 ), it has been shown that suspension dynamics becomes very different from that in the Stokes limit [33, 39, 85, 118, 193]. The effects of the fluid inertia on suspension dynamics and rheology is important in many biological and engineering flows. For example, while the bulk Reynolds number of blood flow in arteries is O(102 ), the particle-scale Reynolds number remains O(0.1) ∼ O(1). However, still it is not clearly understood how mechanical properties of the bulk fluid are affected by inertia. Since the pioneering studies by Schowalter [130] and Acrivos [114, 173], the dynamics of a single sphere in a shear flow under finite fluid inertia has been studied extensively [11, 19, 154, 171, 204]. Compared to the suspension dynamics in the dilute limit, i.e. without considering hydrodynamic interaction between the suspended particles, there are only a limited number of studies about the Reynolds-number effects on semi-dilute to concentrated suspensions in a linear shear flow. One of the main reasons behind the slow progress is 206 207 that numerical simulation of suspension flows is computationally challenging because the numerical method should be able to resolve the small gap between suspended particles, which is in many cases as small as O(10−3 ) ∼ O(10−2 ) of the particle radius, as well as long-range multibody hydrodynamic interactions on the lengthscale of O(1) ∼ O(10) of the particle radius. Nguyen & Ladd [162] developed a lattice-Boltzmann method (LBM) supplemented with a near-field correction scheme, which facilitated the study of finite- inertia suspensions. Yan et al. [226] and, subsequently, Kulkarni & Morris [118] have studied hydrodynamic interactions between two spherical particles in a wall-bounded linear shear flow from LB simulations. In particular, Kulkarni & Morris [118] have shown that the topology of the relative trajectory changes from an open or closed trajectory in Stokes flow to an open, reversing, or spiralling trajectory under finite fluid inertia. In their two- dimensional LB simulations of cylinders suspended in a wall-bounded linear shear flow, Kromkamp et al. [117] investigated the Reynolds number effects, particularly, focused on the self-diffusion of the cylinders. Verberg & Koch [215] performed three-dimensional LB simulations of suspensions of inertial particles in a wall-bounded linear shear flow to study the effects of both fluid and particle inertia on suspension rheology. In the LB simulations of suspensions of neutrally buoyant particles in a wall-bounded linear shear flow, Kulkarni & Morris [119] have reported the changes in the particle stresses and microstructures under finite fluid inertia. As shown above, most of the studies using LBM have been performed in a wall-bounded linear shear flow. However, Yeo & Maxey [230, 229] have shown that the dynamics of wall-bounded suspensions becomes very complicated due to the coherent structures induced by the wall. To identify the finite-Reynolds-number effect on suspension dynamics clearly, it is necessary to investigate suspensions in a homogeneous linear shear flow. In the present study, we report the changes in rheology and diffusion of concentrated suspensions of neutrally buoyant spheres in a homogeneous linear shear flow under finite fluid inertia. The force-coupling method with a lubrication correction scheme is employed for the numerical simulations of concentrated suspensions in a homogeneous linear shear flow. The details of the simulation parameters are given in 6.2. In 6.3, the effects of fluid inertia on the velocity statistics and diffusivity are reported. The changes in suspension rheology under finite fluid inertia are shown in 6.4 and the microstructures are analyzed in 6.5. 208 6.2 Simulation parameters The force-coupling simulations for mono-disperse suspensions at finite Reynolds numbers are performed in a tri-periodic cubic domain. A Fourier spectral method is used to solve the Navier-Stokes equations. The computational mesh is deformed with the shear flow to satisfy the periodic boundary condition. Details of the numerical method are given in 2.4 and 2.5.1. The size of the computational domain is fixed, (Hx × Hy × Hz ) = (2π × 2π × 2π), and the number of Fourier modes used in the simulations is (Nx × Ny × Nz ) = (64 × 64 × 64). Here (x, y, z), or (x1 , x2 , x3 ), denotes the flow, velocity gradient, and vorticity directions. The radius of the suspended particles is a = 0.3. The numbers of the suspended particles for each volume fractions (φ, Np ) are (0.2, 438), (0.3, 657), and (0.4, 876). The time step ˙ = 1 × 10−3 for Reγ˙ = 0.005 to γδt size is changed from γδt ˙ = 5 × 10−5 for Reγ˙ = 2, in ˙ 2 ργa which γ˙ is the shear rate and Reγ˙ = µ0 is the particle Reynolds number. The initial particle configurations are obtained from molecular dynamics simulations. The systems are sheared for 50 ∼ 100 γt ˙ to reach a stationary state. Then, the simulations are performed for T ∗ = 500 ∼ 1000γt ˙ after the system reaches a stationary state to gather the statistics. Since the system is ergodic, the time average is used instead of an ensemble average. As a model for non-hydrodynamic interaction between particles, an inter-particle po- tential force is used. The potential force to particle β from particle α is given as  µ 2 ¶6   2 Rref −|r|2 r −6πµ0 γa ˙ Fref 2 2 |r| if |r| < Rref Rref −4a FPαβ = (6.1)   0 otherwise, in which r = Y α − Y β , Rref is the cut-off distance, and Fref is a constant. To minimize the effects of the potential force on the suspension dynamics, the force parameters are chosen to make the potential force behave close to the hard-sphere potential; Rref = 2.004a and Fref = 5 × 104 . The effects of the potential force on the suspension rheology are briefly discussed in 6.4. 209 (a) 0.4 V1 V2 V3 0.3 r.m.s 0.2 0.1 -3 -2 -1 0 10 10 10 10 Reγ (b) 0.5 0.4 r.m.s 0.3 0.2 10-3 10-2 10-1 100 Reγ (c) 0.6 0.5 r.m.s 0.4 0.3 10-3 10-2 10-1 100 Reγ Figure 6.1: Root-mean-square velocity fluctuation normalized by aγ; ˙ (a) φ = 0.20, (b) φ = 0.30, and (c) φ = 0.40. 210 6.3 Velocity statistics Figure 6.1 shows root-mean-square (r.m.s) velocity fluctuations normalized by the shear rate and the particle radius (γa). ˙ Root-mean-square velocity fluctuation is computed by · Z T ¸1/2 1 2 vi0 = (Vi (s) − γY ˙ 2 (s)δi1 ) ds , (6.2) T 0 in which V is the particle velocity, Y is the position of the particle, and the overline indicates the average over the suspended particles. It is shown that v 0 does not change significantly when Reγ˙ < 0.1. Consistent with the Stokesian Dynamics simulations of Drazer et al. [59] at zero Reynolds number, v20 is the largest and v20 > v10 > v30 for small Reγ˙ . For Reγ˙ > 0.1, v20 and v30 are decreasing functions of the Reynolds number, while v10 slightly increases when Reγ˙ > 0.5. As the volume fraction φ increases, the increase of v10 with Reγ˙ becomes less pronounced. The ratio of v10 (Reγ˙ = 2) to v10 (Reγ˙ = 0.005) changes from 0.087 to 0.051 and, then, to 0.009 for φ = 0.20, 0.30, and 0.40, respectively. Hence, the overall “suspension temperature” T sus = (v10 2 + v20 2 + v30 2 )/3 normalized by γa ˙ becomes a decreasing function of Reγ˙ . Similar decrease of T sus with increasing Reγ˙ has been observed by Verberg & Koch [215]. However, since they considered the wall-bounded suspensions at high Stokes numbers (high particle inertia), a quantitative comparison is not feasible. The auto-correlation function for the particle velocity is defined as hVi (0)Vi (τ )i ρVi (τ ) = , (6.3) vi0 2 in which τ is a time-lag. Figure 6.2 shows the auto-correlation functions for φ = 0.30. The auto-correlation function shows three distinctive regions. Near the origin γτ ˙ < 0.2, the auto-correlation functions for larger Reγ˙ decays faster. The effects of the finite inertia is more pronounced in the intermediate timescale, 0.5 < γτ ˙ < 4. In contrast to the early-time dynamics, the decay rate of the auto-correlation functions in the intermediate timescale is smaller at larger Reγ˙ and, hence, the negative loop of ρV becomes smaller. Finally, the auto-correlation functions for γτ ˙ > 4 collapse onto one curve. A Lagrangian integral timescale can be defined by Z ∞ TiL = ρVi (s)ds. (6.4) 0 211 (a) 1 Re = 0.005 Re = 0.5 Re = 1.0 Re = 2.0 0.5 ρVy (τ) 0 0 2 4 6 τ (b) 1 0.5 ρVz (τ) 0 0 2 4 6 τ Figure 6.2: Auto-correlation functions for the (a) velocity-gradient- and (b) vorticity- direction velocities for φ = 0.30. The time-lag τ is normalized by the shear rate γ. ˙ 212 (a) 0.05 Dyy * 0 -2 -1 0 10 10 10 Reγ (b) 0.06 0.04 Dzz * 0.02 0 -2 -1 0 10 10 10 Reγ Figure 6.3: The self-diffusivities normalized by γa ˙ 2 in the (a) velocity-gradient and (b) vorticity directions as functions of Reγ˙ : ¥, φ = 0.20; N, φ = 0.30; •, φ = 0.40. 213 (a) Re = 0.005 0 10 Re = 0.5 Re = 1.0 ∆Y22 / a2 -1 10 -2 10 10-3 10-1 100 101 τ (b) 100 10-1 ∆Y23 / a2 10-2 10-3 10-1 100 101 τ Figure 6.4: Mean-square displacements normalized by the particle radius a2 for φ = 0.20 in the (a) velocity-gradient and (b) vorticity directions. The integral timescale is related to the self-diffusivity of the suspended particles as 2 Dii = vi0 TiL . (6.5) ∗ ) is shown in The self-diffusivity normalized by the shear-rate and the particle radius (Dii figure 6.3. For 0.2 ≤ φ ≤ 0.4, the diffusivity remains almost a constant for Reγ˙ < 0.1. The diffusivity for Reγ˙ < 0.1 is quantitatively consistent with the Stokesian Dynamics simula- tions of Sierou & Brady [198] at zero Reynolds number. Although the velocity fluctuation v 0 is attenuated at larger Reγ˙ , the diffusivity is shown to be an increasing function of Reγ˙ for Reγ˙ > 0.1. As shown in figure 6.2, the particle motion in finite-Reγ˙ flows has longer correlation than in Stokes flows, which contributes to the increase in the integral timescale and, consequently, increase in the diffusivity. Figure 6.4 shows the mean-square displacements (MSD) for φ = 0.20. The central limit theorem dictates that the mean-square displacement of the particles exhibiting stochas- 214 tic motion should be proportional to the travel time τ in the long-time limit (τ À T L ); h∆Yi2 (τ )i ∼ τ , in which ∆Yi (τ ) = Yi (τ ) − Yi (0). The diffusivity can be obtained from this relation as, 1 d ¯ 2 ¯ Dii = h∆Yi (τ )i¯ for T D À T L . (6.6) 2 dτ τ =T D The diffusivities computed from (6.5) and (6.6) should be the same. We estimated the diffusivities from those two relations and the differences between them were always less than 10%, which is due to the statistical noise. In the short-time limit τ ¿ T L , the particle motion is still strongly correlated to the initial velocity and, hence, MSD is proportional to τ 2 . In the τ 2 regime, MSD is smaller at larger Reγ˙ , which is consistent with the decrease of v 0 with increasing Reγ˙ . In the intermediate timescale γτ ˙ ∼ O(1), the growth-rate of MSD for Reγ˙ = 0.005 becomes smaller than 1; MSD ∼ τ ν and ν ' 0.5, which is related with the negative loop in the velocity auto- correlation. In figure 6.2, it is shown that the velocity auto-correlation functions decay more slowly at larger Reγ˙ . In agreement with the velocity auto-correlations, the range of the intermediate region decreases and the growth-rate ν in the region increases as Reγ˙ increases. Hence, although the initial MSD in the τ 2 regime for larger Reγ˙ is smaller, in the diffusive regime, MSD becomes an increasing function of Reγ˙ . The probability density functions (PDF) of the particle velocity P (Vi ) are shown in figure 6.5. The probability density functions are normalized by the corresponding standard deviations (vi0 ). Drazer et al. [58] have shown that P (Vy ) at zero Reynolds number changes from an exponential distribution P (V ) ∼ exp(−α|V |) at low φ to a Gaussian distribution P (V ) ∼ exp(−αV 2 ), in which α is a fitting coefficient. In figure 6.5 (a, c), P (Vy ) for φ = 0.20 shows a slower decay for |Vy /vy0 | > 3 compared to the Gaussian distribution, while P (Vy ) for φ = 0.40 is nearly Gaussian. The flatness factor of P (Vy ) for Reγ˙ = 0.005 changes from 3.17 at φ0.20 to 3.02 at φ = 0.40. For the particle velocity in the vorticity direction, the core of P (Vz ) (|Vz /vz0 | < 3) is well represented by the Gaussian distribution whereas large magnitude events (|Vz /vz0 | > 3) show a ∼ exp(−α|V |) behavior. Both in P (Vy ) and P (Vz ), the velocity PDFs becomes wider at larger Reγ˙ , indicating that the system becomes more intermittent at larger Reγ˙ . Using the lattice-Boltzmann simulations, Kulkarni & Morris [119] showed that the velocity PDF becomes narrower at larger Reγ˙ . 215 (a) 100 Re = 0.005 (b) 100 Re = 0.5 -1 Re = 1.0 -1 10 Re = 2.0 10 P(V*y) P(V*z) 10-2 10-2 -3 10 10-3 10-4 10-4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 * V V* (c) 100 (d) 10 0 10-1 10-1 P(V*y) P(V*z) 10-2 10-2 -3 10 10-3 10-4 10-4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 V* V* Figure 6.5: The probability density functions (PDF) of particle velocity for (a, b) φ = 0.20 and (c, d) φ = 0.40; (a,c) are PDFs of the particle velocity in the velocity-gradient direction and (b, d) are PDFs for the vorticity direction. The dashed line is the Normal distribution. Each pdf is normalized by its standard deviation. This contradictory result between the present simulations and [119] may come from the difference in normalization. In the present study, the particle velocity is normalized by the standard deviation to investigate the changes in the intermittency, while Kulkarni & Morris [119] showed PDFs for the particle velocity normalized by γa. ˙ 6.4 Particle stresses 6.4.1 Shear stress Following Batchelor [13], the bulk stress of homogeneous suspension flows is given by *Z + D E B p hσij i = −δij pdV + 2µ0 Eij + σij − ρhu0i u0j i, (6.7) ΩD \Ωp in which ΩD is the control volume, ΩP is the volume occupied by the particles, ρ is the density of the fluid, E B is the ensemble averaged strain-rate tensor of the bulk fluid, u0 is the fluctuating velocity of the bulk fluid, and σ p is the particle stress. In homogeneous B = γ/2(δ suspensions, the bulk strain rate is Eij ˙ i1 δj2 + δi2 δj1 ). The last term of (6.7) is 216 8 Re = 0.005 Re = 0.1 Re = 0.5 6 Re = 1.0 Re = 2.0 µeff 4 2 0 0 0.2 0.4 φ Figure 6.6: The changes in the effective viscosity with φ and Reγ˙ . The dashed line is the Eilers’ fit [203]. the Reynolds stress. Since we consider only small Reγ˙ , the Reynolds stress is not expected to contribute significantly to the total stress. Kulkarni & Morris [119] suggested that the Reynolds stress is negligibly small compared to the particle stress term when Reγ˙ is O(1) or smaller. The particle stress tensor is computed from hσ p i = n (hSi − hr ⊗ Fp i) . (6.8) Here, n is the number density, Sij is the stresslet, and Fp is the interparticle potential force. Here, the effective viscosity is computed from the bulk viscosity of the suspension flow, µef f hσ P i = 1 + 12 . (6.9) µ0 µ0 γ˙ 217 0 10 -1 10 ~Reγ σ12-1 10-2 * -3 10 -4 10 -2 -1 0 10 10 10 Reγ Figure 6.7: The normalized particle shear stress as a function of Reγ˙ . The particle stress is normalized by its value at Reγ˙ = 0.005. ¥, φ = 0.20; N, φ = 0.30; •, φ = 0.40. The Reynolds stress term is not considered in the present study. The effective viscosity is shown in figure 6.6 together with the Eilers’ fit [203], à !2 1 µef f 2 [η]φ = 1+ . (6.10) µ0 1 − φ/φm Here, the fitting parameters are chosen as η = 2.5 and φm = 0.63. It is shown that µef f estimated for Reγ˙ = 0.005 agrees well with the empirical relation. In agreement with Kulkarni & Morris [119], µef f increases with the increase in Reγ˙ . P with Re further, the particle shear stress is normalized To investigate the changes of σ12 γ˙ its value at Reγ˙ = 0.005 and one is subtracted from the normalized particle shear stress: ∗ = σ P (φ, Re )/σ P (φ, 0.005) − 1. In the dilute limit, Lin et al. [130] have shown ana- ∆σ12 12 γ˙ 12 3/2 lytically that the particle shear stress is proportional to Reγ˙ . Mikulencak & Morris [154] 218 showed, in their finite element simulations of a single neutrally buoyant particle in finite- 3/2 Reynolds-number shear flow, that the particle shear stress does follow the Reγ˙ scaling at low Reynolds numbers (Reγ˙ < 10−1 ) with a coefficient different from that suggested by Lin et al. [130]. In their simulations, it is shown that the increase of the particle shear stress 3/2 with Reγ˙ changes from Reγ˙ to Reγ˙ around Reγ˙ ' 10−1 . Using the lattice-Boltzmann simulations, Verberg & Koch [215] have shown that the stresslet for φ = 0.30 grows pro- portional to Reγ˙ for 10−2 < Reγ˙ < 1 and then it shows ∼ Re0.6 γ˙ behavior. Similar to their ∗ is proportional to Re in 0.1 < Re < 2. In the results, in figure 6.7, it is shown that ∆σ12 γ˙ γ˙ present study, the Reγ0.6 ˙ scaling is not observed. Again, it should be noted that the numeri- cal simulations of [215] are high-Stokes-number suspensions while, in the present study, we consider neutrally buoyant particles. ∗ ) The probability density functions for the shear component of the particle stresslet P (S12 is shown in figure 6.8. Here, the particle stresslet is estimated from the sum of hydrodynamic stresslet and interparticle potential contributions; ∗ 1X S12 = S12 − r ⊗ FP , (6.11) 2 nb in which nb is the number of particles in the cut-off distance; Rref < 2.004a. The pdf is standardized by its mean and standard deviation s012 so that P (S12 ∗ ) has zero mean ∗ ) is strongly skewed to the right. and standard deviation of 1. For Reγ˙ = 0.005, P (S12 The skewness factors at Reγ˙ = 0.005 are 1.25, 1.31, and 1.63 for φ = 0.2, 0.3, and 0.4, ∗ becomes highly intermittent. The probability of the respectively. As Reγ˙ increases, S12 ∗ /s0 < 5 decreases at larger Re , while the positive intermediate magnitude events 0 < S12 12 γ˙ probability of intermittent events, more than 5 times larger than s012 , rapidly increases with ∗ ) the increase in Reγ˙ . In the inset of figure 6.8, it is shown that the positive tail of P (S12 ∗ |) at Re = 0.005 to an algebraic decay changes from an exponential decay ∼ exp(−α|S12 γ˙ ∗ −β at Re = 2. The fitting parameters (α, β) are (1.2, 2.9) for φ = 0.2 and (1.1, 2.0) ∼ S12 γ˙ ∗ /s0 > 20. It is not for φ = 0.4. In all cases, the tail of pdfs drops quickly to zero after S12 12 ∗ is because of the insufficient sample size clear whether this exponential decay at large S12 to resolve such an intermittent event or of a physical mechanism. It is worthwhile to noted ∗ −2 behavior for all S ∗ ∈ (L, ∞), in which L is a constant, that if the pdf exhibits a ∼ S12 12 even the first moment is divergent, which is not physical. 219 (a) 10 0 0 10 10-1 -2 10 10-1 10 -3 P(S*12) -4 10 0 1 10-2 10 10 10-3 -4 10 -5 0 5 10 15 * S12 (b) 10 0 0 10 10 -1 10-2 -1 10 10 -3 P(S*12) -4 10 -2 0 1 10 10 10 -3 10 -4 10 -5 0 5 10 15 * S 12 Figure 6.8: The standardized probability density functions of the shear component of the particle stresslet for (a) φ = 0.20 and (b) φ = 0.40: ¥, Reγ˙ = 0.005; N, Reγ˙ = 0.5; ¨, Reγ˙ = 1.0; •, Reγ˙ = 2.0. The dashed line is the Normal distribution. The inset shows the positive tails of the pdfs for Reγ˙ = 0.005 and Reγ˙ = 2.0. 220 0.5 Π / µeffγ 0.4 0.3 0.2 0.1 0 -3 -2 -1 0 10 10 10 10 Reγ Figure 6.9: The particle pressure normalized by the effective viscosity: ¥, φ = 0.20; N, φ = 0.30; •, φ = 0.40. The hollow symbols are the numerical results of Sierou & Brady [197] at zero Reynolds number. 6.4.2 Normal stresses Figure 6.9 shows the particle pressure normalized by the effective viscosity and the shear rate. The particle stress is defined as 1 P P P Π = − hσ11 + σ22 + σ33 i. (6.12) 3 The normalized particle pressure is almost uniform for Reγ˙ < 0.1 and then begins to increase with the increase in Reγ˙ , indicating that the particle pressure increases faster than the shear stress. The hollow symbols are the Accelerated Stokes Dynamics simulations of Sierou & Brady [197]. It is shown that the present simulation results at low Reγ˙ agree well with the results of [197] at zero Reynolds number. 221 (a) 5 0 σii -5 σ11 -10 σ22 σ33 -15 -2 -1 0 10 10 10 Reγ (b) 0 -10 σii -20 -30 -2 -1 0 10 10 10 Reγ (c) 0 -20 σii -40 -60 -2 -1 0 10 10 10 Reγ Figure 6.10: Normal stresses normalized by µ0 γ˙ as functions of Reγ˙ for (a) φ = 0.2, (b) φ = 0.3, and (c) φ = 0.4. 222 (a) (b) 0.5 100 0 - N1 N2 10-1 -0.5 10-2 -3 -1 -3 10 10-2 10-1 100 10 10-2 10-1 100 Reγ Reγ Figure 6.11: First (a) and second (b) normal stress differences normalized by µ0 γ˙ as func- tions of Reγ˙ : ¥, φ = 0.20; N, φ = 0.30; •, φ = 0.40. Each components of the particle normal stresses are shown in figure 6.10. In Stokes-flow suspensions, all three normal stress components are negative [240]. It is shown that the P normal stresses at Reγ˙ = 0.005 are all negative. In the range of φ in the present study, σ11 P are always negative and the magnitude (|σ P | and |σ P |) increases with Re . On the and σ33 11 33 γ˙ P for φ = 0.2 is negative at Re = 0.005 and σ P increases with Re , which other hand, σ22 γ˙ 22 γ˙ eventually becomes positive for Reγ˙ > 10−1 . Similarly, for φ = 0.3, σ22 P increases with Re γ˙ P becomes at first and, then, starts decreasing for Reγ˙ > 1. For φ = 0.4, the behavior of σ22 P and σ P . These changes in the normal stresses a monotonic function of Reγ˙ similar to σ11 33 will be discussed further in 6.5. The first (N1 ) and second (N2 ) normal stress differences normalized by µ0 γ˙ are shown in figure 6.11. The normal stress differences are defined as P P N1 = σ11 − σ22 , (6.13) P P N2 = σ22 − σ33 . (6.14) Consistent with the Stokes-flow suspensions, both N1 and N2 are negative at Reγ˙ = 0.005. In the range of Reγ˙ of the present study, N1 is always negative. For Reγ˙ < 0.1, N1 looks 223 to converge to an asymptotic value. For Reγ˙ > 0.1, N1 grows proportional to Reβγ˙ , in which β is a growth factor. For comparison, the theory in the dilute limit (φ → 0) given in 1/3 Lin et al. [130] predicts ∼ Reγ˙ growth for large Reγ˙ . The growth factor in concentrated suspensions is shown to be a decreasing function of φ; β = 0.63, 0.56, and 0.37 for φ = 0.2, 0.3, and 0.4, respectively. Unlike N1 , it is shown that N2 ’s for φ = 0.2 and 0.3 become positive for Reγ˙ > 0.1 and Reγ˙ > 0.5, respectively. This behavior of N2 is qualitatively similar with Kulkarni & Morris [118]. However, in their lattice-Boltzmann simulations of a wall-bounded Couette-flow suspensions, N2 becomes positive at larger φ for φ ≤ 0.2 and N2 stays negative for φ = 0.3, whereas, in the present simulation, the positive N2 is observed for the volume fraction up to 0.3. The quantitative difference between the present study and [118] may come from the effects of the confinement. The channel height of [118] is 20a. Yeo & Maxey [230] have shown in Stokes-flow suspensions that the near-wall structure has a significant effects on suspension rheology in concentrated suspensions φ ≥ 0.3. It is also important to note that the normal stress differences are sensitive to the choice of the interparticle potential force. 6.5 Microstructure Here, the changes in the suspension microstructure is investigated by analyzing the pair- distribution functions, g(r, θ, ψ), in which r is the radial distance, θ and ψ are, respectively, the azimuthal angle measured from the flow direction and the polar angle measured from the positive z-direction. In essence, the pair-distribution function is related with the probability to find a particle with respect to the reference particle at the origin. Figure 6.12 shows the pair distribution in the shear plane, i.e. for ψ = π/2. Both for φ = 0.2 and 0.4, the thickness of the boundary layer near the compressive principal axis of shear flow, θ = 3/4π, is reduced at Reγ˙ = 1.0 compared to that of Reγ˙ = 0.005. At the same time, the wake in the expansive principal axis direction, θ = 1/4π, is thickened at larger Reγ˙ . The near-contact pair-distribution in the plane of shear gnb (θ) is estimated by averaging g(r, θ) over 2 < r < 2.1. As shown in figure 6.12, gnb (θ) rapidly decreases near the expansive principal axis θ = 1/4π. Consistent with Kulkarni & Morris [118], the width of this wake region increases with Reγ˙ . It is shown that the separation point, where gnb starts decreasing 224 (a) 4 Re = 0.005 (c) 4 Re = 0.005 4 4 3.5 3.5 3 3 2.5 2.5 2 2 2 2 1.5 1.5 1 1 ∆y / a ∆y / a 0.5 0.5 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 ∆x / a ∆x / a (b) 4 Re = 1.0 (d) 4 Re = 1.0 4 4 3.5 3.5 3 3 2.5 2.5 2 2 2 2 1.5 1.5 1 1 ∆y / a ∆y / a 0.5 0.5 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 ∆x / a ∆x / a Figure 6.12: The pair-distribution function in the plane of shear for (a, b) φ = 0.2 and (c, d) φ = 0.4. The line denotes g(r, θ) = 1. 225 (a) 8 Re = 0.005 Re = 0.1 6 Re = 0.5 Re = 1.0 g nb (θ) 4 2 01 0.8 0.6 0.4 0.2 0 θ/π (b) 8 6 g nb (θ) 4 2 01 0.8 0.6 0.4 0.2 0 θ/π (c) 12 8 g nb (θ) 4 01 0.8 0.6 0.4 0.2 0 θ/π Figure 6.13: The near-contact pair-distribution function in the plane of shear for (a) φ = 0.2, (b) φ = 0.3, and (c) φ = 0.4. 226 rapidly, moves towards the upstream (larger θ). Around the compressive principal axis, gnb is found to be an increasing function of Reγ˙ . The increase of gnb depends also on φ and, at φ = 0.4, the changes of gnb is not so conspicuous as in φ = 0.2. In Kulkarni & Morris [118], they found a local peak of gnb near θ = 0, or π, which grows rapidly with Reγ˙ . However, in the present study, we could not find such a local peak near θ = 0. Considering that Yeo & Maxey [230] have been shown that gnb of the particles in the near-wall particle layer has a distinctive peak at θ = 0, it is possible that the local peak in [118] is due to the contribution from the near-wall particle layer. To investigate the relation between the microstructural changes and the rheology further, we define a pair-distribution weighted by the particle stress tensor as P Ns * ∗ 1 (∆Y p ) + Sij Ωr ∗ p=1 Sij g(r) = , (6.15) nNs |Ωr | in which Ωr is a small volume centered at r, 1Ωr (x) is an indicator function of Ωr , Ns is the number of particles in the domain of interest centered at the reference particle, ∆Y p is ∗ is the particle stresslet of the reference particle. If S ∗ is 1, the separation vector, and Sij ij the pair-distribution function is recovered from (6.15). ∗ and normalized by the mean Figure 6.14 shows gnb weighted by the shear component S12 ∗ . For Re = 0.005, g of S12 γ˙ ng shows a plateau in 0.4 < θ/π < 0.8, while the weighted gnb shows two local peaks near θ ' 0.75 and the separation point, around which the ∼ 1/² ∗ g normal lubrication interaction is important. There is a local minimum of S12 nb near θ = π/2, where the lubrication interaction between two particles becomes ∼ log ² of the sliding motion. The near-contact pair-distribution functions show a nearly uniform growth in 0.4 < θ/π < 0.8 with Reγ˙ . However, the weighted pair-distribution clearly indicates the ∗ from the events around the compressive direction increased contribution to the total S12 (θ/π = 0.75). On the other hand, the contribution in θ/π < 0.6 to the separation point does not change noticeably with Reγ˙ . ∗ is shown in figure 6.15. Similar to The near-contact pair-distribution weighted by S11 ∗ , two local minima are observed around θ/π ' 0.85 and 0.45. The absolute values that of S12 of the two local minima increase with Reγ˙ . The second peak becomes more pronounced at larger Reγ˙ and the location of the second peak is shifted towards the upstream as Reγ˙ 227 (a) 15 Re = 0.005 Re = 0.1 〈S12 g nb 〉 / 〈S12〉 Re = 0.5 * 10 Re = 1.0 5 * 01 0.8 0.6 0.4 0.2 0 θ/π (b) 15 〈S12g nb 〉 / 〈S12〉 * 10 5 * 01 0.8 0.6 0.4 0.2 0 θ/π (c) 15 〈S12g nb 〉 / 〈S12〉 * 10 5 * 01 0.8 0.6 0.4 0.2 0 θ/π Figure 6.14: The near-contact pair-distribution function weighted by S12∗ for (a) φ = 0.2, ∗ (b) φ = 0.3, and (c) φ = 0.4. The weighted gnb is normalized by hS12 i. 228 (a) 0 〈S11g nb 〉 * -100 Re = 0.005 Re = 0.1 Re = 0.5 Re = 1.0 -200 1 0.8 0.6 0.4 0.2 0 θ/π (b) 0 〈S11g nb 〉 * -200 -400 1 0.8 0.6 0.4 0.2 0 θ/π (c) 0 〈S11g nb 〉 * -500 -1000 1 0.8 0.6 0.4 0.2 0 θ/π Figure 6.15: The near-contact pair-distribution function weighted by S11∗ for (a) φ = 0.2, (b) φ = 0.3, and (c) φ = 0.4. The weighted gnb is normalized by µ0 γa ˙ .3 229 increases. ∗ -weighted g . In S ∗ at Re = 0.005, a positive peak is Figure 6.16 shows the S22 nb 22 γ˙ observed around θ/π ' 0.35 for φ = 0.2, θ/π ' 0.38 for φ = 0.3, and θ/π ' 0.40 for φ = 0.4. As Reγ˙ increases, the magnitude of the positive peak is augmented and the ∗ g width of the positive S22 nb becomes larger. This changes in the microstructure explain the P at larger Re observed for φ = 0.2 and 0.3. At θ = 0, S ∗ is very close to zero positive σ22 γ˙ 22 ∗ at θ = 0 should be zero when only for φ = 0.2 and becomes negative at larger φ. Since S22 ∗ at θ = 0 comes a particle-pair hydrodynamic interaction is considered, this negative S22 from the mean field. The effects of the fluid inertia is pronounced only around the local ∗ g minimum and maximum and S22 nb is not sensitive to Reγ˙ near the equator (0 < θ/π < 0.3 and 0.9 < θ/π < 1). ∗ g In the normal stress in the vorticity direction S33 nb for φ = 0.2 at Reγ˙ = 0.005 (figure 6.17 a), a positive peak is observed near the compressive principal axis whereas there is a local minimum around the separation point near the expansive principal axis. The magni- tude of the positive peak decrease with φ and eventually becomes negative at φ = 0.4. For φ = 0.2, with the increase in the Reynolds number, the magnitude of the negative peak be- ∗ g near the peak (0.75 < θ/π < 15) comes larger and moves towards the upstream, while S33 nb ∗ g remains almost unchanged. On the contrary, for φ = 0.4, S33 nb decreases with Reγ˙ almost uniformly in 0.4 < θ/π < 1. The three-dimensional structure of the weighted gnb for φ = 0.3 and Reγ˙ = 0.005 is ∗ , S ∗ , and S ∗ , the major contribution to the mean stresses shown in figure 6.18. For S11 22 12 ∗ has a more comes from the events near the shear-plane (ψ = π/2). On the other hand, S33 complex three-dimensional structure. Around ψ = π/4 and 3π/4, there are two negative ∗ regions in the upper hemisphere and at the same ψ positive S ∗ regions also observed S33 33 ∗ in the upper hemisphere is in the lower hemisphere. The magnitude of the negative S33 ∗ in the lower hemisphere. almost a order of magnitude larger than that of the positive S33 6.6 Concluding remarks In this chapter, the rheology and dynamics of concentrated suspensions under finite fluid inertia have been studied by using the force-coupling method described in 2.4. The force- coupling simulations are performed for suspensions in a homogeneous linear shear flow for 230 (a) 100 〈S*22g nb 〉 0 Re = 0.005 Re = 0.1 -100 Re = 0.5 Re = 1.0 1 0.8 0.6 0.4 0.2 0 θ/π (b) 200 〈S*22g nb 〉 0 -200 1 0.8 0.6 0.4 0.2 0 θ/π (c) 0 〈S*22g nb 〉 -500 1 0.8 0.6 0.4 0.2 0 θ/π Figure 6.16: The near-contact pair-distribution function weighted by S22∗ for (a) φ = 0.2, (b) φ = 0.3, and (c) φ = 0.4. The weighted gnb is normalized by µ0 γa ˙ .3 231 (a) 40 〈S33g nb 〉 0 * Re = 0.005 Re = 0.1 -40 Re = 0.5 Re = 1.0 1 0.8 0.6 0.4 0.2 0 θ/π (b) 0 〈S33g nb 〉 * -40 -80 1 0.8 0.6 0.4 0.2 0 θ/π (c) 0 〈S33g nb 〉 * -100 -200 1 0.8 0.6 0.4 0.2 0 θ/π Figure 6.17: The near-contact pair-distribution function weighted by S33∗ for (a) φ = 0.2, (b) φ = 0.3, and (c) φ = 0.4. The weighted gnb is normalized by µ0 γa ˙ .3 232 Figure 6.18: The three-dimensional near-contact pair-distribution function gn b(θ, ψ) ∗ for φ = 0.3; (a) S ∗ , (b) S ∗ , (c) S ∗ , and (d) S ∗ . S ∗ g (θ, ψ) is shown in weighted by Sij 11 22 33 12 ij nb the y − z plane. The flow direction is into the paper; the upper hemisphere represents the compressional axis and the low hemisphere is in the extensional axis. 233 the volume fraction 0.2 ≤ φ ≤ 0.4. To investigate the effects of finite fluid inertia on suspension dynamics, the Reynolds number Reγ˙ is changed between 0.005 and 2. It is shown that the velocity fluctuations normalized by the shear rate and the particle radius γa ˙ in the velocity-gradient and vorticity directions are decreasing functions of Reγ˙ , while that in the flow direction is augmented at larger Reγ˙ . Under finite inertia, the particle velocity has a longer correlation, which results in the increase in the integral timescale. In the velocity auto-correlation functions, it is found that the effects of the finite inertia are the most pronounced in the intermediate timescale, 0.5 < γt ˙ < 4. The tails of the velocity auto-correlation (γt ˙ > 4) for different Reγ˙ collapse onto one curve, indicating that the memory lost in this later time is closely related to the multiple encounters with the suspended particles. Even though the velocity fluctuation is a decreasing function of Reγ˙ , the diffusivities in the velocity-gradient and vorticity directions increase with the increase in Reγ˙ due to the changes in the velocity auto-correlation. Consistent with Kulkarni & Morris [119], the effective viscosity is an increasing function of the Reynolds number. In the range of the simulation parameters (0.2 ≤ φ ≤ 0.4 and p 0.005 ≤ Reγ˙ ≤ 2), the growth of the shear component of the particle stress σ12 exhibits ∼ Reγ˙ behavior. Although the mean shear stress shows a monotonic increase with Reγ˙ , the probability density function indicates that the dynamics is changed significantly with ∗ changes the increase in Reγ˙ . The positive tail of the probability density function for S12 ∗ ) at Re = 0.005 to an algebraic decay ∼ S ∗ −β at from the exponential decay ∼ exp(−αS12 γ˙ 12 Reγ˙ = 2, implying that the intermittency of the particle stresslet grows rapidly with Reγ˙ . It is shown that the particle pressure grows faster than the effective viscosity as Reγ˙ increases. However, the changes of the individual normal stress components are not a monotonic function of Reγ˙ . The normal stresses in the flow and vorticity directions (σ11 , σ33 ) are negative and decreasing functions of Reγ˙ . On the other hand, for φ = 0.2, σ22 is negative at Reγ˙ = 0.005 and increases with Reγ˙ , which eventually becomes positive for Reγ˙ > 0.1. Similarly, σ22 for φ = 0.3 increases at first and then starts decreasing for Reγ˙ > 1. It is shown that the first normal stress difference N1 is always negative as in Stokes-flow suspensions. N1 exhibits ∼ Reβγ˙ behavior for Reγ˙ > 0.1 and the growth factor p β is shown to be a decreasing function of φ. Because of the increase of σ22 with Reγ˙ , the second normal stress difference for φ ≤ 0.3 becomes positive at larger Reγ˙ . To investigate the changes in suspension rheology further, a weighted pair-distribution 234 function is introduced. From the weighted pari-distribution function, it is shown that the increase of the effective viscosity is mainly related to the increase in the frequency and magnitude of the events near the compressive principal axis of shear flow. Even though the probability of the particle encounter near the pole (θ = π/2) increases at larger Reγ˙ , the weighted pair-distribution functions for different Reγ˙ collapse onto one curve, indicating the contributions to the total effective viscosity from the region remains unchanged. The ∗ shows that S ∗ near the expansive principal axis pair-distribution function weighted by S22 22 ∗ grows with Re , which explains the becomes positive and the region of the positive S22 γ˙ P at larger Re . The three-dimensional weighted pair-distribution function increase of σ22 γ˙ ∗ has a complicated structure and the maximum magnitude events are not shows that S33 observed in the plane-of-shear ψ = π/2. Instead, the maximum magnitude events of negative ∗ are observed around ψ = π/4 and 3π/4 in the upper hemisphere and those of positive S33 ∗ are around the same ψ in the lower hemisphere. S33 Chapter 7 Turbulent flow: Modulation of isotropic turbulence seeded with finite size bubbles or particles 7.1 Introduction In the context of dispersed two-phase flows in liquids, the study of turbulence seeded with bubbles or small particles is a longstanding topic of research. Engineers involved in the chemical industry, oil extraction or materials handling need to predict both aspects of the coupled dynamics. Of the many applications, there have been a number of recent experiments to investigate the dynamics of drag reduction in boundary layer and channel flows through the injection of gas microbubbles [161, 186, 212]. The typical volume fractions encountered in these experiments are in the range of 5 − 30%. Microbubble drag reduction has been investigated too using various direct numerical simulations [55, 69, 70, 106, 137, 224]. These illustrate that a range of mechanisms are involved and that the dynamics are a complex interplay of the bubbles with the turbulence and near-wall interactions. Kiger & Pan [109] have made detailed measurements of the turbulence dynamics for solid particles in horizontal channel flow. Locally homogeneous turbulence is a simpler context in which to investigate the two- phase flow dynamics and is relevant to the small-scale dynamics of more general turbulent flows. The experiments of Lance & Bataille [123] provide measurements of the energy spec- 235 236 trum of the turbulent velocity fluctuations with a uniform suspension of bubbles. Changes in the spectrum were noted, as compared to single-phase turbulence, that depended on the bubble size and the bubble concentration. These have been followed by more recent exper- iments by Rensen et al. [181] and Berg et al. [213], where a reduction of the spectrum of the energy at large scales and an increase at small scales is observed with a corresponding increase of the dissipation spectrum at the smallest scales. These changes are linked to the bubble size, relative to the Kolmogorov length scale, and the “bubblance” parameter that characterizes the kinetic energy associated with the bubble motion relative to that of the turbulent liquid phase. Experiments on turbulent two-phase flows are challenging as optical access and phase discrimination are not easy, especially as the volume fraction increases. Computing power has steadily increased over the past two decades making it possible for simulations to ap- proach actual “numerical experiments”. Although the direct numerical simulation of homo- geneous isotropic turbulence is now well established [237], only a few numerical approaches are able to handle simultaneously all the length scales in a two-phase flow ranging from the surface boundary layers and wakes of individual particles to the largest coherent structures of the flow. In many two-way coupling simulations, particles have been modeled as point-force source terms in the Navier-Stokes equations, averaged locally over some numerical cell volume of size ∆x much larger than a particle. Inherently only a portion of the overall flow is resolved and the disturbance flow generated by individual particles is modeled [21, 93]. The dispersed phase motion is simulated in a Lagrangian framework using a particle tracking, force balance equation to predict each particle trajectory. The fluid forces on a particle are explicitly estimated in terms of a slip velocity between the particle motion and the resolved (larger scale) flow field of the surrounding fluid. The underlying assumption is that the typical size of the particles or bubbles is significantly smaller than all the significant scales of the turbulent fluid flow. This approach is relevant when the turbulent modulation is induced by collective effects and when direct hydrodynamic interactions between particles can be neglected. Such conditions may occur for very low volume fractions of the dispersed phase. Even with these limitations, some basic features have been revealed on modulation of homogeneous turbulence in gas-solid flows [21, 62, 65, 202] and bubble-laden turbulence [148, 149]. The results of the latter simulations are in qualitative agreement with the 237 experiments [181, 213] as summarized by Berg et al. [214] In many liquid-solid flows, the typical Reynolds number of the carrying flow is often high and the smallest scales of the velocity field are generally smaller than or comparable to the particle (or bubble) diameter. In these contexts, point-force approximations are less applicable because the scale separation is no longer effective. Only a few studies have considered the finite size of the particulate phase in a turbulent flow. Sundaram & Collins [206] among others took this constraint into account by including elastic collisions between particles but the hydrodynamic forcing terms were still assumed to be local point (Dirac delta) functions. More recently, ten Cate et al. [208] carried out a fully resolved simulation of a liquid-solid turbulent suspension. By means of the Lattice-Boltzmann method they simulated moderately concentrated two-phase flows, at volume fractions of 2% to 10%, in a sustained homogeneous turbulence. Finite size particles with a solid to fluid density ratio varying from 1.15 to 1.73 generated perturbations in the three-dimensional homogeneous turbulent flow. Fluctuating motions were observed to be enhanced at small scales and the energy spectra showed a weak reduction in the lower wavenumber content. The range of scales involved in the dissipation of fluid kinetic energy are clearly increased when particles are seeded in the flow. Using a new modeling approach, namely PHYSALIS, Zhang & Prosperetti [243] simulated the modulation of decaying homogeneous isotropic turbulence seeded with particles four times larger than the Kolmogorov scale. Neglecting the effect of gravity, they showed in their preliminary study that the presence of particles enhances dissipation whereas the turbulence decay rate of the turbulence is slightly increased. At low volume fractions a characteristic feature of particle motion in turbulence is for local particle accumulations to develop in response to the local vorticity or rate of strain in the flow. This depends on the inertial response time of the particle and is generally strongest for small particles when the response time matches the Kolmogorov time scale [219, 220]. Particles denser than the surrounding fluid tend to collect in regions of high strain-rate, while particles (or bubbles) less dense tend to collect in regions of high vorticity and lower fluid pressure [145]. Calzavarini et al. [30] have shown that the tendency for bubbles to cluster is significantly stronger than for denser particles. For larger particles, where the response time of the particle falls within the inertial sub-range, one may expect particles to exhibit clustering in response to turbulent motions with time scales comparable to those of the particle response time. This has been explored through numerical simulations for gas-solid 238 turbulent flows by Yoshimoto & Goto [238], where the response time is large compared to Kolmogorov scales but the physical dimensions of the particles remain very small. In general though, there is both an effective temporal filtering of the particle response to the turbulence and an effective spatial filtering associated with the finite size of the particle in comparison to the spatial scales of the turbulence, as noted for example by Xu & Bodenschatz [223]. These are still open questions. Recent experiments for isolated particles by Qureshi et al. [180], Volk et al. [216] and Xu & Bodenschatz [223], based on earlier work by Voth et al. [217], have investigated the response of finite size particles to isotropic turbulence. These experiments cover almost neutrally buoyant particles, bubbles and particles with specific density ratios up to 1.4. The experiments are able to achieve a higher Reynolds number, and hence establish more of an inertial sub-range, than is generally possible in numerical simulations. By examining the particle acceleration statistics it is generally observed that for neutrally buoyant particle with diameters five times the Kolmogorov scale or less the response is essentially the same as a Lagrangian fluid element. For larger particles there is a transition in the response as the diameter falls within the inertial sub-range. Qureshi et al. [180] observe that for particle diameters larger than fifteen times the Kolmogorov scale, the acceleration variance of neutral particles is consistent with an inertial range scaling. In this chapter, we investigate the modulation of homogeneous isotropic turbulence in- duced by spherical bubbles, neutrally buoyant particles and slightly inertial particles at volume fractions of 3 − 6%. We exclude the effects of gravitational settling or buoyancy and focus on the effects associated with the finite size (and volume) of the particles, within the range of 6 − 12 Kolmogorov scales in diameter. The force-coupling method provides a bridge between the point-particle methods used for gas-solid flows and the more detailed resolution of LBM, immersed boundary method or PHYSALIS simulations. FCM is an effi- cient scheme for simulating large numbers of small particles in dispersed two-phase flow and has been applied previously to simulations of microbubble drag reduction, Xu et al. [224] and sedimentation at finite Reynolds numbers, Climent & Maxey [38]. It is worthwhile to note that our goal is not to provide definitive answers to the dynamics of two-phase flow in homogeneous turbulence but to point to the many open issues that still need further investigation. 239 7.2 Simulation method 7.2.1 Inertial particle simulation If mP (mB for bubbles) and mF denote the mass of a particle and the mass of displaced fluid, respectively, the force experienced by the fluid due to the presence of the particle is à ! dV(n) (n) F(n) = (mP − mF ) g − + FC . (7.1) dt This force is the sum of the net external force due to buoyancy of the particles (or bubbles), the inertia excess over the corresponding volume of displaced fluid due to the density dif- ference and a contact force between particles. In the present study we neglect the effect of gravity g and focus on the inertial interactions between the phases as well as the effects of finite particle size. These conditions may be achieved in experiments under microgravity or where the terminal settling velocity is very small compared to the turbulent velocity scales. For gas bubbles in liquids, the mass is negligible (mB ¿ mF ) and the monopole term mF dV/dt represents the primary interphase coupling through momentum transfer. Results on bubbly turbulence will be compared to simulations with neutral particles (mP = mF ), (n) where only the force dipole contribution Gij is nonzero, and also with inertial particles (mP > mF ). (n) The term FC represents the effects of short-range hydrodynamic interactions and a rigid-body contact force that prevents bubbles or particles from overlapping. In the present simulations, and as in [224], we assume that bubbles are small enough that they remain spherical and due to surface contamination respond approximately as rigid spheres (no- slip boundary condition). Also, when bubbles come into contact they do not coalesce but eventually separate again. We use an effective repulsion or contact force between bubble or particle surfaces. The contact force for each pair i and j, is a function of the relative position vector xij = Y(i) − Y(j) and the distance rij = kxij k. If rij < Rref , the cut-off length scale for the barrier, then the contact force acting on particle i due to particle j is " #2 2 − r2 Rref Fref ij Fij C = 2 − 4a2 xij (7.2) rij Rref Otherwise the contact force is zero. This force acts along the line of centers of each pair, it 240 is elastic and conserves momentum. We fixed the force scale as Fref = mF (vK /τK ) (see later in the section for definitions and the values in Table7.1) and the cut-off scale as Rref /2a = 1.2 in the present simulations. The actual value of the force barrier during a collision is set in response to the proximity of the bubbles (or particles) and the relative turbulent motion leading to the contact. The total force is obtained through a pairwise summation, (i) X FC = Fij C (7.3) i6=j It is important to include these collisions or contact effects in order to maintain the volume fraction of the dispersed phase. Further details on these short-range interactions are given in Dance et al. [49]. At higher volume fractions it is necessary to provide a more detailed representation including viscous lubrication forces and solid-body contact forces. In princi- ple, viscous lubrication forces will prevent contact of perfectly smooth particles but contact occurs in practice through surface roughness and other variations. Tests were made, varying the magnitude of the force Fref and the cut-off distance Rref . The results presented here were found to be insensitive to the specific value of Fref within a broad working range. There was a more obvious effect of varying Rref but for the chosen value this was still not significant, especially at the average concentration levels of 6% or less considered here. Any transient overlap between particles was negligible. (n) The symmetric part of the force dipole Gij , the stresslet term, is set through an iterative procedure to ensure that the strain-rate within the fluid volume occupied by the dispersed phase is zero (when spatially averaged on the appropriate bubble or particle scale σ 0 ). The details of the iterative solver used are given in section 2.4. For a laminar flow the additional stresslet term leads to an enhanced viscous dissipation. Even for a neutrally buoyant particle moving with the surrounding fluid, any local rate of strain in the flow will be deflected around the particle, see for example Figure 3 of Lomholt & Maxey [135], leading to a locally higher rate of strain and dissipation near the particle. As described in section 4.11 of Batchelor [16], this may be represented by an effective viscosity in a suspension of many particles. Summing the stresslet contributions in a dilute sheared suspension of rigid spheres gives the classical Einstein estimate of the effective viscosity 241 under conditions of Stokes flow, µef f = µ(1 + 2.5φ + αφ2 ), (7.4) where φ is the volume fraction of the particles. The second-order coefficient α depends on the specific particle configuration [14]. These effects are captured by FCM at low volume fractions, and at larger volume fractions when the appropriate lubrication forces are included [1, 233]. The anti-symmetric part of the dipole corresponds to the sum of an external torque, Text and the excess inertia for rotation of the particle (IP ) or bubble (IB ) over the corresponding, appropriately chosen, volume of fluid (IF ) as given by à ! (n) dΩ(n) T(n) = Text − (IP − IF ) . (7.5) dt dΩ/dt is the rate of variation of the particle (or bubble) rotation rate. No external torques are applied to the particles in the present simulations, Text = 0. The rotation rate of a particle changes in response to the viscous stresses exerted by the surrounding fluid, summed over the particle surface, and the moment of inertia of the particles. Neutrally buoyant particles respond quickly and their rotation rate is in equilibrium with the rotation rate of the surrounding fluid at that length scale. The (rigid) bubbles would show the largest discrepancy due to differences in inertia but were found still to come to equilibrium quite quickly. We monitored the amplitude of both symmetric and anti-symmetric parts (n) of the dipole Gij in selected simulations of bubble motion involving both motion in a single vortex and in stationary isotropic turbulence. The antisymmetric contribution is significantly weaker than the stresslet term. It represents less than 8% when averaged over all bubble trajectories. While (7.5) is correct in principle the specification of IF for FCM is not trivial. We decided to exclude this small effect rather than introduce a possibly inaccurate fluctuating term in the simulations. Hence throughout the simulations T(n) = 0. Examples of the application of FCM to the motion of particles at low to moderate, finite Reynolds numbers, together with comparisons to fully resolved direct numerical simulations, are given in Liu et al. [132] and Maxey et al. [147]. A discussion of the response to unsteady flow conditions, including the ability of FCM to capture Basset history forces, is reported 242 Table 7.1: Simulation parameters of the sustained homogeneous isotropic turbulence (single phase flow). Parameter definitions are given in sections 7.2.2 and 7.2.3. The standard Kolmogorov velocity, length and time scales are vK , η and τK . Grid u0 ² ν Reλ λ Te u0 Te η τK vK kmax η 1923 19.8 5562 0.12 58.7 0.356 0.070 1.39 0.0236 0.0046 5.08 2.1 in Maxey [146]. The formulation of FCM provides a consistent energy budget for both the fluid and particle phases. Viscous dissipation of fluid kinetic energy is evaluated by integration over the whole domain including the volume occupied by the particles. This is required since the representation of the particles involves a spatial smoothing or filtering of the near-surface conditions for each particle and the interior of the particle volume remains an active part of the flow [135, 144]. 7.2.2 Turbulent flow simulation The flow is simulated in a fully periodic domain, of size 2π ×2π ×2π, using a Fourier pseudo- spectral code. The turbulent flow is sustained by a random forcing at low wavenumbers to achieve a statistically stationary state. The simulation procedure is similar to that used by Wang & Maxey [220] with the random forcing term applied over a shell of small wavenumbers [67]. The forcing term is applied to nonzero wavenumbers in the range 0 < k ≤ 3. This unsteady forcing will drive the large length scales of the flow. When a sufficiently large separation between production and dissipation scales occurs the small-scale structures exhibit the universal characteristics of a turbulent flow. Basic data on the single phase turbulence are given in Table 7.1. The simulations are based on a 1923 grid resolution, for which kmax η = 2.1. Simulations on a 1283 grid yielded the same results but with a lower resolution, kmax η = 1.4. Both are adequately resolved, with kmax η > 1 [205]. The forcing parameters are the same in all simulations and a balance between the average energy input from the forcing and the viscous dissipation rate ² is achieved. The turbulent kinetic energy K, or turbulence intensity u0 , is obtained by averaging over space and time after the initial transient (two to three integral time scales) necessary to develop stationary turbulence. This can be also evaluated in the spectral space by 243 Table 7.2: Parameters of the flow simulations: Bubbles (B, mB = 0), Neutrally-buoyant particles (N, mP = mF ) and Solid particles (S, mP = 1.4mF ). Particle radius a = 0.1305 (1) or a = 0.091 (2). All simulations include force monopole (M) and dipole terms (D), except S2-M which is based on the monopole only. Parameter definitions are given in sections 7.2.2 and 7.2.3; NP is the number of particles. Case Grid NP Reλ u0 ² a/η a/λ St τi /τK B1 1283 1600 59.7 20.18 5827 5.59 0.37 0.23 3.5 B2 1923 4508 59.2 20.14 5876 3.90 0.26 0.11 1.7 N1 1283 1600 57.3 19.44 5436 5.50 0.37 0.68 10.1 N2 1923 4508 58.7 19.77 5574 3.84 0.26 0.33 5.0 S1 1283 1600 57.0 19.44 5490 5.51 0.37 0.87 12.8 S2 1923 4508 57.3 19.54 5552 3.84 0.26 0.42 6.3 S2-M 1923 4508 58.3 19.78 5629 3.86 0.26 0.42 6.3 integrating the energy spectrum function E(k) as kZmax 3 1 ­ 0 0® K = u02 = uu = E(k)dk. (7.6) 2 2 i i 0 The rate of kinetic energy dissipation per unit volume, ² is balanced by the volumetric power input of the turbulence forcing and it is directly evaluated by integration of the dissipation spectrum D(k) over all the wavenumbers kZmax kZmax ²= D(k)dk = 2ν k 2 E(k)dk, (7.7) 0 0 where ν = µ/ρ is the kinematic viscosity. Among others, two typical length scales character- ize the turbulent fluid motion, namely the Taylor microscale, λ, defined by λ2 = 15νu0 2 /², and the Kolmogorov length scale, η, defined by η = (ν 3 /²)1/4 . The Reynolds number of the single phase flow, based on the Taylor microscale Reλ = u0 λ/ν is 59. 7.2.3 Simulation parameters The primary group of simulations are for bubbles (B), neutrally buoyant particles (N) and inertial solid particles (S) at a volume fraction of approximately 6%. The parameters for these simulations are listed in Table 7.2. Two particle or bubble sizes are considered, with particle radius a = 0.1305 or 0.091. For the larger particles, N = 1600 and corresponds 244 to a volume fraction of 6.0% while for the smaller particles, N = 4508 and corresponds to a volume fraction of 5.7%. In order to resolve the motion of the particles using FCM, the minimum spatial resolution should be between 5 to 6 grid points to a particle diameter to accurately represent the effect of both the force monopole and dipole terms. The larger particles are simulated adequately with a numerical grid of 1283 points for which then a/∆x = 2.66. The smaller particles are simulated on a 1923 grid for which a/∆x = 2.78. Particles or bubbles are initially seeded at random positions in the homogeneous turbu- lent flow and statistics are computed after a transient period, typically one or two integral time scales. The particle concentrations remain uniform on average but local fluctuations develop in response to the turbulence. These fluctuations depend on the relative inertial response time of the particles (or bubbles) to the turbulence time scales. The bubble re- sponse time is defined in terms of the mass of displaced fluid and the added-mass effect. The Stokes drag law provides a convenient reference estimate for defining the response time, so that τB = a2 /9ν. For inertial solid particles, with relative density ρ∗ = ρP /ρ, the particle relaxation time scale is τP = 2(ρ∗ + 1/2)a2 /9ν and ρ∗ = 1 for neutral particles (index N) while ρ∗ = 1.4 for the inertial particles (index S). We define the Stokes number St by com- paring τB (or τP ) to the turbulence time scale Te = u0 2 /², for the large eddy turnover time or eddy lifetime. This definition does not account for any specific modification to the drag law based on instantaneous Reynolds number, but it is appropriate as an a priori estimate for the bubble or particle response to the turbulence in the absence of buoyancy effects. The Stokes number (7.8) is also equivalent to comparing the bubble or particle radius a to the Taylor microscale λ, since the rate of viscous dissipation per unit mass ² is equal to 15νu0 2 /λ2 in homogeneous isotropic turbulence, τi 5 a2 St = = αi 2 . (7.8) Te 3 λ In (7.8) the coefficient αi is equal to 1 for bubbles (B), 3 for neutral particles (N) and 3.8 for inertial particles (S). The primary data related to the two-phase simulations are summarized in Table 7.2. The case index is a combination of the particle type (B, N or S) and the size of the particle (1 or 2) corresponding respectively to a radius a = 0.1305 or a = 0.091. The particles or bubbles are larger than the Kolmogorov scale with a length scale ratio a/η varying from 245 3.8 to 5.6. On the other hand, they are smaller than the Taylor microscale, typically from λ = 2.7a to λ = 4a meaning that the disperse phase will be interacting with the full range of the most energetic structures of the turbulent flow. For all the simulations, the Stokes number St is low or moderate, based on the estimate (7.8) using the integral time scale. We also provide the values of the particle relaxation time compared to the Kolmogorov time scales. 7.3 Turbulence modulation In this section we consider the ways in which the Eulerian characteristics of the fluid flow are modified by the presence of the bubbles or particles. As buoyancy forces are excluded, the main issues are the inertial forces associated with a particle (or bubble), which determine the force monopole, and the stresslet component of the force dipole in response to local velocity gradients, specifically the rate of strain. Both the monopole and the stresslet are associated with the finite volume effect of the particles. Generally, a particle moves with the surrounding fluid and motion relative to an otherwise uniform far-field flow arises from inertial forces for a bubble or solid particle. The finite volume effect of a neutrally buoyant particle is associated with the stresslet. The questions we consider are the length scales at which the particle phase and fluid phase interact and how the evolution of the energy spectrum of the fluid phase is modified by the particles. These interactions may be individual or the collective effect of many particles locally. We compare several different simulations with bubbles, neutrally buoyant particles and solid slightly inertial particles. The important parameters are collected in Table 7.2. The single-phase flow characteristics are identical on the two grids used (1283 , 1923 ) and the turbulent Reynolds number Reλ stays in the range of 57 to 60. This is similar to the conditions for the simulations by [208] of inertial particles in homogeneous turbulence. In order to compare directly with these simulations, we simulate a specific configuration S2 with solid inertial particles of density ρP = 1.4ρ. All the particles are small compared to the Taylor microscale but larger than the Kolmogorov dissipation length scale η, with particle radius a varying between 3.8 and 5.6η. Analyzing the results collected in Table 7.2, we see that both the turbulent intensity of the flow and the rate of kinetic energy dissipation by viscosity are essentially constant 246 with small variations of 4% for u0 and 7% for ². A general comment is that the turbulence intensity in all cases is similar because the forcing parameters are kept constant. The total energy input by the forcing is balanced by the turbulent dissipation rate. This is a particular feature of the simulation where all the scales of the flow are resolved. Departures from this may occur due to limited scale separation between the range of forced wavenumbers and those in the dissipation range or where there is a direct interaction of the particle phase and the forcing scales. In two-way coupling simulations using point particles [21, 202], the particles act as a sink of turbulent kinetic energy. Unresolved scales of the order of the particle diameter are modeled by an interphase coupling term. The dissipation by the particles is governed by the slip velocity of the particles relative to the local fluid velocity of the resolved part of the flow. In the fully resolved simulations by [208], they also obtained the result that the rate of energy dissipation is constant over the various two-phase flow conditions studied and that this balances the random forcing power input. In Figures 7.1 and 7.2, we plot the energy spectra resulting from the simulations with the two different sizes of particles, corresponding to scale ratios a/η = 5.5 (B1, N1, S1) and a/η = 3.9 (B2, N2, S3). The wavenumbers are scaled by kd , based on the particle diameter d = 2a and kd d = 2π, following the proposal by [208]. We observe that the large scale kinetic energy content (low wavenumbers) behave similarly because their dynamics are fixed by the forcing scheme and the level of kinetic energy input at low wavenumbers is essentially the same for all the simulations. We expect that the presence of bubbles or particles will lead to interactions with smaller scales. Indeed, a significant increase of kinetic energy at higher wavenumbers is seen for k/kd > 0.78 for the larger particles (a = 0.37λ) and k/kd > 0.65 for the smaller particles (a = 0.26λ). Similar features are obtained in the dissipation spectra also shown in Figures 7.1 and 7.2. This enhancement is more closely related to finite size effects than momentum transfer as it is seen equally with bubbles, neutrally buoyant and inertial solid particles. Simulations with only the force monopole term lead to only a weak modification of the energy spectrum, see case S2-M in Figure 7.2. It was unexpected that all three types of particle would induce similar modifications of energy and dissipation spectra regardless of their particular dispersion characteristics and net momentum transfer (monopole). The dynamics instead are more directly controlled by the stresslet contribution. This enforces a zero local rate of strain within the spherical Gaussian envelope so that the flow responds as a solid body within the fluid volume occupied 247 2 10 101 5/3 E(k)/ε η 2/3 0 10 10-1 10-2 10-3 -1 0 10 10 k/kd 101 0 10 D(k)/εη 10-1 10-2 -1 0 10 10 k/kd Figure 7.1: Energy (a) and dissipation (b) spectra (grid 1283 ). Solid line, single phase flow; dashed line with circles, bubbles B1; dash dot line with triangles, neutral particles N1; dotted line with diamonds, solid particles S1. 248 2 10 1 10 5/3 E(k)/ε η 0 10 2/3 10-1 10-2 10-3 10-4 -1 0 10 10 k/kd 101 0 10 D(k)/εη 10-1 10-2 -1 0 10 10 k/kd Figure 7.2: Energy (a) and dissipation (b) spectra (grid 1923 ). Solid line, single phase flow; dashed line with circles, bubbles B2; dash dot line with triangles, solid particles S2; dotted line with diamonds, solid particles S2-M; Long dash line, data set S2 from [208]. 249 by bubbles or particles. This in turn leads to a significant enhancement of small scale kinetic energy and rate of viscous dissipation near each particle surface as local flow variations are deflected around the particle. The local perturbation of the flow by the dispersed phase occurs on the scale of the particle diameter. At larger scales, k/kd < 0.6, the kinetic energy level is slightly reduced over a range of intermediate wavenumbers. This behavior has already been observed by many authors [21, 148, 208] although the physical origins differ. Buoyancy, for example, plays a major role in the studies of Mazzitelli & Lohse [148]. It is important to note that the pivoting wavenumber kp , where the transition between enhanced and reduced kinetic energy is different when the size of the bubbles changes. Ten Cate et al. [208] investigated distinct configurations varying the volumetric concentration and particle inertia but with constant characteristics of the turbulence (Reλ = 61) and a fixed particle size a/η = 3.95. Varying the density ratio and the particle concentration, led to the striking result that the pivoting wavenumber kp always corresponded to kp /kd = 0.72. This is very similar to our results, where for case S2: kp /kd = 0.65 and for this particle size the results for B2, N2 and S2 give kp /kd clustered around this value. The question is whether kd characterizes the value of kP as the particle size varies. In Figure 7.3, we show the dissipation spectra for the neutral particles N1 and N2 with the wavenumbers scaled now by the Kolmogorov length scale η. Here the pivoting wavenumber is kp η = 0.45 for the larger particles (N1) and kp η = 0.56 for the smaller particles (N2). As expected, the value of kp η is larger for the smaller particle, but the increase is only a factor of 1.25 compared to the 1.43 factor for the particle sizes. The observed values of kp /kd are 0.78 (N1) and 0.68 (N2) and these differ by a factor of 0.87. The outcome then is that the scaling of kp by the particle size is a better correlation than a simple scaling with η but the results are not conclusive. Our values for kp /kd bracket those of [208] so there may be fluctuation errors. But it is likely that there is a more subtle Reynolds number effect, based on the particle size, which modifies the correlation with particle size. The simulations of ten Cate et al. [208] were performed using a lattice-Boltzmann ap- proach. In analyzing the energy and dissipation spectra, see Figure 7.2, we note that significant oscillations occur in the high wavenumber content that are related to discon- tinuities in the velocity gradient across particle interfaces. Discontinuities lead to a k −4 scaling of the tail of the energy spectrum and k −2 for the dissipation spectrum. Although 250 101 0 10 D(k)/εη 10-1 10-2 -1 0 10 10 kη Figure 7.3: Dissipation spectra D(k) plotted against wavenumber scaled as kη: solid line, single phase flow; dashed line, neutral particles N1; dash dot line, neutral particles N2. 251 the authors argued that this part of the spectrum contains only 1% of the total energy or dissipation, such oscillations may affect the precise determination of the pivoting wavenum- ber. There are inherent challenges to measuring and interpreting Fourier spectra in the spatial domain for dispersed two-phase flow. In the present FCM simulations, the spectra remain smooth as the velocity and velocity gradients are continuously defined throughout the whole domain. (Post-processing of the computed volumetric flow field may be used to reconstruct the discontinuous variations.) As we identify the stresslet as being the main contribution to flow modulation, we consider how this may contribute to the energy transfer between scales. We can use the momentum equation with FCM to derive an evolution equation for the energy spectrum E(k) in wavenumber space. Since the flow is statistically stationary, there is a balance of energy input at large scales by the random forcing FR (k), direct viscous dissipation D(k), the usual nonlinear energy transfer T (k) and the contributions from the force monopoles M (k) and the force dipoles H(k). The latter two depend on the correlation, in spectral space, of the particle body force f (x, t) and the fluid velocity u(x, t). The balance is 0 = FR (k) − D(k) + T (k) + M (k) + H(k) (7.9) where the dissipation spectrum function is D(k) = 2νk 2 E(k). The values of M (k) and H(k) can be evaluated directly from the simulation data. First we write Np X fiM (x) = Fin ∆(x − Yn ), (7.10) n=1 Np X ∂ 0 fiD (x) = Gnij ∆ (x − Yn ). (7.11) ∂xj n=1 Then, the monopole M (k) and the dipole energy transfer functions H(k) are defined as X M (k) = fˆiM (k)ˆ ui (−k), (7.12) k− 21 ≤|k| 15, the rate of decay for neutrally-buoyant particles becomes similar to bubbles, while ρv (t) of solid particles is always larger than the two others. Once trapped in a vortex, bubbles move toward the vortex core exhibiting rotational motion that results in the faster decay of ρv (t) at early time. The inertial solid particles are swept outside these small-scale structures of the turbulence. As a result, ρv decays more slowly as compared to bubbles and neutrally-buoyant particles. Collision contacts between bubbles are more likely than between the other particles due to the effects of local accumulations and this would further influence the relative correlation time scales. Figure 7.6 shows the probability density function (pdf) of the acceleration. The pdf has 257 100 10-1 10-2 P(A/σA) 10-3 -4 10 10-5 10-6 -15 -10 -5 0 5 10 15 A/σA Figure 7.6: Probability density function of particle acceleration. Legend for symbols: circle, B1; triangle, B2; nabla, N1; diamond, S1. very long tails compared to a Gaussian distribution indicating a high level of intermittency of the acceleration [174]. When scaled by corresponding rms accelerations σa the pdf’s are very similar in the central region, |a| /σa < 3. This indicates that the differences in the ac- celeration statistics between the different particles or bubbles, and different sizes, come from the intermittent events, which are closely related to the coherent vortical structures [124]. These features of the pdf’s are consistent with the experimental observations, obtained at higher Reynolds numbers [180, 216, 223]. The Lagrangian acceleration correlation function for fluid tracers is known to decay much faster than the velocity correlation [235] and has zero integral time scale. Yeung & Pope [235] found that the acceleration auto-correlation crosses zero at about 2τK and this varies only slightly with the turbulent Reynolds number. Mordanta et al. [157] showed that the auto-correlation of the acceleration magnitude decays slowly and argued that this long- time correlation is a key feature of intermittency in turbulence. Lee et al. [124] suggested that this different behavior of acceleration and acceleration magnitude comes from the rotational motion of particles in a vortex in that the zero-crossing time of the acceleration auto-correlation is related to the rotational time-scale of a particle as it is swept around in a vortex. The acceleration auto-correlation function ρA (t) is presented in Figure 7.7. The zero- crossing time of the auto-correlation function for smaller bubbles (B1) and larger bubbles 258 1 0.5 ρA(τ) 0 -0.5 0 5 10 15 t/τK 1 0.5 ρA(τ) 0 -0.5 0 5 10 15 t/τK Figure 7.7: Acceleration auto-correlation function. (a) Solid line, B1; dashed line, N1; dash dot line, S1. (b) Solid line, B2; dashed line, N2; dash dot line, S2. 259 3 2 D(t)/εt 1 0 -1 10 100 101 102 t/τK Figure 7.8: Normalized velocity structure function. Solid line with solid circles, B1; solid line with solid triangles, N1; solid line with solid diamonds, S1; dashed line with open circles, B2; dashed line with open triangles, N2; dashed line with open diamonds, S2. (B2) are, respectively, 2.33 and 2.50τK . These values are smaller than for neutrally-buoyant and solid particles. At early times (t < 2τK ), ρA (t) for the neutrally-buoyant and the solid particles are only slightly different. Due to the effects of inertia, ρA (t) for the solid particles decays more slowly than for the neutrally-buoyant particles. Both Volk et al. [216] and Xu & Bodenschatz [223] have measured the acceleration autocorrelations for a range of isolated particles and particle sizes at high Reynolds numbers. The first zero-crossing for bubbles, a/η = 4.4, is earlier than for neutral or denser particles and for larger neutrally buoyant particles, a/η = 7.4 the autocorrelation extends further [216]. Both observations are consistent with the present simulations but it is not possible to make a quantitative comparison. Finally we examine the Lagrangian velocity structure function. Figure 7.8 shows the 260 velocity structure function DL (t), D E DL (t) = (V (s + t) − V (s))2 , (7.15) where DL (t) is normalized by ²t. In a high Reynolds number inertial sub-range DL (t)/²t would be a constant C0 . The results shown for the two different sizes of bubbles are similar. For both neutrally buoyant and solid particles, the maximum value of DL (t) is greater for the smaller particle size while the time at which the maximum is attained is shorter. For the present simulations there is no inertial sub-range as the Reynolds number is too low. From the normalized velocity structure function, we can provisionally evaluate C0∗ as a low Reynolds number analog to the Kolmogorov constant C0 [235, 190] for comparison purposes. Estimates of C0∗ are given in Table 5, based on the maximum values shown in Figure 7.8. The estimates of C0∗ for the bubbles (cases B1 and B2) are about 2.6, which is similar to that of a fluid particle [235]. The peak value for the bubbles is observed at approximately 4.3τK . 7.5 Conclusions In this chapter, we have examined a number of issues relating to the dynamics of finite-size particles at low to moderate volume fractions in liquid-solid or liquid-bubble turbulence. The results for the dissipation spectra in Section 7.3 show that the particles enhance the small scale turbulence at length scales below the particle size. This is determined primarily by the finite size of the particle and not by the specific density of the particle. The scaling of the pivoting wavenumber kP by the particle diameter, through kd , as proposed by ten Cate et al. [208], is consistent with the present results but does not fully match them. There may be an additional dependence for example on d/η. In describing the dynamics of the energy spectra, it would appear possible to represent the transfer of energy from large scales to small scales by the particle phase through the use of an effective suspension viscosity, at least for scales substantially larger than the particle. The characteristics of this viscosity are more complex than a simple application of the Stokes-Einstein result for dilute suspensions. There is limited fundamental information at present on the rheology of particle suspensions at finite Reynolds numbers. Recent numerical simulations by Kulkarni & Morris [119] for neutrally buoyant particles in a steady Couette 261 flow indicate that the Stokes-Einstein estimate should still apply at volume fractions of 5%. The contribution H(k) from the force dipoles to the spectral energy transfer in (7.9) represents a sum over the stresslet components of all the particles. The particle contribution to the bulk stress in a finite Reynolds number flow depends not only on this but also a local volume average of the symmetric moments of the particle accelerations and a Reynolds stress term [13, 119]. The latter is for the smaller-scale disturbance flow around each particle relative to the local volume averaged flow velocity. In the present formulation these other terms are included in the nonlinear inertial transfer term T (k). The effective viscosity of viscous suspension flows depends too on the relative positions of particles and localized concentrations of particles should have an observable effect [89]. In the present context, the bulk concentration of 6%, or less, is still relatively low and such effects may be limited as seen in the small variation of νef f in Table 7.3, even though the bubbles and solid particles have some tendency to cluster. The dynamics of viscous suspensions are governed by the velocity fluctuations created by the disturbance flows of the individual particles randomly dispersed in the flow and the correlations between these flows is important. Particle suspensions in turbulent flow are subject to the underlying turbulence and the correlations of the disturbance flows with the turbulence are more significant. There is evidence too for the spatial filtering effect from the finite particle size on the response to the turbulent motion. For example, for a neutrally buoyant particle, this is a function of the particle size relative to the Kolmogorov scale, or to the Taylor microscale. As indicated in (7.8), the Stokes number St is closely related to the ratio of the particle size to the Taylor microscale. Similarly the ratio τP /τK is closely related to the ratio of the particle size to the Kolmogorov length scale. The present simulations are at too low a Reynolds number to establish an inertial subrange but the particle diameters are 7 to 11 times larger than the Kolmogorov scales and so fall within the transition range described by Qureshi et al. [180]. This is in addition to any inertial response linked to the specific density of the particle. Chapter 8 Concluding Remarks 8.1 Summary and discussions The main objective of this thesis was to enhance our understanding on fundamental physics of solid-liquid multiphase flows under a wide range of flow conditions. The study of bulk rheology and dynamics of suspensions of solid particles in Stokes flows, i.e. at zero Reynolds number, dates back to 1950s. However, most studies have been focused on homogeneous system without any solid boundaries. In this research, we introduced one of the most funda- mental inhomogeneities, namely suspensions bounded by two parallel walls, and thoroughly studied the effects of the basic inhomogeneity on the mechanical property and dynamics of suspension flows. In high-Reynolds-number turbulent-flow suspensions, we demonstrated the turbulence modulation by the suspensions of heavy, neutrally-buoyant, or negatively buoyant particles of finite size, larger than the Kolmogorov scale, which is an extension of the conventional approaches of assuming suspensions of point particles. As a first step to fill the gap in the previous studies of the suspension flows in zero-Reynolds-number Stokes and high-Reynolds-number turbulent flows, we studied the effects of fluid inertia on rheol- ogy and suspension dynamics for the particle-scale Reynolds number of O(10−3 ) ∼ O(1). One of the most important contribution of this study is that we established a robust and general simulation technique to study suspensions of solid spherical particles of any volume fractions under a geometrical confinement for a wide range of Reynolds numbers. In this chapter, the major results of each chapters are briefly discussed. In this study, the force-coupling method (FCM) was used for the numerical simulations of suspensions of rigid spherical particles. In Chapter 2, we showed that the force-coupling 262 263 method for Stokes flow is a matrix free method to solve the grand mobility matrix. From the positive semi-definiteness of some Galerkin-type discretizations, such as spectral or spectral-element method, it is shown that, even with solid boundaries, the FCM grand mobility matrix is symmetric positive semi-definite. Based on the symmetric positive semi- definiteness, a conjugate gradient procedure for FCM is proposed. As a particle is repre- sented by regularized multipoles, instead of dirac delta functions, the force-coupling method underestimates the near-field viscous lubrication interactions between particles, while the far-field solution matches the analytical solution exactly. We developed a robust and effi- cient method, lubrication-corrected FCM (LC-FCM), to solve the far-field multibody and near-field lubrication interactions simultaneously. Using a preconditioned conjugated gra- dient method, we showed that the near-field calculation can be replaced by a few vector summations and the overall computational cost is almost the same with the standard FCM. LC-FCM has been tested thoroughly and shown to be able to simulate suspensions of rigid spheres for a wide range of volume fractions with a very good accuracy. We extended the numerical method for the simulations of suspension flows of non-zero Reynolds number, i.e. Navier-Stokes flows. First, we studied the effects of a geometric confinement on rheology and particle dy- namics of concentrated suspensions of hard spheres through the FCM simulations of wall- bounded Couette-flow suspensions at zero Reynolds number in Chapter 3. We found that coherent structures develop near the solid boundary, namely near-wall particle layers. The correlations of the wall layer extend 6-7 particle radii into the flow for φ ≥ 0.3 and this near-wall density fluctuation is independent of the channel height Hy . Due to the changes in the suspension microstructure near the wall, rheology of the bulk fluid becomes a func- tion of both the volume fraction and the channel height. Once a particle moves into one of these near-wall particle layers, the particle resides around an equilibrium position of the particle layer for a long time compared to the diffusive timescale, which results in an anoma- lous diffusion. By investigating the mean-square displacement of the suspended particles near the core of a channel for a range of volume fraction and channel height, we suggest that the subdiffusivity observed for the particles in the quasi-homogeneous core region is a consequence of the restriction on the largest dynamic lengthscale of suspension flows by the available lengthscale set by the channel height. To support the argument, we estimate the particle shear stress from the momentum conservation equations shown in Chapter 4, 264 Table 8.1: Changes in the particle shear stress with the channel height. Hy /a 20 30 40 Sierou & Brady[197] p σ12 4.18 4.45 4.50 4.49 ± 0.037 (4.14). In 3.2, we computed the particle shear stress from a volume average. However, in an inhomogeneous system, the volume-averaged shear stress is not the shear stress measured on the wall. Integrating the momentum conservation equations over a control volume, of which upper boundary is located on the center of the channel and lower boundary is on the liquid-wall interface, and summing the particle- and fluid-phase shear stress, the wall shear stress τ w observable in the experiments is given by τ w = hχp σ12 i|y=Hy /2 + µγ. ˙ (8.1) This relation indicates that the measured shear stress depends on the particle shear stress in the quasi-homogeneous region near the core of the channel, which explains the discrepancy in the wall shear stress and the volume-averaged stress shown in Kulkarni & Morris [119]. Table 8.1 shows the particle-phase shear stress for the volume fraction φ = 0.40 at various channel heights. It is shown that the particle shear stress is an increasing function of the channel height and it approaches the value computed in the homogeneous linear shear flow by Sierou & Brady [197]. The particle shear stress for the channel height of Hy /a = 40 is almost the same as that in [197], at which channel height the particles in the core of the channel exhibit a normal diffusive behavior. In the second half of Chapter 3, we investigated a non-equilibrium phase transition observed in the non-Brownian suspensions at high volume fractions. At high volume frac- tions, the suspended particles are self-assembled into linear strings which are organized as a hexagonal array in the plane perpendicular to the flow direction. The hexagonal structure starts to form near the wall first and extends to the core of the channel as the volume frac- tion increases. It is shown that the ordering transition and crystal structures are dependent on both the volume fraction and the channel height. We showed that, by introducing a disturbance in the crystal structure, the mechanical property of the suspension flows can be altered. Here, we investigated the effects of an external torque on the suspended particle as a model for a DC electro-rotation of suspended particles (Quincke rotation). It is found 265 that the response of the order structure to the external torque is asymmetric to the sign of the torque and also nonlinear to the magnitude of the torque. In Poiseuille flows, the stress is a function of the distance from the wall. As a conse- quence of this inhomogeneous stress field, the suspended particles migrate to the core of the channel. To investigate the stress system of Poiseuille-flow suspensions, we performed FCM simulations for the volume fractions 0.2 ≤ φ ≤ 0.4 and the simulation results are shown in Chapter 4. It is shown that the velocity and local volume fraction profiles com- puted from FCM agree very well with previous experiments. Consistent with the previous stress induced migration hypothesis, the particle normal stresses are almost uniform except very near the wall and core of the channel. We estimated the local particle pressure and shear stress from the simulation data and showed that the local rheology assumption in the suspension balance model is valid, again, except in the wall and core regions. The particle layering and the following changes in the dynamics and rheology in the near wall region are expected from the results in Chapter 3. On the other hand, the suspensions near the core of the channel exhibit very interesting anomalous behavior. We found that the parti- cle shear stress becomes negative near the channel core, even though the local shear rate remains positive. Around the core of the channel, the local volume fraction is close to the maximum random packing and the local strain rate becomes very small, which violate the local rheology and local homogeneity assumptions in continuum theories. The anomalous behavior around the core of the channel is a subject of further investigation. We studied the hydrodynamic interaction between spinning particles at finite Reynolds number in Chapter 5. It is shown that a secondary meridional circulation develops around a spinning particle at finite Reynolds numbers and, due to this secondary flow, there is a net repulsive hydrodynamic force between a pair of rotating coplanar spheres. Although qualitatively correct, we found that FCM underestimates the magnitude of the secondary flow. It is shown that adding a correction force based on the perturbation analysis can improve the result. However, development of a general and robust correction scheme needs further studies. In Chapter 6, the effects of fluid inertia on rheology and particle dynamics are studied for the volume fractions 0.2 ≤ φ ≤ 0.4 for the range of the particle Reynolds numbers 0.005 ≤ Reγ˙ ≤ 2. Unexpectedly, the velocity fluctuation is found to be a decreasing function of Reγ˙ . However, as the motion of the particles has a longer correlation at finite Reγ˙ , the 266 diffusivity increases at larger Reγ˙ . It is shown that the particle shear stress increases with the increase in Reγ˙ . On the other hand, the particle normal stresses exhibit non-monotonic behaviors. The normal stresses in the flow and vorticity directions decrease as Reγ˙ increases. The normal stress in the velocity-gradient direction is shown to be an increasing function of Reγ˙ at φ = 0.2, while that for φ = 0.3 increases at first and starts to decrease when Reγ˙ ≥ 0.1, which changes to a monotonically decreasing function of Reγ˙ at φ = 0.4. We introduced a weighted pair-distribution function in the plane of shear to investigated the microstructural changes at finite Reynolds number in more details. The normal stress in the flow direction shows a monotonic decrease at the increase in Reγ˙ over all azimuthal angles θ. The normal stress in the velocity-gradient direction has positive and negative peaks around the compressive and expansive principal axes, respectively, at Reγ˙ = 0.005. For φ = 0.2, the positive peak grows rapidly with the increase in Reγ˙ , while the growth of positive peak is significantly attenuated at larger φ, which explains the non-monotonic behavior of the normal stress in the velocity-gradient direction. It is important to note that all the variables in this study is normalized by the shear rate γ. ˙ At zero Reynolds number, the only timescale is the shear rate γ. ˙ In other words, a suspension quantity obtained in a Couette device with a shear rate γ˙ 1 should be identical to the same quantity obtained in a same Couette device with a different shear rate γ˙ 2 , once normalized by the corresponding shear rates. However, this is no longer true at finite Reynolds numbers. There may exist multiple timescales. For example, it is shown that, at short time, the decaying rate of the velocity auto-correlation is an increasing function of Reγ˙ , which changes to a decreasing function of Reγ˙ at the intermediate timescale. Determining the characteristic timescale of suspensions under finite fluid inertia is a subject of further investigation. In Chapter 7, we considered the modulation of turbulence by suspensions of finite- sized heavy particles, neutrally-buoyant particles, or bubbles in isotropic turbulence. The particle size is chosen to be in between the Kolmogorov scale and the Taylor microscale. To maintain turbulence, a stochastic forcing is applied at small wave numbers. It is shown that the major effect of the finite-size particles is to redistribute the turbulent kinetic energy and dissipation rate, instead of simply increasing or decreasing a certain wave number mode. The turbulent kinetic energy at the lengthscales smaller than the particle diameter is noticeably augmented, the turbulent energy is attenuated over a range of intermediate wavenumbers, lengthscales larger than the particle size. It is shown that the Eulerian statistics, such as 267 the turbulent kinetic energy and dissipation rate spectra, are not altered noticeably by the types of the suspended particles, positively, neutrally, or negatively buoyant particles, while the effects of the particle inertia are pronounced in the Lagrangian statistics. It is shown that both the velocity and acceleration auto-correlations of the bubbles decay faster than those of the neutrally buoyant particles. The heavy particles have the longest correlation time among the types of the particles considered. Near a vortex tube, bubbles tend to spiral inward to the core of a vortex core, while neutrally-buoyant particles rotate around the vortex following the streamline. On the other hand, heavy particles, which has a long relaxation time, tend to penetrate the vortex. These different responses near a vortex explain the differences observed in the Lagrangian statistics. 8.2 Future directions In this thesis, we developed LC-FCM to simulate wall-bounded suspension flows. As a first step to understand fundamental physics of wall-bounded suspensions, we focused on an idealized system, suspensions of monodisperse hard spheres with a short-ranged repulsive potential force. However, it should be noted that still there is a gap between the ideal- ized numerical and laboratory experimental systems. One of the open questions in the numerical simulation of suspension flows is the effects of the interparticle potential force. Mathematically, spheres immersed in liquid can never contact because the lubrication force becomes singular as the gap between two spheres goes to zero. In experiments, the sus- pended particles are not perfect spheres and small asperities on the particle surface result in an actual contact between the suspended particles. Most, if not all, of the numerical techniques use a short-ranged repulsive potential force to model such a non-hydrodynamic effect. Even though it is now well known that the computed rheology depends on the form of this short-ranged force, there has been no systematic study to find a physically sound model of non-hydrodynamic forces. For a more realistic computer simulation, one needs to construct a more robust contact force model. Another important issue in the computer simulations of suspension flows is that the sus- pended particles in physical systems are polydispersed, while most simulations have been working on monodisperse systems. Even for the bi-disperse suspensions, there are only very limited number of numerical studies and most of them are either in two-dimensional simu- 268 lations [31, 32] or in semi-dilute limit [2]. In LC-FCM, provided that the analytical solution is given in literature, it is very straightforward to extend to polydisperse suspensions. In the standard FCM, the polydisperse simulation can be done simply by changing the length- scales (σM and σD ) of the regularized multipoles according to the particle radius. The only additional effort to use LC-FCM for polydisperse suspensions is computing two-body FCM resistance matrices for each particle size ratio numerically. Including the new two-body lubrication matrix into the LC-FCM formulation given in 2.3.1 is trivial. In this thesis, we considered only one of the most fundamental confinements, suspension flows bounded by two parallel walls. However, in most engineering flows, the flow geometry is much more complicated. For example, many microfluidic mixing devices use chaotic mixing strategies, in which three-dimensional flow structure is important. In LC-FCM, one can simulate these three-dimensional suspension flows without any additional complications. In 2.4, LC-FCM for finite inertia suspensions was developed and we studied the effects of finite fluid inertia on suspension dynamics in Chapter 6. The suspension dynamics at finite Reynolds number is a virtually unexplored field. Here, we investigated the dynam- ics in a homogeneous linear shear flow, in which the bulk lengthscale is indefinite. In a wall-bounded system, there is a clear separation of the scales. The bulk dynamics is re- lated to the Reynolds number based on the characteristic length of the confinement and the bulk flow rate, while the particle-scale local dynamics is determined by the particle- scale Reynolds number. However, there is no study about the interplay between the bulk dynamics and the particle-scale dynamics. LC-FCM simulations for the finite-inertia wall- bounded suspensions can be conducted easily by modifying the current LC-FCM code for the wall-bounded Stokes flows. In Chapter 7, we showed that the Eulerian statistics are not sensitive to the types of the suspended particles, whether they are heavy, neutrally-buoyant, or light particles, while the effects of the particle inertia are reflected in the Lagrangian statistics. In the same context, it would be instructive to investigate the modulation of turbulence by particle suspensions from the Lagrangian statistics of fluid element (massless and volumeless tracer particle). Study of the Lagrangian dynamics of tracer particles is important in understanding the turbulent mixing and heat transfer. This study can be readily conducted by using the standard FCM technique with some additional functions to compute the fluid velocity at the location of a tracer particle from an interpolation from the neighboring grid points. 269 When we performed the simulations, the particle acceleration term was treated explicitly. Due to the severe CFL condition, the simulations were limited only to the low Stokes-number particles, of which density ρp is close to that of the fluid density ρf , ρp /ρf < 2. In 2.4, we presented an implicit discretization of the particle acceleration term, which can be used for any density ratio. It will be interesting to investigate the turbulence-particle interaction for suspensions of large Stokes-number particles, i.e. gas-solid multiphase turbulent flows. Appendix A Analytical resistance functions The theoretical, asymptotic expressions for the particle-wall resistance functions can be found in the literature [42, 22, 50]. Due to the typographical errors in some sources, we list the near-field forms of the resistance functions used in the simulation. For the particle-wall interactions, as the gap ² → 0, 1 1 1 XA = − log ² + 0.8193 − ² log ², ² 5 21 8 64 YA = − log ² + 0.9557 − ² log ², 15 375 2 86 YB = log ² + 0.2568 + ² log ², 15 375 1 XC = 1.2021 + ² log ², 2 2 66 YC = − log ² + 0.3720 − ² log ², 5µ 125 ¶ 2 7 221 YG = − log ² + 0.923 + ² log ² , 3 10 250 1 2 YH = log ² + 0.0916 − ² log ². 10 250 The near-field form of the particle-particle resistance functions is given in Table A.1. 270 271 Table A.1: Near field forms of scalar resistance functions A 1 −1 9 3 X11 = 4² − 40 log ² − 112 ² log ² + 0.9946 + O(²) A 1 −1 9 3 X12 = − 4 ² + 40 log ² + 112 ² log ² − 0.3510 + O(²) C 1 X11 = 8 ² log ² + 1.0518 + O(²) C 1 X12 = − 8 ² log ² − 0.1503 + O(²) G 1 −1 9 39 X11 = 4 ² − 40 log ² − 280 ² log ² − 0.3127 + O(²) G 1 −1 9 39 X12 = − 4 ² + 40 log ² + 280 ² log ² + 0.1300 + O(²) A Y11 = − 61 log ² + 0.9983 + O(²) A 1 Y12 = 6 log ² − 0.2737 + O(²) B 1 1 Y11 = 6 log ² + 12 ² log ² + 0.1594 + O(²) B 1 1 Y12 = − 6 log ² − 12 ² log ² − 0.0011 + O(²) C 1 47 Y11 = − 5 log ² − 200 ² log ² + 0.7029 + O(²) C 1 31 Y12 = − 20 log ² − 500 ² log ² + 0.0274 + O(²) G 1 1 Y11 = − 12 log ² − 24 ² log ² − 0.0947 + O(²) G 1 1 Y12 = 12 log ² + 24 ² log ² + 0.0687 + O(²) H 1 137 Y11 = − 40 log ² − 2000 ² log ² − 0.0741 + O(²) H 1 113 Y12 = − 10 log ² − 2000 ² log ² − 0.0294 + O(²) Bibliography [1] M. 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