Abstract of “Continuum Modeling of Flow and Size-Segregation in Dense Granular
Materials”
by Daren Liu, Ph.D., Brown University, May 2019.
Granular materials are mixtures of discrete, macroscopic particles. They are ubiquitous
in nature as well as in everyday life – in forms such as sand, gravel, pharmaceutical pills,
food grains, and industrial powders. However, certain mechanical behaviors of these mate-
rials are still beyond the community’s understanding, including flow and size-segregation
phenomenon. Compared to other common engineering materials such as elastic solids and
viscous fluids, modeling of granular flow has remained a persistent challenge, especially
from a continuum perspective. This thesis addresses several problems related to the con-
tinuum modeling of dense granular materials:
• Dense granular heap flows (e.g., avalanches and landslides in nature) consist of a
rapid flow regime localized near the free surface and a creeping flow region deep
beneath the surface. We apply a scale-dependent continuum approach – the non-
local granular fluidity model – to a representative steady, dense granular heap flow
setup and successfully capture the salient features of both regions, which existing
continuum models have not been able to simultaneously predict.
• The flow threshold of dense granular materials exhibits a size effect that cannot be
captured by conventional local, stress-based criteria. Using two-dimensional discrete
element method calculations, we explore the configurational generality of the size-
dependence of the flow threshold – i.e., additional strengthening with smaller system
size – and show that the nonlocal granular fluidity model is capable of quantitatively
capturing this effect.
• Dense granular systems consisting of particles of disparate sizes segregate based on
size during flow, resulting in complex, coupled segregation and flow patterns. We
study the two driving forces of size-segregation – pressure-gradients and shear-strain-
rate-gradients – in dense flows of bidisperse disks and show that both the flow fields
and the segregation dynamics may be simultaneously captured by coupling a segre-
gation model with the nonlocal granular fluidity model.
Continuum Modeling of Flow and Size-Segregation in
Dense Granular Materials
by
Daren Liu
B. S., University of Science and Technology of China, 2013
Sc. M., Brown University, 2015
A dissertation submitted in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy
in School of Engineering at Brown University
Providence, Rhode Island
May 2019
c Copyright 2019 by Daren Liu
This dissertation by Daren Liu is accepted in its present form by
the School of Engineering as satisfying the dissertation requirement
for the degree of Doctor of Philosophy.
Date
David Henann, Ph.D., Advisor
Recommended to the Graduate Council
Date
Kenneth Kamrin, Ph.D., Reader
Date
Kyung-Suk Kim, Ph.D., Reader
Date
Tom Powers, Ph.D., Reader
Approved by the Graduate Council
Date
Andrew G. Campbell, Ph.D.
Dean of the Graduate School
iii
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Curriculum Vitæ
Daren Liu was born on January 16th, 1992 in Ji’an, Jiangxi Province of southeast China.
He started his undergraduate study at the University of Science and Technology of China in
2009, and received a Bachelor of Science degree in Theoretical and Applied Mechanics in
2013. In the fall of that year, he started graduate work in the Mechanics of Solids program
at Brown University, and attained a Master of Science degree in Solid Mechanics in 2015.
Refereed Journal Publications
(1 = These authors contributed equally.)
Daren Liu and David L. Henann. “Size-dependence of the flow threshold in dense granular
materials.” Soft Matter (2018).
Daren Liu and David L. Henann. “Non-local continuum modelling of steady, dense gran-
ular heap flows.” Journal of Fluid Mechanics 831 (2017): 212-227.
Conference Proceedings
(1 = talk, 2 = poster)
“Modeling shear-rate-gradient-driven size-segregation in dense, bidisperse granular flows”.
Gordon Research Conference on Granular Materials, Stonehill College, Jul. 2018.2
“Modeling shear-rate-gradient-driven size-segregation in dense, bidisperse granular flows”.
16th Annual Northeastern Granular Materials Workshop, Yale University, Jun. 8th, 2018.1
v
“Modeling shear-rate-gradient-driven size-segregation in dense, bidisperse granular flows
down a vertical chute”. APS March Meeting 2018, Los Angeles, California, Mar. 5th - 9th,
2018.1
“Size dependence of the flow threshold in dense granular materials”. 7th New England
Workshop on Mechanics of Materials and Structures, Harvard University, Oct. 22th, 2016.1
“Size dependence of the flow threshold in dense granular materials”. Society of Engi-
neering Science 53rd Annual Technical Meeting, University of Maryland, Oct. 2nd - 5th,
2016.1
“Size dependence of the yield threshold in dense granular materials”. Gordon Research
Conference on Granular Materials, Stonehill College, Jul. 2016.2
“Continuum modeling of granular flow down heaps”. 6th New England Workshop on
Mechanics of Materials and Structures, Boston University, Oct. 3rd, 2015.2
Teaching Experience at Brown
Undergraduate
Advanced Mechanics of Solids, TA (Fall ’14)
High School
Introduction to Engineering and Design, Instructor (Summer ’18)
vi
Acknowledgements
“The academic journey is as much about finding truth in books and experiments as it is
about finding truth in yourself.” — “Piled Higher and Deeper”.
My own graduate journey would certainly not be completed without standing on shoul-
ders of mountains of people. Lessons, meetings, experiences, even brief encounters or
conversations that sometimes seem small, shape me into who I am today.
I am so very proud and fortunate to be the first student of my advisor/role model, Prof.
David Henann, a true mentor and dedicated scholar. I see from you the pure love and
passion towards science and research. Over the years I have been privileged to have the
freedom to explore, to experience, and your constant and consistent kindness, support,
patience, positiveness and respect have made this journey quite enjoyable and memorable.
I don’t know how many times I walked in your office with unpleasant results in frustration,
while you would always respond with smile, patiently dive into the details with me, and
use your insights to point me in the right direction. I truly appreciate these positive impacts
you’ve had on me.
I would like to express my gratitude towards my thesis committee for willingly taking
the time to read my thesis and provide valuable feedback. Prof. Kyung-Suk Kim and
Prof. Tom Powers, I also enjoyed your classes. Prof. Ken Kamrin, this thesis would not
be possible without you paving the fundamentals and your continuous inputs. A bunch of
thanks also goes to Prof. Huajian Gao, Prof. Haneesh Kesari, Prof. Allan Bower, among
other faculty and staff members in the School of Engineering.
With five years and a half spent here, the distinguished Brown University with its plenti-
ful resources and openness, and the historic Providence with its bitter New England winter,
vii
have in a sense become my second home. To be surrounded by so many smart and interest-
ing people from diverse backgrounds is most fantastic. I am happy to see the Arnold 322
family growing over the years: Shihong, Yuhao, Mike, Xiuqi, Natassa and Harkirat. I will
cherish our exhilarating office environment, and I wish we could had done those dinner and
movie nights more often. To my peer students in Solids and our remarkable Engineering
community, many of you assisted me in various occasions. I know I would definitely miss
someone if I start listing names, so I will just say I am so much obliged to you for all the
help and support that carried me along the way. Thank you. You know who you are.
To the many wonderful friends, you kept reminding me to not always bury my face into
papers and codes and instead take a break once in a while and live a balanced life. Special
thanks to Chao, Xizheng, Zhiyu, Li, An, for the long-lasting friendship. Zheng, Wenzhe,
Boda, Xi, Yinsui, Shihong, Siyi, Guangyao, Zhi, our trips to Britain, Italy, Turkey, Mexico,
Alaska really opened up my eyes. Connie, thank you for praying for me and pushing me to
dream big when I lacked will. To the Brown Chinese Dragon Soccer Team and generations
of intramural teammates, it is my pleasure to have played with you. And to my remote
friends in China, thanks for the wishes from the other side of the Earth.
Most importantly, none of this would be possible if not for the love from my parents,
Yicheng and Xiaoqing. I can only imagine how you hid your feelings and sacrificed your
need of companion from your only child in lunar new years seeing other families in reunion.
Thank you mom and dad for letting me do what I love and always cheering for me. I love
you.
viii
D EDICATED TO MY TEACHERS , MENTORS , MY DEAR
PARENTS , AND ALL OF THOSE WHO LOVE AND SUPPORT
ME AS ALWAYS .
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x
Contents
List of Tables xiii
List of Figures xv
1 Introduction 1
1.1 Background on dense granular materials and continuum modeling . . . . . 1
1.2 Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 The nonlocal granular fluidity model on steady, dense heap flows . 4
1.2.2 Size-dependent flow threshold of dense granular materials . . . . . 4
1.2.3 Modeling size-segregation in dense, bidisperse granular flows . . . 5
1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Nonlocal continuum modeling of steady, dense granular heap flows 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The nonlocal granular fluidity model . . . . . . . . . . . . . . . . . . . . . 11
2.3 Numerical simulations of dense, steady granular heap flow . . . . . . . . . 16
2.3.1 Flow configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Flow fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.3 Volume flow-rate and maximum velocity . . . . . . . . . . . . . . 23
2.3.4 Two length scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Size-dependence of the flow threshold in dense granular materials 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Discrete-element method simulations . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Simulated granular system . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Planar shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Granular rheology and the flow threshold . . . . . . . . . . . . . . . . . . 36
3.4 Flow threshold in other configurations . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Planar shear flow with gravity . . . . . . . . . . . . . . . . . . . . 43
3.4.2 Annular shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.3 Vertical chute flow . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Modeling shear-rate-gradient-driven and pressure-gradient-driven size-segregation
in dense, bidisperse granular flows 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Continuum framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
xi
4.2.1 Bidisperse systems . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Granular rheology – from monodispersity to bidispersity . . . . . . 65
4.2.3 Segregation model . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Diffusion flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 Shear-rate-gradient-driven segregation flux . . . . . . . . . . . . . . . . . . 71
4.4.1 Vertical chute flow . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4.2 Annular shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5 Validation of shear-strain-rate-gradient-driven segregation flux in transient
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5.1 Vertical chute flow . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5.2 Annular shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Addition to the segregation model – Pressure-gradient-driven segregation
flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.7 Inclined plane flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.8 Planar shear flow with gravity . . . . . . . . . . . . . . . . . . . . . . . . 95
4.9 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Conclusions and outlook 107
5.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A Technical details of DEM in LAMMPS 113
B Averaging method – coarse graining 115
B.1 Slices technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.2 Sliding bins technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C Diffusion flux consistency check 121
xii
List of Tables
4.1 Summary of the quantities introduced to describe dense, bidisperse granu-
lar mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
xiii
xiv
List of Figures
1.1 Examples of granular materials and common applications . . . . . . . . . . 2
2.1 Heap flow schematic and computational domain . . . . . . . . . . . . . . . 18
2.2 Flow fields in a narrow channel of width W /d = 19 for steady, dense gran-
ular heap flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Flow fields in a wide channel of width W /d = 142 for steady, dense gran-
ular heap flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Relationships between free surface inclination angle, maximum velocity,
and volume flow-rate in steady, dense granular heap flow . . . . . . . . . . 24
2.5 Calculated dependence of the size of the flowing zone and the creeping
zone decay length on the volume flow-rate in steady, dense granular heap
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Configuration and flow threshold locus for inclined plane flow . . . . . . . 30
3.2 Configuration for planar shear flow and calculated local inertial rheology . . 34
3.3 Configuration for planar shear flow with gravity, dependence of the stress
ratio at the wall on wall speed, flow threshold locus, and steady velocity
fields in the quasi-static regime . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Configuration for annular shear flow, dependence of the stress ratio at the
wall on wall speed, flow threshold locus, and steady velocity fields in the
quasi-static regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Configuration and flow threshold locus for vertical chute flow . . . . . . . . 52
3.6 Comparison of the analytical flow thresholds predicted by the NGF model
for inclined plane flow, planar shear flow with gravity, annular shear flow,
and vertical chute flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 A representative sub-domain of a dense, bidisperse granular system con-
sisting of two-dimensional disks. . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Configuration for planar shear flow and calculated local inertial rheology
for dense, bidisperse granular systems . . . . . . . . . . . . . . . . . . . . 66
4.3 Binary diffusion coefficient versus γ˙d¯2 in DEM simulations of homoge-
neous, steady planar shearing of dense, bidisperse granular systems . . . . . 70
4.4 DEM simulation of bidisperse vertical chute flow . . . . . . . . . . . . . . 72
4.5 Collapse of diffusion flux versus segregation flux in quasi-steady vertical
chute flow of bidisperse disks . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 DEM simulation of bidisperse annular shear flow . . . . . . . . . . . . . . 77
4.7 Collapse of diffusion flux versus segregation flux in quasi-steady annular
shear flow of bidisperse disks . . . . . . . . . . . . . . . . . . . . . . . . . 79
xv
4.8 Comparisons of the coupled continuum model predictions with correspond-
ing DEM simulation results during transient flow and for five vertical chute
flow test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.8 Continued: Comparisons of the coupled continuum model predictions with
corresponding DEM simulation results during transient flow and for five
vertical chute flow test cases . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.9 Comparisons of the coupled continuum model predictions with correspond-
ing DEM simulation results during transient flow and for five annular shear
flow test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.9 Continued: Comparisons of the coupled continuum model predictions with
corresponding DEM simulation results during transient flow and for five
annular shear flow test cases . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.10 DEM simulation of bidisperse inclined plane flow . . . . . . . . . . . . . . 89
4.11 Collapse of the difference between diffusion flux and shear-rate-gradient-
driven segregation flux and the pressure-gradient-driven segregation flux in
quasi-steady inclined plane flow of bidisperse disks . . . . . . . . . . . . . 92
4.12 Comparisons of the coupled continuum model predictions with correspond-
ing DEM simulation results during transient flow and for five inclined plane
flow test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.12 Continued: Comparisons of the coupled continuum model predictions with
corresponding DEM simulation results during transient flow and for five
inclined plane flow test cases . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.13 DEM simulation of bidisperse planar shear flow with gravity . . . . . . . . 97
4.14 Collapse of the difference between diffusion flux and pressure-gradient-
driven segregation flux and the shear-rate-gradient-driven segregation flux
in quasi-steady planar shear flow of bidisperse disks with gravity . . . . . . 99
4.15 Comparisons of the coupled continuum model predictions with correspond-
ing DEM simulation results during transient flow and for five planar shear
flow with gravity test cases . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.15 Continued: Comparisons of the coupled continuum model predictions with
corresponding DEM simulation results during transient flow and for five
planar shear flow with gravity test cases . . . . . . . . . . . . . . . . . . . 101
4.16 Collapse of diffusion flux versus fluidity-gradient-driven or granular-temperature-
gradient-driven segregation flux in quasi-steady vertical chute flow and an-
nular shear flow of bidisperse disks . . . . . . . . . . . . . . . . . . . . . . 102
C.1 DEM simulation of planar shear flow of an initially-segregated bidisperse
granular system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
xvi
Chapter 1
Introduction
1.1 Background on dense granular materials and contin-
uum modeling
A granular material is a collection of discrete, macroscopic particles. Granular materials
are a common class of materials appearing in everyday life as well as playing a vital role in
a variety of industries, such as geophysical, pharmaceutical, food processing, and energy, in
forms of sand, soil, debris, pills and powders, as well as food grains (illustrated in Fig. 1.1).
In fact, granular material is second only to water as the most common industrial material
[1].
Due to their ubiquity and broad applications, the flow behavior of granular materials
has received much attention from both the scientific and industrial communities. One major
focus of the research on granular media is understanding the physics and mechanics behind
the many flow behaviors and then formulating models aimed at capturing the salient flow
features, which remains a challenge. The lack of predictive models for granular flow is
a long-standing and costly problem in engineering design involving granular materials,
and the benefits from improved granular flow modeling are abundant. For example, it is
estimated that 40% of capacity is wasted due to over-design of equipment and handling
1
2
Figure 1.1: Examples of granular materials and common applications: (a) geophysics, (b)
pharmaceutical industry, (c) Mars rover ”Spirit” in sandy terrain, and (d) food industry.
procedures in many industrial plants processing granular materials [2].
For a dry granular system, grain-grain interactions (repulsive forces) only occur when in
grains are in contact, and the evolution of the positions of constituent grains follows New-
ton’s Law without being affected by thermal fluctuations. From that perspective, one might
conclude that the behavior of an assembly of grains is very intuitive and deterministic, and
this first impression could lead one to model granular materials using the Discrete Ele-
ment Method (DEM) [3], which simulates the whole granular system particle-by-particle.
Indeed, DEM can serve as a good tool in many situations, but computation time quickly
becomes a bottleneck as the system size scales up, making it difficult to handle many real
industrial problems involving granular materials. Also, the predictivity of DEM is based
on the fidelity of the grain-grain contact model, and it is users’ responsibility to make sure
the applied model physically reflects the problem.
On the other hand, continuum models, which treat the granular media as a continuous
mass instead of considering discrete particles, are desirable for predicting granular flows
– just as for other materials in engineering design – since a continuum approach enables
numerical implementation utilizing the finite-element method (FEM), which is speedier
3
than DEM. In fact, researchers have been working on obtaining continuum models for the
mechanical behavior of granular materials for many years [4–6]. Some models do succeed
in predicting certain features but are often only applicable to specific flow conditions. Many
complicating factors arise, and an effective general model remains elusive. It turns out, even
for the simplest granular system - a collection of dry, hard spheres that are roughly all the
same size (monodisperse) - flow in a slightly complicated geometry can be very complex
and difficult to predict. For example, the inertial rheology [7–10] is a common and popular
continuum model that fully describes a monodisperse granular system in planar shear flow.
Nevertheless, issues such as nonlocality and size-effects [11–17] become apparent in other
flow conditions.
The situation becomes yet more complex in non-monodisperse granular systems, in
which the constituent particles can differ in size, shape, density, or surface properties, lead-
ing to segregation based on species during flow and resulting in strikingly complicated
segregation patterns [18]. Segregation during flow is a phenomena related to the “Brazil
nut effect,” which occurs when a non-monodisperse granular system is subjected to shak-
ing [19–21]. In industry, segregation can be a undesirable effect when uniform blending is
a necessary processing step in food, chemical, and pharmaceutical applications. Granular
size segregation is also an interested topic in geophysics, such as observations involving
sediment transport or landslides, in which larger grains segregate to the top of a flowing
granular media and end up doing the most damage.
1.2 Thesis contributions
The overarching motivation of this thesis is to develop a more general and easy-to-use
continuum model that can be applied to study the flow of dense, monodisperse granular
materials and the size-segregation in dense, bidisperse granular systems. Specific problems
addressed are summarized as follows.
4
1.2.1 The nonlocal granular fluidity model on steady, dense heap flows
Dense granular heap flows are common in nature, such as during avalanches and landslides,
as well as in industrial flows [7, 9, 12, 22–26]. In granular heap flows, rapid flow is local-
ized near the free surface with the thickness of the rapidly-flowing layer dependent on the
overall flow-rate. In the region deep beneath the surface, exponentially-decaying creeping
flow dominates with characteristic decay length depending only on the geometry and not
the overall flow-rate. Existing continuum models for dense granular flow based upon local
constitutive equations are not able to simultaneously predict both of these experimentally-
observed features – failing to even predict the existence of creeping flow beneath the sur-
face. In this work, we apply a scale-dependent continuum approach – the nonlocal gran-
ular fluidity (NGF) model [27, 28] – to steady, dense granular flows on a heap between
two smooth, frictional side walls. We show that the model captures the salient features of
both the flow-rate-dependent, rapidly-flowing surface layer and the flow-rate-independent,
slowly-creeping bulk under steady flow conditions.
1.2.2 Size-dependent flow threshold of dense granular materials
The flow threshold in dense granular materials is typically modeled by local, stress-based
criteria. However, grain-scale cooperativity leads to size effects that cannot be captured
with local conditions. In a widely studied example, flows of thin layers of grains down an
inclined surface exhibit a size effect whereby thinner layers require more tilt to flow [11].
In this work, we consider the question of whether the size-dependence of the flow thresh-
old observed in inclined plane flow is configurationally general. Specifically, we consider
three different examples of inhomogeneous flow – planar shear flow with gravity, annu-
lar shear flow, and vertical chute flow – using two-dimensional discrete-element method
(DEM) calculations and show that the flow threshold is indeed size-dependent in these
flow configurations, displaying additional strengthening as the system size is reduced. We
then show that the nonlocal granular fluidity model is capable of quantitatively capturing
5
the observed size-dependent strengthening in all three flow configurations.
1.2.3 Modeling size-segregation in dense, bidisperse granular flows
Dense granular systems, consisting of particles of disparate sizes, segregate based on size
during flow, resulting in complex, coupled segregation and flow patterns [18]. According
to current understanding, there are two driving mechanisms of size-segregation in dense
granular flows: (i) pressure-gradient (gravity) and (ii) shear-strain-rate-gradient. The mech-
anism of shear-rate-gradient-driven segregation has received less attention and, as such, is
not as well understood. In this work, we start by studying this segregation mechanism in
dense flows of bidisperse disks in two flow configurations that eliminates pressure-gradient-
driven segregation – vertical chute flow and annular shear flow. Specifically, we perform
two-dimensional discrete element method (DEM) simulations, which inform the develop-
ment of a continuum constitutive model for the segregation dynamics. Similar processes
are then repeated for two more configurations – inclined plane flow and planar shear flow
with gravity, in which both mechanisms coexist, thereby arriving at a complete segregation
model. Coupling this segregation model with the nonlocal granular fluidity (NGF) model,
we show that both the flow field and segregation dynamics may be simultaneously captured,
which no existing models have achieved.
1.3 Thesis outline
This thesis is outlined around the above three problems. Chapter 2 demonstrates the ne-
cessity of a nonlocal rheology, introduces the nonlocal granular fluidity (NGF) model in
detail, and applies it to the problem of heap flows. Another application of the NGF model
finds itself in Chapter 3, addressing the problem of the size-dependent flow threshold in
dense granular materials. In Chapter 4 we couple the NGF model with a segregation model
to fully describe the flow and segregation of dense, bidisperse granular mixtures. We close
6
in Chapter 5 with the major findings of the entire thesis and future research directions.
Chapter 2
Nonlocal continuum modeling of steady,
dense granular heap flows
Note: A version of this chapter is published in the Journal of Fluid Mechanics. Data and
figures have been used with all co-authors’ consent.
Daren Liu and David L. Henann. “Non-local continuum modelling of steady, dense gran-
ular heap flows.” Journal of Fluid Mechanics 831 (2017): 212-227.
2.1 Introduction
Dense granular flows display many manifestations of grain-size dependence in which coop-
erative effects at the microscopic grain-level have an observable effect on the macroscopic
flow phenomenology – such as wide shear bands in split-bottom flow [13–16], secondary
rheology of intruders [17, 29], and the thickness-dependence of the flow threshold in shal-
low inclined plane flow [11, 30, 31]. Another class of phenomenology affected by coop-
erativity is dense granular heap flow, such as the flows that arise during avalanches and
landslides, in flows down chutes, or in rotating drums. Dense granular heap flow is notable
because both rapid, fluid-like flow and slow, quasi-static flow are evident in a single con-
figuration. Steady flow fields are characterized by a rapidly-flowing layer localized at the
7
8
free surface with very slow flow beneath [7, 9, 12, 22–26]. The thickness of the rapidly-
flowing surface layer is strongly dependent on the overall flow-rate. Due to cooperative
effects, the rapid surface flow induces creeping flow beneath [12].1 The creeping flow de-
cays exponentially with the distance from the free surface. In contrast to the thickness of
the surface layer, the characteristic decay length of the creeping flow is independent of the
overall flow-rate [25].
Simultaneously capturing both the rapid surface flow and the quasi-static, creeping flow
with a continuum model has posed a significant challenge and remains an open research
question in granular physics. A well-regarded approach, which serves as our starting point,
is the inertial rheology [7–9, 32], which may be understood through basic dimensional
arguments. Consider a quasi-monodisperse granular system with mean grain diameter, d,
and grain material density, ρs , subjected to homogeneous planar shear with pressure, P,
and shear stress, τ. The consequent shear strain-rate, γ,
˙ may be non-dimensionalized as
I = γ˙ d 2 ρs /P – referred to as the inertial number. The inertial number operates as a
p
normalized strain-rate and represents the ratio of the microscopic timescale associated with
particle motion, d 2 ρs /P, to the macroscopic timescale of applied deformation, 1/γ˙ [7].
p
The local inertial rheology – local in the sense that it relates the local stress state to the
local state of strain-rate at a point – then asserts that the stress ratio, µ = τ/P, and the
inertial number, I, are related through a one-to-one constitutive relationship, µ = µloc (I).
A common functional form for the local inertial rheology – appropriate for modeling the
rapidly-flowing surface layers described above – was proposed by Jop et al. [9]:
µ2 − µs
µ = µloc (I) = µs + , (2.1)
I0 /I + 1
where µs = µloc (I → 0) is the static yield value, µ2 = µloc (I → ∞) is the maximum value
1 Following the terminology of Komatsu et al. [12], we use the term creep to describe very slow, quasi-
static flow.
9
of µ asymptotically approached as I increases, and I0 is a dimensionless parameter charac-
terizing the nonlinear, rate-dependent response.
The subsequent work of Jop et al. [10] generalized the local rheology (2.1) to three-
dimensional flow settings as follows.2 Denote the symmetric strain-rate tensor as γ˙i j =
(1/2)(∂ vi /∂ x j +∂ v j /∂ xi ) with vi the velocity vector and xi the spatial coordinate. Jop et al.
[10] began by assuming that steady, dense flow proceeds at approximately constant volume,
so that the strain-rate tensor is deviatoric and γ˙kk = 0. The equivalent shear strain-rate is
then defined as γ˙ = (2γ˙i j γ˙i j )1/2 . Next, to generalize the stress-related quantities involved
in the inertial rheology (2.1) to three dimensions, denote the symmetric Cauchy stress as
σi j = σ ji , and define the pressure P = −(1/3)σkk , the stress deviator σi0j = σi j + Pδi j ,
the equivalent shear stress τ = (σi0j σi0j /2)1/2 , and the stress ratio µ = τ/P. Jop’s three-
dimensional form of the local inertial rheology then takes the following form:
γ˙i j
σi j = −Pδi j + 2µloc (I)P , (2.2)
γ˙
which assumes that the strain-rate tensor and stress deviator are codirectional. When this
constitutive model is combined with the equations of motion, the resulting set of govern-
ing equations may be used to obtain predictions of steady, dense granular flow in general,
three-dimensional flow configurations and has been applied to a variety of problems, such
as heap flows [10, 25, 32], silo drainage [32–35], granular column collapse [35, 36], and
projectile impact into dense granular media [37]. In the context of steady, dense granular
heap flow, aspects of the rapidly-flowing surface layer may be accurately captured; how-
ever, as evidenced by the calculations of Jop et al. [10, 25] and Kamrin [32], the model also
predicts a static zone beneath the rapidly-flowing surface layer. This is because the consti-
tutive relation (2.2) possesses a yield condition in the form of a Drucker-Prager criterion
2 Notation: We use standard component notation, which supposes an underlying set of Cartesian basis
vectors {ei |i = 1, 2, 3}, and in which the components of vectors, v, and tensors, σ , are denoted by vi and σi j ,
respectively. The Einstein summation convention is employed, and the Kronecker delta, δi j , is utilized to
denote the components of the identity tensor.
10
[38], in which flow does not occur for µ < µs . Since µ < µs in the region deep beneath
the free surface in dense granular heap flow, the rheology (2.2) predicts that this region
is frozen, contrary to experimental observations of exponentially-decaying creep flow. In
addition to dense heap flow, a decaying flow profile is experimentally observed in creeping
regions where µ < µs across a variety of configurations, such as annular shear flow [39–
42], planar shear flow with gravity [43], and split-bottom flow [13–16]. When applied to
these cases, the inertial rheology (2.2) will continue to predict frozen regions with a sharp
flow cutoff occurring where µ = µs . The failure of the local inertial rheology to capture
creeping flow when µ < µs stems from the fact that it does not mathematically account for
nonlocal, cooperative effects, which become dominant in this regime.
Motivated by this shortcoming, in recent years, nonlocal continuum constitutive models
for dense granular flow have been developed, aimed at accounting for cooperative effects
[e.g., 44–54]. While these approaches incorporate nonlocality and some have had success
in individual geometries, none have been shown to capture the salient aspects of steady,
dense granular heap flow. Recently, Kamrin and Koval [27] proposed a nonlocal con-
tinuum model – called the nonlocal granular fluidity (NGF) model – inspired by related
modeling work in the emulsion community [55, 56], which is capable of quantitatively
describing a diverse set of slow, boundary-driven inhomogeneous flows [27, 28], the sec-
ondary rheology of intruders [57], and the size-dependence of the flow threshold in inclined
plane flow [58]. The purpose of this chapter is to show that the nonlocal granular fluidity
model is capable of capturing steady, dense granular heap flow – in particular, the coex-
istence of a rapidly-flowing surface layer and a creeping bulk. The chapter is organized
as follows. In Section 2.2, we discuss the specifics of the NGF model, and in Section
2.3, we apply the model to steady, dense granular flows on a heap between two smooth,
frictional side walls. Specifically, in Section 2.3.2, we compare velocity fields predicted
by the model with experimentally-measured flow fields from the literature for glass beads
[24, 25]. Importantly, the material parameters appearing in the NGF model have previously
11
been determined for glass beads [9, 28, 59], allowing the NGF model to be applied without
parameter adjustment. Then, in Section 2.3.3, we examine the relationships between total
flow-rate, maximum velocity, and surface inclination angle, and in Section 2.3.4, we ex-
plore the length scales that characterize the coexistent rapidly-flowing and creeping zones.
Throughout, our results demonstrate a level of agreement between model predictions and
experiments that has not been previously reported. We close with discussion and conclud-
ing remarks in Section 2.4.
2.2 The nonlocal granular fluidity model
In this section, we summarize the nonlocal granular fluidity (NGF) model for dense, steady
granular flow [see 27, 28, 57, 58, 60]. Similar to the local inertial rheology, the NGF
model is valid for granular systems consisting of grains that are (i) spherical, (ii) quasi-
monodisperse with constant mean grain diameter d, and (iii) stiff3 and is intended for
steady, well-developed flow conditions. The model consists of three main ingredients:
(i) a state parameter, called the granular fluidity, (ii) the flow rule, and (iii) the nonlocal
rheology, described in detail below.
1. Granular fluidity: Central to the model is the granular fluidity field – a positive,
scalar state variable, denoted as g. This state variable is kinematic in nature and
characterizes microscopic fluctuations in a flowing granular media. More precisely,
Zhang and Kamrin [64] and Bhateja and Khakhar [65] have established that the gran-
ular fluidity has a kinematic definition that holds across a variety of inhomogeneous
flow configurations for both two- and three-dimensional dense granular systems and
is given through the relation g = (δ v/d)F(φ ), where δ v is the velocity fluctuation,
φ is the solid volume fraction, d is the grain size, and F(φ ) is a function of only φ .
3 Incorporating particle stiffness into granular rheology is an open research problem. For recent progress
on this point, see the works of Campbell [61], Singh et al. [62], and Roy et al. [63].
12
2. Flow rule: For the case of homogeneous planar shear, the granular fluidity relates
the stress ratio, µ = τ/P, to the consequent shear strain-rate, γ,
˙ through the following
constitutive relation:
γ˙ = gµ. (2.3)
This one-dimensional version of the flow rule may be generalized to three dimen-
sions following a procedure analogous to that used by Jop et al. [10] to generalize
the inertial rheology, discussed in Section 2.1. We define the strain-rate tensor as
γ˙i j = (∂ vi /∂ x j + ∂ v j /∂ xi )/2 – where vi is the velocity field and xi is the spatial co-
ordinate. Next, we make the common approximation that well-developed, steady
flow proceeds at constant volume, so that γ˙kk = 0 [10, 32, 41, 66]4 and define the
equivalent shear strain-rate as γ˙ = (2γ˙i j γ˙i j )1/2 . The Cauchy stress tensor and stress
deviator are denoted as σi j = σ ji and σi0j = σi j − (1/3)(σkk )δi j . Then, the equivalent
shear stress, pressure, and stress ratio are defined through stress-tensor invariants
as τ = (σi0j σi0j /2)1/2 , P = −σkk /3, and µ = τ/P, respectively. Finally, assuming
that the strain-rate tensor and stress deviator are codirectional [10, 32, 66], the one-
dimensional flow rule (2.3) may be expressed in tensorial form as
P
σi j = −Pδi j + 2 γ˙i j . (2.4)
g
We note that codirectionality is an approximation, and slight deviations from codirec-
tionality have been observed in discrete element simulations – namely ∼ 1-2% non-
coaxiality between γ˙i j and σi0j [68] and the normal stress differences [69]. However,
codirectionality remains a useful simplifying assumption and enables a reasonably
good description of experimental data. For recent work incorporating these effects
into a rheological model, see Weinhart et al. [68].
4 Recent
work has shown that dilatancy can have an effect on steady flows, leading to secondary flow with
magnitude on the order of ∼ 5% of the primary flow [67]. Since this effect has not yet been reported for
steady, dense granular heap flow, we neglect it in the present work.
13
3. Nonlocal rheology: With the introduction of the granular fluidity field, g, an ad-
ditional field equation is required to relate the fluidity field to the stress field. In a
local constitutive approach, the fluidity would be given as an algebraic function of
the stress through the stress invariants µ and P. In contrast, in the nonlocal approach,
the granular fluidity, g, is governed by the following partial differential equation [60]:
" #
∂ 2g µ µ ρ
r
Dg − d 2
s s
t0 = A2 d 2 − ∆µ g+b µg2 , (2.5)
Dt ∂ xk ∂ xk µ2 − µ P
where D(•)/Dt is the material time derivative.5 In (2.5), t0 is a constant timescale
associated with the dynamics of g, A is a constant dimensionless material parameter
characterizing nonlocal effects called the nonlocal amplitude, and the constants ∆µ =
µ2 − µs and b = ∆µ/I0 are given through the three constant material parameters
{µs , µ2 , I0 } appearing in the local inertial rheology (2.1). The mean grain diameter
and grain material density continue to be denoted by d and ρs , respectively, and are
constants. During steady, homogeneous flow, the nonlocal rheology (2.5) reduces to
the local inertial rheology, (2.1) and (2.2). To see this, consider homogenous flow in
the absence of spatial gradients in g (∂ 2 g/∂ xk ∂ xk = 0) and for a fixed state of stress
(µ, P). Then, at sufficiently long time (t t0 ), g evolves to a stable Lagrangian
steady state (Dg/Dt ≈ 0), which we denote as gloc (µ, P). The nature of the stable
steady state, gloc (µ, P), depends on the sign of (µs − µ). For µ ≤ µs , the only steady
solution is gloc = 0, and it is stable under perturbation in g. For µ > µs , gloc = 0
remains a steady solution; however, it becomes unstable under perturbation in g.
The stable, steady solution for µ > µs is gloc = P/ρs d 2 I0 (µ − µs )/(µ(µ2 − µ)).
p
5 Note that the nonlocal rheology (2.5) shows mathematical similarities to the order-parameter-based rhe-
ological approach of Aranson and Tsimring [46].
14
Putting the stable solutions together, we have
P/ρs d 2 I0 (µ−µs ) if µ > µs ,
p
µ(µ2 −µ)
gloc (µ, P) = (2.6)
0 if µ ≤ µs .
We call the stress-dependent function gloc (µ, P) the local fluidity, since if the granular
fluidity, g, is taken to be given through the function gloc (µ, P) and combined with the
flow rule (2.4), we recover the local inertial rheology, (2.1) and (2.2). Then, in the
case of inhomogeneous flows involving spatial gradients in g, the Laplacian term
appearing in (2.5) introduces an intrinsic length scale given through d, and hence,
the rheology is nonlocal in character. In the present work, we are concerned with
dense granular flows that are steady in a Lagrangian sense, in which t t0 so that
Dg/Dt ≈ 0. However, reducing the dynamical PDE (2.5) to the steady-state case is
not as simple as setting the left-hand side to zero because the stability of the g = 0
solution depends on the sign of (µs − µ). In order to obtain a differential relation for
g specialized to the case of steady flow, we allow for gradients in g but limit attention
to small deviation of g from gloc for a given state of stress (µ, P). The result of such
a simplification (see Section 4.3.1 of Henann and Kamrin [60] for the details of this
calculation) is
∂ 2g
g = gloc + ξ 2 , (2.7)
∂ xk ∂ xk
where gloc (µ, P) is the local fluidity function, given through (µ, P) as in (2.6), and
ξ (µ) is a stress-dependent length-scale called the cooperativity length, given by
µ2 − µ
r
ξ (µ) = A d. (2.8)
∆µ|µ − µs |
15
Note that in the absence of any stress or flow gradients (∂ 2 g/∂ xk ∂ xk = 0), the steady-
state form (2.7) simply reduces to g = gloc (µ, P), and the local inertial rheology is re-
covered, as should be the case for homogeneous, steady flow. However, for inhomo-
geneous flow, the presence of the Laplacian term gives rise to nonlocal effects – with
predictions of the NGF model being size-dependent due to the cooperativity length
(2.8) being directly proportional to the grain size, d. In particular, in regions where
µ ≤ µs , the local fluidity (2.6) is zero, and (2.7) reduces to g = ξ 2 (∂ 2 g/∂ xk ∂ xk ),
which gives rise to creeping flow predictions with a decay length determined by ξ
rather than a sharp flow cutoff. In the simulations of Section 2.3, with a few noted
exceptions, we utilize the steady-state form of the NGF model, (2.7) with (2.6) and
(2.8), rather than the dynamical form (2.5). We note that the dimensionless parameter
A appearing in (2.8) is the only new material parameter introduced in the steady-state
form of the NGF model beyond the local parameters, {µs , µ2 , I0 }.
The system of equations described above are then closed by the standard equations of
motion
∂ σi j Dvi
+ φ ρs Gi = φ ρs (2.9)
∂xj Dt
with Gi the acceleration of gravity (denoted as G to differentiate it from the granular flu-
idity g) and φ the solid volume fraction, taken to be 0.62 for randomly-closed-packed,
quasi-monodisperse, spherical grains [70]. Consistent with our assumption of steady, well-
developed flow, we neglect the effect of macroscopic inertia, so that (2.9) reduces to
∂ σi j
+ φ ρs Gi = 0i . (2.10)
∂xj
The boundary conditions accompanying this boundary-value problem are discussed in the
next section in the context of the specific problem of steady, dense granular heap flow.
In previous work, the NGF model has been shown to quantitatively describe a broad
set of experimental steady flow data, including all variations of the complex split-bottom
16
family of geometries in addition to annular shear flow and planar shear flow with gravity
[28]. The model has also been shown to correctly capture other nonlocal phenomena such
as nonlocally-induced material weakening, i.e., “secondary rheology” [17, 29], whereby
the motion of a boundary removes the yield strength of the material everywhere, permitting
far-away loaded objects to creep through the grains when otherwise they would remain
static [57]. Finally, in Kamrin and Henann [58], the model has also shown itself able to
describe the size-dependent strengthening seen in experiments of gravity-driven flow down
a rough inclined plane [11, 30, 31] – a problem that we will consider further in Chapter 3.
When discussing the successes of the NGF model, it is also important to note its limits.
Although the NGF model can model developing flows and the approach to steady state, it
has not been designed to be quantitatively predictive in this regard. In particular, it is not
yet known whether the time-dependent term appearing in the dynamical form (2.5) is able
to quantitatively capture transient effects, such as those reported by Ries et al. [71]. Hence,
at present, the NGF model is only appropriate for steady-state conditions. It is also limited
to granular systems, in which grains are spherical, quasi-monodisperse, and stiff. In what
follows, we will apply the model to situations in which these assumptions are valid.
2.3 Numerical simulations of dense, steady granular heap
flow
In this section, we turn to applying the NGF model to the problem of dense, steady granular
heap flow. We will compare predictions of the NGF model to experiments in the literature
using glass beads [24, 25]. Regarding the dimensionless material parameters for glass
beads {µs , µ2 , I0 , A}, the local parameters are taken from well-established data [9, 59]:
µs = 0.3819, µ2 = 0.6435, and I0 = 0.279.
17
In previous work [28], the nonlocal amplitude for glass beads was calibrated to be
A = 0.48
by fitting NGF model predictions to experimental flow fields for dense, quasi-static granular
flow in the split-bottom cell [13, 14]. Since the heap flows reported by Jop and coworkers
[24, 25] also involve dense flows of spherical glass beads, we expect that these material
parameter values remain valid, and in what follows, we continue to use these parameters
without adjustment. Further, we take ρs = 2450 kg/m3 and d = 0.5 mm in our calculations;
however, all results in subsequent sections will be presented in dimensionless form, and
hence, the numerical values of ρs and d are inconsequential.
2.3.1 Flow configuration
To study gravity-driven, dense granular heap flow, we consider the system shown in Fig.
2.1(a), consisting of a granular layer of depth, H, at an inclination angle, θ , flowing be-
tween two smooth, frictional side walls separated by a distance, W . The coefficient of slid-
ing friction between the granular media and the side walls, µwall , is taken to be a constant
value and the floor is rough. The component of the acceleration of gravity in the y-direction
– G sin θ – then drives flow down the channel – which is very long in the y-direction. The
expected consequent steady-state flow field – which is invariant along the length of the
channel, i.e., the y-direction, and decays with depth beneath the surface, i.e., the z-direction
– is sketched schematically in Fig. 2.1(a). The total volume flow-rate of granular material
down the channel is denoted by Q.6 Note that in experiments the flow-rate, Q, is typically
prescribed, and the free surface inclination angle, θ , follows as a consequence [9].
To apply the NGF model to flow in this configuration, we take advantage of the invari-
ance of the steady flow field in the y-direction and make an antiplane shear idealization,
6 Inthe present work, we use Q to denote the total volume flow-rate, rather than the flow-rate per unit
width as in Jop et al. [9, 25].
18
(a) Frictional walls (b) W
W G x
x Simulated plane
z Free surface
y Free surface
z
H
Frictional
H
walls, µwall
vsurf (x)
vwall (z)
Rough bottom, v = 0
Rough bottom
Q
✓ Simulated plane
Figure 2.1: (a) Schematic and (b) computational domain for steady, dense granular heap
flow, following the flow configurations used in the experiments of Komatsu et al. [12] and
Jop et al. [9, 25].
in which the only non-zero component of velocity is in the y-direction, denoted by v, and
only depends on the x and z coordinates, i.e., v(x, z), thereby reducing the problem to two
dimensions. Under the antiplane shear assumption, the constitutive equation (2.4) implies
that the non-zero stress components are σxx = σyy = σzz = −P, σxy = (P/g)∂ v/∂ x, and
σyz = (P/g)∂ v/∂ z, and hence for steady (Dvi /Dt = 0i ) antiplane shear flow, the pressure
field is simply hydrostatic, P(z) = φ ρs G(cos θ )z, and the one remaining equilibrium equa-
tion (2.10) reduces to
∂ σxy ∂ σyz
+ + φ ρs G(sin θ ) = 0. (2.11)
∂x ∂z
The fields v(x, z) and g(x, z) may be solved through the coupled partial differential equa-
tions (2.7) and (2.11) along with appropriate boundary conditions. Solutions to this bound-
ary-value problem are obtained via the custom finite-element approach, described in Henann
and Kamrin [72] and implemented in the commercial finite-element program Abaqus [73].
In the implementation, a stiff elastic response is included, which does not affect the cal-
culated steady flow fields, and as such, the finite-element degrees-of-freedom are the dis-
placement in the y-direction – rather than v – and the granular fluidity. For details on
this point, see Henann and Kamrin [72]. A sample computational domain for the case of
W = 19d and H = 20d is shown in Fig. 2.1(b). Based on the mesh convergence study for
19
the finite-element implementation described in Henann and Kamrin [72], we choose a fine
mesh resolution of 0.125d to ensure accurate calculation results.
Regarding mechanical boundary conditions – indicated in Fig. 2.1(b) – the top surface
is traction-free, σyz (x) = 0 at z = 0, the side walls are frictional, σxy (z) ≤ −µwall σxx (z) =
µwall P(z) at x = 0 and x = W , and the bottom is rough, v(x) = 0 at z = H. Regarding the
frictional side wall boundary condition, as in the bulk, there is a stiff elastic component
to the wall slip, so that the shear traction on the granular media at a point on the sur-
face, σxy (z), may be less than µwall P(z); however, for steady wall slip, σxy (z) = µwall P(z).
Throughout, we take a constant value of the wall friction coefficient of µwall = 0.32, ob-
tained by treating µwall as an adjustable parameter in our subsequent comparisons with
experiments. More specifically, as will be pointed out in Section 2.3.3, µwall is determined
by fitting to the overall flow-rate data of Fig 2.4(a) rather than any specific flow field. Note
that since the material parameters {µs , µ2 , I0 , A} have been previously calibrated to experi-
mental flow fields in other flow configurations and subsequently fixed, µwall represents the
only adjustable parameter in our simulations of dense heap flow. This simple picture of
side wall friction is an idealization [see 74, 75]. In particular, the recent work of Artoni
and Richard [75] has shown that the sliding friction coefficient is not constant. Instead, it
depends on both the slip velocity as well as the velocity fluctuation, δ v. In spite of this
simplification, our comparisons with experiments will show that good agreement may still
be obtained under this idealization. For the granular fluidity boundary conditions, we in-
voke homogeneous Neumann boundary conditions on all boundaries. This is based on past
work, which has shown that such a boundary condition provides an excellent description
of experiments, provided that the size of the flow geometry is greater than approximately
10d (see Henann and Kamrin [72] for an expanded discussion of this point).
20
(a) W = 19d (b) W = 19d
p
v/ Gd x x
2.500
2.292 z z
2.083
1.875
1.667
1.458
1.250
1.042
0.833
0.625
0.417
0.208
0.000
✓ = 26.1 ✓ = 28.0
0 0
(c) (d)
5 5
z/d
Experiment - θ = 26.1◦
z/d
z/d
z/d
10 10
Experiment - θ = 28.0◦
NGF model - Eq. (2.5)
15 15
NGF model - Eq. (2.3)
Inertial rheology - Eq. (1.2)
20 20
0 0.5 1 √
p 1.5 2 10 -3 10 -2 p
-1
√ 10 10 0
vvwall // Gd
wall Gd vwall// Gd
vwall Gd
Figure 2.2: Flow fields in a narrow channel √ of width W /d = 19. Contour plots of the
normalized, steady-state velocity field, v/ Gd, in the x-z-plane for (a) θ = 26.1 and (b)
◦
θ = 28.0◦ calculated
√ using the steady-state NGF model (2.7). Normalized velocity field at
the wall, vwall / Gd, as a function of the depth beneath the free surface, z, for each surface
inclination angle, θ , in (c) linear and (d) semi-logarithmic scale. Symbols represent the
experimental data of Jop et al. [25]; solid lines are the calculated results of the steady-
state NGF model (2.7); dotted lines are the calculated steady results of the dynamical NGF
model (2.5); and dashed lines are the calculated results of the local inertial rheology (2.2).
2.3.2 Flow fields
In this section, we compare simulated flow fields with experiments. To do so, we must first
specify the geometry – most importantly, the dimensionless channel width, W /d. Through-
out, we take the depth of the granular layer, H/d, to be large enough that it does not affect
the flow field. Second, we must specify either the inclination of the free surface, θ , or the
√
volume flow-rate, non-dimensionalized as Q/W d Gd.7
Following the experiments of Jop et al. [25], we begin by considering narrow channels
of width W /d = 19 for two values of the free surface inclination angle – θ = 26.1◦ and θ =
28.0◦ – and calculate steady flow predictions using the steady-state form of the NGF model
(2.7) and the computational setup of Fig. 2.1(b). Contour plots of the normalized velocity
7 This dimensionless flow-rate is denoted as Q∗ in Jop et al. [9, 10].
21
√
field at steady state, v/ Gd, in the x-z-plane for each case are shown in Figs. 2.2(a) and
(b). As expected, flow is at its most rapid at the free surface with the velocity decaying into
the bulk (in the z-direction). In the cross-channel direction (the x-direction), the velocity
field has a clear bowed shape due to the frictional side walls. As is clear from the contour
plots of Figs. 2.2(a) and (b), flow for the θ = 28.0◦ case is considerably more rapid. For
the purpose of comparing to experiments, we introduce the wall velocity field, vwall (z), at
an x-position of x = d – one grain diameter inside the side wall. Figure 2.2(c) displays the
good quantitative agreement between the simulated wall flow fields (solid lines) and the
experimental data of Jop et al. [25] (symbols) for the two inclination angles considered. In
particular, the NGF model captures both the rapid flow near the free surface as well as the
exponentially-decaying, creeping velocity field deep beneath the surface – made clear in
the semi-logarithmic-scale plot of the wall velocity field in Fig. 2.2(d).
For comparison, we have also calculated steady flow predictions of the dynamical form
of the NGF model (2.5) in a narrow channel of width W /d = 19. To obtain steady flow
predictions for the dynamical form of the NGF model, an initial condition for the fluidity
field, g, is required due to the presence of the time derivative in (2.5). Since steady (t t0 )
solutions of the dynamical form of the NGF model (2.5) are expected to be similar to the
solutions of the steady-state NGF model (2.7), we utilize the respective solutions of the
steady-state form as the initial condition for the dynamical form for each case. Then, we
choose an arbitrary value of the timescale t0 and run the simulation to a final time of 1000t0
to ensure that the steady state is attained. The material parameters and boundary conditions
remain unchanged. The calculated steady wall velocity fields for the dynamical form of the
NGF model for the cases of θ = 26.1◦ and θ = 28.0◦ are included in Figs. 2.2(c) and (d) as
dotted lines. For both free surface inclination angles, the calculated steady predictions of
both forms of the NGF model are quite similar in both the rapidly-flowing surface layer and
the creeping bulk. Since solutions are much more easily obtained through the steady-state
form of the NGF model (2.7), it is typically preferable to use this form in practice.
22
As a second comparison, we have calculated steady flow predictions of the local in-
ertial rheology (2.2) in a narrow channel of width W /d = 19. Steady flow predictions of
the local inertial rheology (2.2) are obtained by setting the nonlocal amplitude to zero, i.e.,
A = 0, and leaving all other material parameters and boundary conditions unchanged. The
calculated wall velocity fields for the local inertial rheology for the cases of θ = 26.1◦
and θ = 28.0◦ are plotted in Figs. 2.2(c) and (d) as dashed lines. In the rapidly flow-
ing surface layer, the NGF model and the local inertial rheology offer similar flow field
predictions, which are both in agreement with experimental data. However, beneath the
surface, the local inertial rheology predicts a sharp flow cutoff, clearly observed in the
semi-logarithmic-scale plot of Fig. 2.2(d), whereas the NGF model is capable of capturing
the exponentially-decaying, creeping bulk. The NGF model clearly provides significantly
improved predictions of steady, dense granular heap flow fields compared to the local iner-
tial rheology.
Next, returning to the steady-state NGF model (2.7), we consider a wider channel with
W /d = 142 and compare to the experimental data of Jop [24]. For this set of experiments,
√
the flow-rate, Q/W d Gd, was reported rather than the inclination angle, θ . Accordingly,
√
we consider flow-rates of Q/W d Gd = 4.4, 15, 40, and 91.8 Contour plots of the normal-
√
ized, steady-state velocity field, v/ Gd, in the x-z-plane are shown in Fig. 2.3(a) and (b)
√
for the least rapid and most rapid flow-rates, Q/W d Gd = 4.4 and 91, respectively. Again,
we observe that the velocity decays with the depth beneath the surface and is greater in the
center of the channel than at the walls. In the contour plots, it is clear that the size of the
rapidly-flowing surface layer is larger for the higher flow-rate. Figure 2.3(c) then shows
the comparison of the calculated wall velocity field, vwall (z), to the experimental data of
Jop [24] for all four volume flow-rates, and again the quantitative agreement is good both
for the rapid flow near the surface as well as the creeping bulk. Defining the surface flow
8 In
our calculations, the inclination angle, θ , is prescribed, so for the W /d = 142 simulations, θ is itera-
tively adjusted to achieve the desired volume flow-rate. We utilize a channel depth of H/d = 140 and a mesh
resolution of 0.5d.
0.5 23
5.5
p p
(a) v/ Gd W = 142d (b) v/ Gd W = 142d
0.500 x 5.500 x
0.458 5.042
0.417 4.583
0.375 4.125
0.333 z 3.667 z
0.292 3.208
0.250 2.750
0.208 2.292
0.167 1.833
0.125 1.375
0.083 0.917
0.042 p 0.458 p
0.000 Q/W d Gd = 4.4 0.000 Q/W d Gd = 91
0 6
(c) (d)
p
10 p 5 Q/W d Gd = 91
Q/W d Gd = 4.4
Gd
20 4
vsurf/ Gd
p
Q/W d Gd = 40
p
z/d
z/d
√
30 3
vsurf
40 2 p
Q/W d Gd = 15
p
Q/W d Gd = 91 p
50 1 Q/W d Gd = 4.4
60 0
10 -4 -2
10√p 10 0 0 20 40 60 80 100 120 140
vwall Gd
vwall / Gd x/d
x/d
Figure 2.3: Flow fields in a wide channel √ of width W /d = 142. Contour plots
√ of the
normalized, steady-state
√ velocity field, v/ Gd, in the x-z-plane for (a) Q/W d Gd = 4.4
and (b) Q/W d Gd √= 91 calculated using the NGF model. (c) Normalized velocity field
at the wall, vwall / Gd, as a function of the depth√ beneath the free surface, z, and (d)
normalized velocity√ field on the surface, vsurf / Gd, as a function of x for each volume
flow-rate, Q/W d Gd. Symbols represent the experimental data of Jop [24], and solid
lines are the calculated results of the steady-state NGF model.
field as vsurf (x, z = 0), Fig. 2.3(d) shows that the calculated surface flow field is also in
agreement with experiments – both at the wall and in the center of the channel. Overall, the
NGF model is able to capture the salient aspects of the flow fields in steady, dense granular
heap flow over a range of channel widths.
2.3.3 Volume flow-rate and maximum velocity
Additional insight may be obtained by comparing the predicted relationships between
global quantities – such as the relationships between volume flow-rate, Q, maximum veloc-
ity, vmax = v(x = W /2, z = 0), and free surface inclination angle, θ – to experiments. Before
making such comparisons, we first consider some consequences of dimensional analysis.
The dense granular heap flow configuration of Fig. 2.1(a) relates the following quantities
– W , d, ρs , ρs G, and θ – to the consequent volume flow-rate, Q, and maximum velocity,
24
(a) 0.6 (b) 10 -1
NGF model - W/d = 19 NGF model - W/d = 19
NGF model - W/d = 142 NGF model - W/d = 142
0.55 Experiment - W/d = 19 Experiment - W/d = 19
Experiment - W/d = 38 Experiment - W/d = 57
Experiment - W/d = 57 10 -2 Experiment - W/d = 142
vvmax
tan θ✓
max
Experiment - W/d = 142
tan
0.5
0′
0.45
10 -3
0.4
10 -4 10 -3 10 -2 10 -4 10 -3 10 -2
Q′0
Q Q′0
Q
Figure 2.4: Relationships between (a) free surface inclination angle, θ , and dimensionless
volume flow-rate, Q0 , and (b) dimensionless maximum velocity, v0max , and dimensionless
flow-rate, Q0 , for steady, dense granular heap flow. Symbols represent the experimental
data of [9], and lines are calculated predictions of the steady-state NGF model.
vmax . However, the analysis of Jop et al. [9] showed that the scaling suggested by the local
inertial rheology (2.1) could account for the observed scalings between Q, vmax , and θ , and
importantly, in the local inertial rheology, the grain size does not appear independently –
instead combined with the grain material density as ρs d 2 . Consequently, the reduced set
of quantities – W , ρs d 2 , ρs G, and θ – should give Q and vmax . Straightforward dimen-
sional analysis leads to the conclusion that the local inertial rheology suggests a one-to-one
relationship between the following three dimensionless quantities:
Qd vmax d
Q0 = , v0max = , and θ .
G1/2W 7/2 G W 3/2
1/2
Such a collapse was observed in experimental data by Jop et al. [9], and their data in the
form tan θ versus Q0 and v0max versus Q0 is reproduced in Figs. 2.4(a) and (b), respectively,
for several channel widths. Also plotted in Fig. 2.4 are the respective calculated relation-
ships obtained from the NGF model for W /d = 19 and 142 – both a narrow and a wide
channel. We note that the constant wall friction coefficient, µwall , was determined by fitting
to the data of Fig. 2.4(a), rather than by fitting to individual flow fields of Section 2.3.2. We
observe first that agreement with experiments may be obtained as a result. Second, while
the local inertial rheology predicts a perfect collapse between Q0 , v0max , and θ , the collapse
for the NGF model is approximate with the differences between the simulated relationships
25
(a) (b)
5 W/d = 19 2.6
2.4
scale
2.2
Length scale
4 h/d 2
/d
λ/d
Length
1.8
3
1.6
2 /d 1.4
1 1.2
10 -4 10 -3 10 -2 10 1 10 2
Q′0
Q W/d
W/d
Figure 2.5: (a) Calculated dependence of the size of the flowing zone, h/d, and the creeping
zone decay length, λ /d, on the dimensionless volume flow-rate, Q0 , for W /d = 19. (b)
Relationship between creeping zone decay length, λ /d, and the channel width, W /d. The
dashed line represents a 1/4 power-law included for reference
for the two channel widths being small, indicating that nonlocal effects – i.e., the presence
of the creeping bulk – are secondary in these scaling relationships. Therefore, the NGF
model is able to account for the most significant deficiency in the predictions of the local
inertial rheology applied to steady, dense heap flow – i.e., the existence of the creeping
bulk – while approximately maintaining its greatest success – i.e., the scalings between Q0 ,
v0max , and θ .
2.3.4 Two length scales
Finally, to explore the qualitative difference between the rapidly-flowing surface layer and
the creeping bulk, we introduce two characteristic length scales. Following Jop et al. [25],
the rapidly-flowing surface layer is characterized by the mean flow depth, h, defined as
h = Q/W vmax . The creeping bulk is characterized by the decay length of the exponential tail
in the mid-channel velocity field, λ , defined through v(x = W /2, z) ∼ exp(−z/λ ) for large
z. Figure 2.5(a) shows both h/d and λ /d versus the dimensionless flow-rate, Q0 , for the
calculated results of the NGF model for W /d = 19. Importantly, h is flow-rate-dependent
while λ is flow-rate-independent, which agrees both qualitatively and quantitatively with
experiments (see Fig. 4(b) of Jop et al. [25] in which λ /d extracted from the wall flow field
is approximately 2 for W /d = 19). This observation may be understood mathematically
26
as follows. In the rapidly-flowing layer where µ > µs , the local contribution to (2.7),
gloc , dominates, introducing the rate-dependence of the local rheology. Conversely, in the
creeping bulk where µ < µs , gloc = 0, rendering (2.7) rate-independent [27] and leading
to the rate-independence of λ . Regarding the dependence of the length scales h and λ on
the channel width W , since h is defined through Q and vmax , its scaling may be understood
through the arguments of Section 2.3.3; however, due to the rate-independence of λ , we
expect that λ /d will only depend upon W /d (see Figure 2(b) of Henann and Kamrin [28]
for the analogous relationship between a flow length scale and geometry in split-bottom
flow). Figure 2.5(b) shows λ /d versus W /d calculated by the NGF model for channel
widths ranging from W /d = 12 to 142. The observed relationship appears close to a 1/4
power-law (plotted as a dashed line for reference); however, appropriate experimental data
to verify this point has not been reported.
2.4 Concluding remarks
In this chapter, we have shown that the NGF model is capable of capturing several im-
portant experimentally-observed aspects of steady, dense granular heap flow, including (i)
wall and surface flow fields for a range of channel widths; (ii) the relationships between
volume flow-rate, maximum velocity, and surface inclination; and (iii) the rate-dependent
thickness of the rapidly-flowing surface layer and the rate-independent decay length of the
exponentially-decaying velocity field in the creeping bed. Importantly, the NGF model
offers improved flow predictions over the local inertial rheology while involving only one
new dimensionless material parameter, the nonlocal amplitude A. In previous work, the
value of A for glass beads was determined by applying the NGF model to slow boundary-
driven flows [28], and in the present work, we have continued to use this numerical value
and demonstrated that good agreement with experiments is maintained in the case of steady,
dense granular heap flow.
27
There remain several avenues for improvement and refinement. A major idealization of
our modeling work is treating the friction coefficient between the granular material and the
side walls as constant – instead of dependent on the slip velocity and velocity fluctuation
as shown by Artoni and Richard [75]. The recent work of [64] relating the granular fluidity
to the velocity fluctuation suggests a path forward on this point, in which the wall friction
coefficient is taken to be dependent on both the slip velocity and the granular fluidity.
We have limited attention to the case of steady, well-developed flow. To apply the NGF
model to transient, developing flows, the incorporation of additional physics is required.
The work of Jop et al. [25] showed that accounting for macroscopic inertia in conjunction
with the local inertial rheology was sufficient to obtain a reasonable description of the
approach to steady state in dense granular heap flow, quantified by the overall flow rate and
free surface velocity as a function of time. Since the inertial rheology and the NGF model
provide comparable predictions in the rapid flow regime, if macroscopic inertia effects are
reintroduced to (2.11), we expect to obtain similar results with the NGF model. Transient
effects are not solely due to macroscopic inertia. For example, Reynolds dilation during
shear initiation induces transient variations in the flow resistance, which are commonly
described using critical-state models [76] but which are not yet included in the NGF model.
Further, Ries et al. [71] recently reported an exponential-type transient during the initiation
of dense simple shearing of spheres, and it is not yet known whether this transient may be
captured using the time-dependent, dynamical form of the nonlocal rheology (2.5).
In closing, the ability of the NGF model to capture steady, dense granular heap flow
in a channel is an encouraging indication that the model might successfully be applied to
other types of surface flow, such as the flows that arise in rotating drums. The essential
physics of steady, dense granular flow in a rotating drum is expected to be the same as that
in channel flows [7], but the geometric complexity of the rotating drum configuration gives
rise to additional characteristics in the flow field, such as the S-shape of the free surface.
Applying the NGF model to steady, dense flows in a rotating drum will possibly require
28
more elaborate numerical techniques than the finite-element-based approach, such as the
material point method [35, 37], and is a task which is left to future work.
Chapter 3
Size-dependence of the flow threshold in
dense granular materials
Note: A version of this chapter is published in Soft Matter. Data and figures have been
used with all co-authors’ consent.
Daren Liu and David L. Henann. “Size-dependence of the flow threshold in dense granular
materials.” Soft Matter (2018).
3.1 Introduction
Recall from the discussion of Section 2.1 that the local inertial rheology relates the inertial
number I and the stress ratio µ through a one-to-one functional relationship µ = µloc (I),
which is empirically fit. A common feature among different functional forms of the inertial
rheology is a static yield value of the stress ratio – µloc (I → 0) = µs . Consequently, the
inertial rheology possesses a flow threshold in which steady flow is not possible when
µ < µs but steady flow becomes possible whenever µ exceeds µs .1
1 In the common generalization of the inertial rheology to three-dimensions [10, 32], the flow threshold
takes the form of a Drucker-Prager yield condition [38].
29
30
(a) (b) 16
H 14
G 12
z 10
/d
s t o p/d
Hstop
8
H
6 Flow
4
q 2 No flow
0
18 20 22 24 26 28 30
qθ (degrees)
( d e gr e e s)
Figure 3.1: (a) Configuration for inclined plane flow. (b) Theoretically predicted flow
threshold locus (–) (3.8) as calibrated for glass beads on a fully rough base, compared
to experimentally determined values (◦) from Pouliquen [11]. Figure reproduced from
Kamrin and Henann [58]
A logical next step is to apply this flow threshold to more complex flow configura-
tions. A dense granular flow configuration that is more complex than homogeneous, planar
shear – but still quite simple – is flow down a rough inclined surface, shown schemat-
ically in Fig. 3.1(a). In inclined plane flow, the ratio of the shear stress to the pres-
sure is a constant value at every point in the granular layer – as in planar shear – and
is given through the inclination angle θ by µ = tan θ . Therefore, the flow threshold as-
sociated with the inertial rheology predicts that flowing and non-flowing states are sepa-
rated by a thickness-independent angle of repose θr = tan−1 µs . However, extensive experi-
ments and discrete-element method (DEM) simulations have shown that this is not the case
[7, 11, 30, 31, 59, 77]. Instead, thin granular layers do not flow for a range of θ greater
than θr with additional strengthening as the layer thickness is decreased. This observation
is illustrated in Fig. 3.1(b) for experimental measurements of glass beads on a fully rough
base, reported by Pouliquen [11].
The inability of the inertial rheology to capture this size effect stems from its local
nature – local in the sense that it relates the stress state to the strain-rate at a point. The
size-dependence of the flow threshold in inclined plane flow arises due to nonlocal, coop-
erative effects at the microscopic grain level, which are not accounted for in the inertial
31
rheology. In a thin granular layer, the proximity of the grains to the fixed, rough, inclined
surface imbues the granular layer with additional strength. The converse manifestation of
cooperativity may also be observed. Flow in one region of a granular medium can induce
flow in far-away regions – even when these regions experience stress states that are beneath
the flow threshold extracted from homogeneous, planar shearing. For example, as dis-
cussed in Chapter 2, in gravity-driven heap flow, a decaying flow field is observed, whereas
the stress-based flow threshold of the inertial rheology would predict a sharp flow cutoff.
An additional example of this effect is the “secondary rheology” of intruders, whereby the
motion of a boundary removes the flow threshold of the material everywhere, permitting
far-away loaded objects to creep through the grains when otherwise they would remain
static [29].
While the effect of cooperativity on steady flow fields has been investigated in a diverse
set of geometric configurations, studies exploring the size-dependence of the flow threshold
in dense granular materials have been limited to inclined plane flow. To better elucidate the
role of the stress field, the first purpose of this chapter is to systematically explore the
size-dependence of the flow threshold in different flow configurations with more complex
stress fields. Specifically, we consider dense two-dimensional flows of stiff, frictional disks
using DEM simulations in three flow configurations: (1) planar shear flow with gravity, (2)
annular shear flow, and (3) vertical chute flow, and show that additional strengthening is
observed as the system size is reduced in all three cases.
The second purpose of this chapter is to rationalize the dependence of the flow threshold
on the system size in the presence of different stress fields with the nonlocal granular flu-
idity (NGF) model. Among the various nonlocal continuum constitutive theories proposed
in the literature, several have been applied to the flow threshold in inclined plane flow,
such as integral equations representing a self-activated process [45]; Ginzberg-Landau the-
ories based on a partial-fluidization order parameter [46] or the inertial number [54]; and
extensions of kinetic theory [48]. The NGF model has also been shown to be capable of
32
capturing the size-dependence of the flow threshold in inclined plane flows of glass beads
[58] – as illustrated in Fig. 3.1(b). In this chapter, we utilize the NGF model to obtain
predictions of the size-dependence of the flow threshold in planar shear flow with gravity,
annular shear flow, and vertical chute flow. Specifically, in each case, we calculate the an-
alytical flow threshold predicted by the NGF model. Importantly, we show that the NGF
model is capable of quantitatively describing the observed size-dependent strengthening in
all three flow configurations, while simultaneously capturing steady flow fields.
The remainder of this chapter is organized as follows. In Section 3.2, we discuss the
specifics of our two-dimensional DEM simulations and verify our simulations against ex-
isting DEM data for stiff, frictional disks in planar shear flow from the literature [8, 78]. In
Section 3.3, we discuss the NGF model and its attendant description of the flow threshold.
Then, in Section 3.4, we present the results of our DEM simulations in planar shear flow
with gravity, annular shear flow, and vertical chute flow along with the predictions of the
NGF model, comparing predictions of both the size-dependent flow threshold and steady
flow fields to DEM data. We close with discussion and concluding remarks in Section 3.5.
3.2 Discrete-element method simulations
In this section, we provide details of our two-dimensional DEM methodology and briefly
describe simulations of planar shear flow in order to verify our simulations against existing
literature data [8, 78].
3.2.1 Simulated granular system
Following several previous works [8, 27, 41, 78], we consider a simulated, two-dimensional,
quasi-mondisperse granular system consisting of a dense collection of dry, stiff, frictional,
circular disks.2 The average disk diameter and the grain-material area-density are denoted
2 This is in contrast to the discussion of Chapter 2, which focused on dense collections of spheres.
33
as d and ρs , respectively, so that we may define a characteristic grain mass as m = ρs πd 2 /4.
In our DEM simulations, the distribution of disk diameters involves a polydispersity of
±20% to prevent crystallization. In the context of disks, the inertial rheology then relates
the stress state – specifically, the pressure P and the shear stress τ (both with units of force
per length in two-dimensional settings) – to the consequent shear strain-rate γ.
˙ Nondimen-
sionalizing the aforementioned quantities gives the inertial number I = γ˙ m/P and the
p
stress ratio µ = τ/P. While the form of the inertial number is slightly modified for two-
dimensional disks when compared to the form used for three-dimensional spheres, it still
represents the ratio of the microscopic time-scale associated with particle motion m/P
p
to the macroscopic time-scale of applied deformation 1/γ.
˙
We utilize a standard DEM grain interaction model [8]. Specifically, there is no force
between non-overlapping grains, but when two grains overlap, they interact through a
spring/dashpot contact law that accounts for elasticity, damping, and sliding friction. With
δn ≥ 0 and δt denoting the normal and tangential components of the contact displace-
ment, the normal contact force Fn is given linearly through the normal contact displace-
ment with stiffness kn and the relative normal velocity with damping coefficient gn , i.e.,
Fn = kn δn + gn δ˙n . The normal damping coefficient is specified through the coefficient of
√ q
restitution for binary collisions e by gn = mkn (−2 ln e)/ 2(π 2 + ln2 e). Tangential inter-
actions are described by a stiffness kt and damping coefficient gt , which we take to be zero,
so that the tangential contact force is Ft = kt δt . Importantly, the tangential contact force is
limited by Coulomb friction, described by the inter-particle sliding friction coefficient µsurf .
Therefore, grain interactions are fully described through the parameter set {kn , kt , e, µsurf }.
In order to simulate stiff, quasi-rigid grains, the normal stiffness is taken to be sufficiently
large throughout, i.e., kn /P > 104 , where P is the characteristic confining pressure for a
given flow configuration. Next, it is well-appreciated that the precise values of kt /kn and e
have a negligible impact on the phenomenology of dense flows of stiff disks [8], and as in
34
(a) Pwall (b) 0.5
vwall
moving wall 0.4
x
z 0.3
H = 60d
µ
flowing
0.2 DEM – da Cruz et al. (2005)
grains
DEM – Kamrin and Koval (2014)
v(z) 0.1 DEM – present work
Linear fit – µloc(I) = µs + bI
fixed wall 0
0 0.05 0.1 0.15 0.2
L = 60d I
Figure 3.2: (a) Configuration for two-dimensional DEM simulations of planar shear flow.
Black grains denote rough walls, and gray grains denote flowing grains. (b) The local iner-
tial rheology – µ versus I – for frictional disks with inter-particle sliding friction coefficient
of µsurf = 0.4. Black symbols denote the DEM data of da Cruz et al. [8] and Kamrin and
Koval [78], and gray symbols denote the DEM data of the present work. The solid line
denotes the fit of (3.1) with µs = 0.272 and b = 1.168.
previous works [41], we take kt /kn = 1/2 and e = 0.1. Finally, among the interaction pa-
rameters, µsurf plays the most important role [78]. Exploring the effect of µsurf on the flow
threshold is beyond the scope of the present work, so we restrict attention to the case of
µsurf = 0.4. The equations of motion for each particle are solved using standard molecular
dynamics techniques using the open-source software LAMMPS [79]. For the most part,
q
we restrict the time step to be 0.01 of the binary collision time, τc = m(π 2 + ln2 e)/4kn ,
to ensure stable, accurate simulation results.3
3.2.2 Planar shear flow
First, we perform simulations of planar shear flow in order to verify our DEM results
against existing data reported in the literature. We consider a configuration consisting
of a rectangular region of length L = 60d in the x-direction and height H = 60d in the
z-direction that is filled with a dense collection of 3806 flowing grains and subjected to
shearing through the relative motion of two parallel, rough walls, as shown in Fig. 3.2(a).
3Alarger time step is used for certain DEM simulations of planar shear flow and planar shear flow with
gravity to save computation time. The time step is never taken to be greater than 0.1 of τc and has been
verified to not affect results.
35
The dense granular system is generated by allowing grains to sediment under the action of
gravity, but gravity is absent in subsequent simulations of planar shear. Each of the two
rough walls on top and bottom consists of a thin layer of touching glued grains, which are
denoted as black in Fig. 3.2(a), while the flowing grains between the walls are denoted
as gray. Regarding wall conditions, the bottom wall is fixed, and the velocity of the top
wall in the x-direction is specified to be vwall . Following da Cruz et al. [8] and subsequent
works [41, 78], the velocity of the top wall in the z-direction is not zero – i.e., H is not
fixed. Instead, in order to maintain a target wall normal stress of σzz (z = 0) = −Pwall , the
vertical position of the wall is continuously adjusted so that the value of H evolves through
H˙ = (−σzz (z = 0) − Pwall )L/gp , where gp is a damping parameter for vertical wall mo-
√
tion. Throughout, we take gp = 100 mkn . Periodic boundary conditions are utilized in the
x-direction.
Next, we extract steady velocity and stress fields from DEM simulations of planar shear
flow for a range of wall velocities, vwall , and fixed wall pressure, Pwall . To ensure that steady
flow is achieved, each simulation of planar shear is first run to a top-wall shear displacement
of at least 400H.4 Then, we consider 1000 system snapshots uniformly distributed in time
over an additional top-wall shear displacement of 250H. Anticipating that the velocity and
stress fields are homogeneous along the x-direction, we average along the x-direction at
discrete z-positions for each snapshot. We utilize the spatial averaging technique described
by Koval and coworkers [41, 78], which is briefly summarized in Appendix B.1. The
instantaneous velocity and stress fields are then arithmetically averaged over all snapshots
to obtain steady fields that depend only upon the z-coordinate. In all cases of planar shear
flow, the steady velocity field vx (z) is linear with very little wall slip, allowing us to define
a corresponding, spatially-constant shear strain-rate γ˙ = |dvx /dz|. All stress components
are spatially constant as well. Moreover, the normal stresses σxx and σzz are approximately
equal. Therefore, the shear stress and pressure may be denoted as τ = |σxz | = |σzx | and P =
4 Experience tells us that flow typically reaches steady state within a top-wall shear displacement of 100H
after a short transient.
36
−σzz ≈ −σxx , respectively, and we may calculate the inertial number I = γ˙ m/P and stress
p
ratio µ = τ/P corresponding to each prescribed wall velocity. The relationship between µ
and I extracted from our DEM simulations is plotted in Fig. 3.2(b) as gray symbols, along
with the DEM data of da Cruz et al. [8] and Kamrin and Koval [78] for stiff, frictional
disks with µsurf = 0.4 as black symbols. The DEM results are consistent, verifying our
methodology. The DEM data for two-dimensional granular systems consisting of disks
may be fit by a simple Bingham-like functional form of the inertial rheology [8]:
µloc (I) = µs + bI, (3.1)
where µs and b are dimensionless material parameters. This functional form for µloc (I)
is simpler than the nonlinear form (2.1) of Jop et al. [9] used in Chapter 2 and is suitable
under conditions in which µ does not increase substantially above µs . The relation (3.1)
– using fitted parameter values of µs = 0.272 and b = 1.168 – is plotted in Fig. 3.2(b),
demonstrating that the linear form (3.1) captures DEM data for homogeneous planar shear.
3.3 Granular rheology and the flow threshold
In this section, we discuss the nonlocal granular fluidity model for steady, dense granular
flow and its attendant description of the flow threshold. Recall from Chapter 2 that in the
NGF model, a positive, scalar field quantity – the granular fluidity – is introduced and
denoted as g. Then, instead of relating the stress state to the strain-rate through a single
constitutive equation as in the inertial rheology (3.1), the NGF model relates the stress state,
the strain-rate, and the granular fluidity through two constitutive equations as follows:
γ˙ = gµ, (3.2)
Dg ∂ 2g m 2
r
t0 = A2 d 2 − (µs − µ)g − b µg , (3.3)
Dt ∂ xk ∂ xk P
37
where t0 > 0 is a constant timescale associated with the dynamics of g, A > 0 is a constant
dimensionless material parameter characterizing nonlocal effects, called the nonlocal am-
plitude, and the dimensionless constants µs and b are the same as those appearing in the
local inertial rheology (3.1). The duel constitutive roles of the granular fluidity become
clear in (3.2) and (3.3). First, in (3.2), g operates as a fluidity-like quantity, relating the
stress ratio µ to the consequent shear strain-rate γ,
˙ and second, in (3.3), g operates as a
nonlocal order parameter governed by a dynamical system. When the flow field is homo-
geneous, the granular fluidity evolves to the stable, steady, stress-dependent solution of the
dynamical system (3.3), which is given by
q
P (µ−µs )
m bµ if µ > µs ,
gloc (µ, P) = (3.4)
0 if µ ≤ µs ,
and referred to as the local fluidity. When the granular fluidity is given through (3.4) and
combined with (3.2), the local inertial rheology (3.1) is recovered, and hence, the NGF
model reduces to the inertial rheology for steady, homogeneous flow. However, when flow
gradients are present, the Laplacian term in (3.3) introduces an intrinsic length-scale given
through the grain size d, and the NGF model produces nonlocal predictions.
Several comments on the dynamical relation (3.3) are in order:
1. As discussed in Henann and Kamrin [60], the last two terms in (3.3) arise through
the derivative of a coarse-grain Ginzburg-Landau-type free energy. Since these terms
then determine the stable, steady solution for g in the absence of flow gradients – i.e.,
the local fluidity (3.4) – the choices of the Ginzburg-Landau-type free energy and the
fitting function for the local inertial rheology are one and the same. Hence, if one
were to utilize a different functional form for the inertial rheology than the Bingham-
like form (3.1) – as we did in Chapter 2 – it would be necessary to modify these
terms in (3.3). Indeed, this is the reason that different forms for the last two terms are
adopted in (2.5) and (3.3). In previous work applying the NGF model to dense flows
38
of spheres down inclines [58], the nonlinear fitting function (2.1) of Jop et al. [9]
was used, and hence, the dynamical relation took on the form (2.5). In this chapter,
focusing on two-dimensional granular systems of disks, the Bingham-like form of
the inertial rheology works well up to a stress ratio of 0.5 (see Fig. 3.2(b)), so we
utilize the corresponding dynamical form (3.3), while limiting attention to situations
in which the stress ratio remains less than 0.5.
2. The dynamics embodied by (3.3) does not involve bistability, in which two stable,
steady-state values of g exist for some range of µ. This feature would lead to a non-
monotonic local rheology and hysteresis of the predicted flow threshold. There is
some experimental evidence for a non-monotonic local rheology in dense flows of
spheres [80]; however, non-monotonicity is not observed in our DEM simulations of
disks, so we neglect the possibility of this effect in the present work.
3. The time-dependent term appearing in (3.3) is not intended to quantitatively describe
the approach to steady state, such as the transient variations in flow resistance that
accompany Reynolds dilatancy or the transient effects reported by Ries et al. [71].
However, the model does provide an accurate description of the long-term dynam-
ical behavior – i.e., steady flow – as well as the flow threshold. In these cases, the
numerical value of the positive parameter t0 appearing in (3.3) is irrelevant.
4. The differential relation (3.3) may be reduced to the steady-state-only form under the
approximation that deviations of the granular fluidity g from the local fluidity gloc are
small. The result of such an approximation – the details of which are discussed in
Henann and Kamrin [60] – is
2 ∂ 2g Ad
g = gloc + ξ with ξ (µ) = p , (3.5)
∂ xk ∂ xk |µ − µs |
where gloc (µ, P) is the local fluidity function (3.4) and ξ (µ) is the stress-dependent
39
cooperativity length. The steady-state form of the NGF model may be straightfor-
wardly applied to obtain accurate predictions of non-uniform steady flow fields in
a variety of geometric configurations – as illustrated in Chapter 2 in the context of
dense heap flows. However, (3.5) cannot capture the size-dependence of the flow
threshold. To see this, note that the local fluidity function (3.4) mathematically acts
as a source term in (3.5) and is non-zero whenever µ > µs . Then, when µ exceeds
µs at any point in a dense granular medium, (3.5) will predict flowing solutions,
regardless of the size of the granular medium. This deficiency arises because the
approximation that g is close to gloc breaks down as the size of a granular medium
decreases. For example, in dense granular flows down an incline, flow arrests in thin
layers when µ is significantly greater than µs and hence gloc is significantly greater
than zero. Therefore, to obtain predictions of the size-dependence of the flow thresh-
old, we utilize the primitive, dynamical form of the NGF model (3.3) exclusively
throughout the remainder of this chapter.
Next, we discuss how predictions of the flow threshold may be obtained from the NGF
model. For the local inertial rheology, the flow threshold is determined by simply com-
paring the maximum value of the stress ratio µ occurring in a flow configuration to the
critical value µs . However, for the nonlocal model (3.3), this method of assessing the flow
threshold is no longer sufficient. Instead, we reframe the question of whether or not steady
flow is possible as whether or not the g = 0 solution is linearly stable under perturbation.
Starting from (3.3) and linearizing about the g = 0 solution renders the g2 term negligible,
and we assume a perturbed solution gˆ of the form g(x,t)
ˆ = C exp(λt/t0 )g(x),
ˇ where λ is
the dimensionless growth rate of the perturbation, g(x)
ˇ is a time-independent function, x is
the spatial coordinate, and C is an arbitrary constant. Substituting the perturbed solution gˆ
into the linearized form of (3.3) and simplifying, we obtain the following linear differential
relation for g:
ˇ
∂ 2 gˇ
A2 d 2 − (λ + µs − µ) gˇ = 0. (3.6)
∂ xk ∂ xk
40
Then, for a given flow configuration, the field µ(x) is specified along with appropriate
ˇ and the growth rate λ may be calculated. If
homogeneous boundary conditions for g,
λ < 0, the perturbation decays, and steady flow is not possible. If λ > 0, the perturbation
grows, and steady flow may occur. The flow threshold may be identified as the case in
which λ = 0. In subsequent discussions of the theoretically predicted flow threshold, we
denote gˇ as g for notational simplicity.
The NGF-model-predicted flow threshold for inclined plane flow has been derived in
previous work [58]. Here, we briefly recap the linear perturbation process that will subse-
quently be applied to the more complex flow configurations in Section 3.4. The µ-field in
inclined plane flow is spatially constant and given through the angle of inclination by tan θ ,
i.e., µ = tan θ . Therefore, (3.6) takes the form of an ordinary differential equation (ODE)
with constant coefficients:
d2g tan θ − λ − µs
+ g = 0, (3.7)
dz2 A2 d 2
where z is the distance from the free surface. Anticipating that the quantity (tan θ − λ − µs )
is positive, the solution to (3.7) is
" p ! # " p ! #
tan θ − λ − µs tan θ − λ − µs
g = C1 sin z +C2 cos z ,
Ad Ad
where C1 and C2 are arbitrary constants. As discussed in previous work [58], the choice
of homogeneous boundary conditions is important. For inclined plane flow, we based this
choice on observations of existing DEM flow data of spheres [30]. In particular, in the
region near the free surface (z = 0), the DEM data of Silbert et al. [30] shows that the strain-
rate approximately levels off, implying a zero strain-rate gradient. Accordingly, we enforce
that dg/dz = 0 at z = 0, which requires that C1 = 0. Second, Silbert et al. [30] observed that
adjacent to a fully rough boundary, the strain-rate approaches an approximately vanishing
state, and hence, we take g = 0 at z = H. The lowest value of tan θ that satisfies this
41
boundary condition corresponds to
H π A
= p . (3.8)
d 2 tan θ − µs
In (3.8), λ has been set to zero, so that (3.8) represents the size-dependent flow threshold for
inclined plane flow. For thick layers, the flow threshold approaches the size-independent
value tan θ = µs ; however, for thinner layers, flow ceases at higher inclination angles. In
Fig. 3.1(b), the predicted flow threshold (3.8) is compared against Pouliquen’s experimen-
tally determined values using glass beads with a fully rough base, using the parameters
for glass beads (µs = 0.3819 and A = 0.48), and the agreement is quite good. To obtain
the flow thresholds corresponding to the more complex flow configurations considered in
Section 3.4, we apply the same linear perturbation process – albeit involving more com-
plex stress ratio fields. The details of these calculations and the resulting theoretical flow
thresholds are given in Section 3.4.
A deeper discussion of the role of fluidity boundary conditions in NGF model predic-
tions of the flow threshold is warranted. In the case of inclined plane flow, it is the choice
of a homogeneous Dirichlet fluidity boundary condition at the rough base that leads to pre-
dictions of size-dependent strengthening. Indeed, if a homogeneous Neumann condition
were employed, a size-independent angle of repose would be predicted. However, as will
be shown in Section 3.4, NGF model predictions of size-dependent strengthening do not
arise solely due to wall conditions. In a flow configuration with a spatially varying µ-field,
in which some spatial regions experience µ > µs while others experience µ < µs , the NGF
model also predicts size-dependent strengthening, regardless of the choice of wall bound-
ary condition. In such a case, the region experiencing µ < µs serves to stabilize the region
experiencing µ > µs through nonlocal effects, while a local model would simply predict
the region experiencing µ > µs to flow. The flow configurations considered in Section 3.4
involve both spatially varying µ-fields and rough walls. In the DEM simulations reported
42
in Section 3.4, we are unable to extract clear evidence justifying a homogeneous Dirichlet
boundary condition for the fluidity at walls. Therefore, for the sake of simplicity, following
our previous work [28, 57, 81], we employ homogeneous Neumann fluidity boundary con-
ditions at walls throughout – both in calculating theoretical flow thresholds and steady flow
fields. We note that a similar choice of wall fluidity boundary condition was employed by
Chaudhuri et al. [82] in their investigation of vertical chute flow of soft, frictionless disks.
In spite of this rather naive choice of fluidity boundary condition, the results of Section 3.4
demonstrate that good agreement between DEM data and NGF model predictions may be
obtained, indicating that the specifics of the stress field may play a larger role than wall
conditions.
3.4 Flow threshold in other configurations
In this section, we present DEM simulation results in three flow configurations – planar
shear flow with gravity, annular shear flow, and vertical chute flow – characterizing the
size-dependence of the flow threshold in each case. To be clear, in the context of our
DEM simulations, the flow threshold refers to the condition for flow cessation and not the
condition for flow start-up, which is typically greater than the flow cessation condition and
is dependent on the preparation history. In this section, we also compare DEM results
with corresponding predictions of the NGF model. Throughout, we use a single set of
material parameters {µs , b, A} in obtaining NGF model predictions. Based on the fit of the
Bingham-like functional form of the inertial rheology (3.1) to DEM data of homogeneous
planar shearing (Fig. 3.2(b)), we utilize local parameter values of µs = 0.272 and b = 1.168.
In contrast, the nonlocal amplitude is not obtained by fitting to a single data set – rather,
we choose a value of A = 0.90 in order to provide the best collective description of all
subsequently reported data. We note that this numerical value is similar to the value A =
0.80, which was reported by Kamrin and Koval [78] for disks with an inter-particle sliding
43
(a) Pwall (b) 0.55
vwall
ℓ/d = 11.75
0.5
ℓ/d = 23.5
x
z vx (z) 0.45 ℓ/d = 47
µwall
H = 60d
ℓ/d = 93.5
0.4
G
0.35
0.3
-4 -3 -2 -1
10 10 10 10
L = 60d v˜wall
(c) 200 (d) 100
1
µ/µwall
150 0.5
vx /vwall
ℓ/d
100 10-1 0
0 10 20 30
Flow z/d
50
No flow
0 10-2
µs 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0 10 20 30 40
µwall z/d
Figure 3.3: (a) Configuration for two-dimensional DEM simulations of planar shear flow
with gravity. (b) The dependence of the stress ratio at the wall µwall on v˜wall for loading
length-scales `/d = Pwall /φ ρs Gd = 11.75, 23.5, 47, and 93.5. (c) Flow threshold locus.
(d) Normalized steady velocity fields in the plateau regime (v˜wall ≈ 10−3 ) for the four
loading length-scales in (b). Inset: Steady stress ratio fields corresponding to each case.
Throughout, symbols represent the steady-state results of DEM simulations. In (b) and (d),
solid lines are the calculated steady results of the NGF model, (3.2) and (3.3). In (c), the
solid line is the analytical flow threshold predicted by the NGF model (3.12). In the inset of
(d), the solid lines are the anticipated µ-fields (3.9) used as input in calculations involving
the NGF model.
friction coefficient µsurf = 0.4.
3.4.1 Planar shear flow with gravity
First, we consider planar shear flow with gravity acting orthogonal to the shearing direc-
tion. We note that size-dependent strengthening in this configuration was first suggested by
Pouliquen and Forterre [45] in the context of their modeling work, but to our knowledge,
this effect has not been reported in experiments or DEM simulations. The DEM set-up
for this case is shown in Fig. 3.3(a) and is achieved by introducing a gravitational body
44
force along the z-direction to the configuration used in planar shear flow, described in Sec-
tion 3.2.2. The parallel, rough walls are separated by a distance H = 60d, and the top wall
moves in the x-direction with a velocity vwall , while the bottom wall remains fixed. The top
wall imposes a compressive normal stress Pwall on the granular material, using the control
procedure described in Section 3.2.2, and the gravitational body force is φ ρs G, where φ is
the mean solid area fraction and G is the acceleration of gravity. Periodic boundary condi-
tions are employed along the x-direction, and the length of the region in the x-direction is
L = 60d.
The expected stress field in this flow configuration may be deduced from a quasi-static
force balance. As in planar shear flow without gravity, the shear stress is spatially constant
and given by the shear stress imparted by the moving wall – i.e., τ(z) = |σxz (z)| = |σzx (z)| =
τwall . The pressure field is a combination of the prescribed wall pressure Pwall and the
gravitational pressure gradient, so that P(z) = −σzz (z) = Pwall + φ ρs Gz. As in planar shear
flow, we assume that σxx (z) ≈ σzz (z), which is consistent with the results of our DEM
simulations. Therefore, the stress ratio field in planar shear flow with gravity varies as
τ(z) µwall
µ(z) = = , (3.9)
P(z) 1 + z/`
where µwall = τwall /Pwall is the maximum value of µ, occurring at the wall (z = 0), and ` =
Pwall /φ ρs G is the loading length-scale, which is defined as the ratio of the wall pressure to
the gravitational body force and is distinct from the dimensions H and L. Importantly, since
the loading length-scale ` is the only length-scale appearing in the stress ratio field (3.9),
` – rather than the dimensions H or L – is the relevant length-scale that characterizes the
system size in this problem. The loading length-scale ` may be interpreted as the distance
beneath the top wall at which the pressure due to gravity φ ρs G` is equal to the pressure
applied by the top wall Pwall – i.e., P(z = `) = 2Pwall . We have verified that the dimensions
H = 60d and L = 60d are sufficiently large so that they do not affect the subsequently
45
reported DEM results.
We run DEM simulations of planar shear flow with gravity for different values of the
top-wall speed vwall and loading length-scale `. Each DEM simulation is run to steady
state through a top-wall shear displacement of at least 5500d, and the steady fields vx (z),
τ(z), and P(z) are calculated using 2000 system snapshots – evenly spaced over an addi-
tional top-wall shear displacement of at least 5500d – as described in Appendix B.1. In
each case, we verify that the resulting shear stress field is indeed constant – thereby ex-
tracting the corresponding value of τwall – and that the pressure field matches the intended
dependence P(z) = Pwall + φ ρs Gz. In presenting results, we utilize a dimensionless wall
velocity v˜wall = (vwall /`) m/Pwall , in which vwall is non-dimensionalized through the load-
p
ing length-scale and the microscopic time-scale associated with particle motion. First, we
probe the dependence of the stress ratio at the wall µwall = τwall /Pwall on v˜wall for loading
length-scales `/d = 11.75, 23.5, 47, and 93.5, which is plotted as symbols in Fig. 3.3(b).
For sufficiently high wall speed – i.e., v˜wall & 3 × 10−2 – the relationship between µwall and
v˜wall is size-independent, indicating that the response is dominated by local, inertial effects.
However, as the wall speed is decreased, a rate-independent plateau emerges, which is de-
pendent on the system-size `/d. Therefore, for a given `/d, steady flow is not possible for
µwall less than the plateau value, enabling the construction of a phase diagram of flowing
and non-flowing states. As shown in Fig. 3.3(c), we create a phase diagram with `/d on
the vertical axis and µwall on the horizontal axis. Then, the DEM-calculated flow threshold
locus is plotted as symbols on the phase diagram – in which each point consists of a given
`/d and the corresponding plateau value of µwall . Steady flow is possible for combinations
of `/d and µwall to the right of the flow threshold locus, while steady flow cannot occur for
combinations to the left of the locus. For a large system-size, the flow threshold approaches
the size-independent value µs ; however, as `/d decreases the flow threshold increases.
We have also numerically computed corresponding steady solutions of the NGF model,
46
(3.2) and (3.3). Model predictions of steady velocity fields are calculated for a given com-
bination of `/d and µwall , by evolving (3.3) to steady state, using finite differences in MAT-
LAB with µ-field given through (3.9), a very fine spatial resolution ∆z d, and the so-
lution of (3.5) as the initial guess. The calculated relationships between the wall stress
ratio µwall and the dimensionless wall speed v˜wall are plotted as solid lines in Fig. 3.3(b)
for `/d = 11.75, 23.5, 47, and 93.5. The NGF model quantitatively captures both the
size-independent but rate-dependent regime observed in DEM simulations at sufficiently
high wall speed and the size-dependent but rate-independent plateau regime – an obser-
vation that may be understood in terms of the dynamical relation (3.3) as follows. At
sufficiently high wall speed, the Laplacian term in (3.3) contributes negligibly, yielding
size-independent model predictions. Consequently, the parameters µs and b – but not A –
set the model predictions in the rate-dependent regime. In contrast, for sufficiently slow
flows, the g2 term plays a negligible role, rendering the dynamical relation (3.3) linear in g
which leads to rate-independent model predictions. The parameters µs and A – but not b –
determine NGF model predictions in the rate-independent regime.
The plateau value of µwall calculated using the NGF model for a given value of `/d
then represents a point on the predicted flow threshold locus. Instead of constructing the
locus using discrete points determined in this way, we have calculated the analytical flow
threshold predicted by the NGF model for planar shear flow with gravity using the linear
perturbation procedure described in Section 3.3. We define a dimensionless transformed
coordinate z˜ and a positive, dimensionless constant α as
(z + `) µwall `
z˜ = 2 λ + µs and α = p
p
. (3.10)
Ad 2 λ + µs Ad
Then, upon substituting the µ-field for planar shear flow with gravity (3.9) into (3.6), the
47
resulting linear ODE for g(˜z) is
d2g 1 α
+ − + g = 0. (3.11)
d z˜2 4 z˜
The solution of (3.11) is g = C1 Mα,1/2 (˜z) + C2Wα,1/2 (˜z), where Mα,1/2 (˜z) and Wα,1/2 (˜z)
are Whittaker functions, and C1 and C2 are constants. The homogeneous boundary con-
ditions consist of a Neumann condition at the moving wall, dg/dz|z=0 = 0, and the far-
field boundary condition, limz→∞ g = 0. Since the function Mα,1/2 (˜z) diverges as z˜ → ∞,
while Wα,1/2 (˜z) → 0, the far-field boundary condition requires that C1 = 0. Then, applying
0
the wall boundary condition, making use of the identity Wα,1/2 (˜z) = ((˜z − 2α)Wα,1/2 (˜z) −
2Wα+1,1/2 (˜z))/2˜z, and simplifying, we obtain
√ √ `
`
µs − α Wα,1/2 2 µs
Ad Ad
√ ` µwall `
−Wα+1,1/2 2 µs =0 with α = √ . (3.12)
Ad 2 µs Ad
In the above expression, we have set λ to the threshold value of λ = 0, so that (3.12) repre-
sents the size-dependent flow threshold for planar shear with gravity. For a given value of
the dimensionless system size `/d and the material parameters µs and A, the smallest, pos-
itive value of µwall that satisfies the transcendental equation (3.12) gives the flow threshold.
The analytical flow threshold (3.12) is plotted as a solid line in Fig. 3.3(c) – displaying a
favorable quantitative comparison with the DEM data.
Finally, we compare steady velocity fields extracted from DEM simulations to cor-
responding NGF model predictions. Steady normalized velocity fields vx (z)/vwall in the
plateau regime (v˜wall ≈ 10−3 ) for loading length-scales `/d = 11.75, 23.5, 47, and 93.5
are shown in Fig. 3.3(d) with symbols denoting DEM data and solid lines representing
NGF model predictions. In the inset of Fig. 3.3(d), symbols denote the corresponding nor-
malized stress ratio fields µ(z)/µwall measured in the DEM simulations, and solid lines
48
represent the anticipated stress ratio fields (3.9) used as input in calculations involving the
NGF model, confirming that the intended stress fields are achieved in the DEM simula-
tions. Overall, the NGF model is able to quantitatively capture the salient aspects of the
flow fields in planar shear flow with gravity over a range of loading length-scales. Impor-
tantly, the nonlocal amplitude A is the operative material parameter that determines NGF
model predictions of steady flow fields in the plateau regime, and using a single numerical
value of A, the NGF model simultaneously captures DEM data of both the size-dependence
of the flow threshold and steady flow fields.
3.4.2 Annular shear flow
Next, we consider annular shear flow – the DEM set-up for which is shown in Fig. 3.4(a)
with inner radius R and outer radius Ro . For the most part, the details of our DEM simu-
lations of annular shear flow follow the procedures of Koval and coworkers [41, 78]. The
walls in our DEM simulations of annular shear flow consist of rings of glued grains of
diameter 2d,5 and the inner radius R corresponds to the radial position of the outermost
points of the inner wall grains. At the inner wall, we prescribe the circumferential wall
velocity vwall , and the radial position of the inner wall grains is fixed. While the outer wall
does not rotate, the value of Ro fluctuates slightly so as to impose a prescribed radial com-
pressive normal stress Pwall on the granular material, using a control procedure analogous
to that used in Section 3.2.2 and described by Koval et al. [41]. We do not utilize periodic
boundary conditions, instead modeling the full annular shear cell, as shown in Fig. 3.4(a)
for the case of R/d = 26. In total, we consider inner wall radii of R/d = 11, 26, 51, and
101. Throughout, we take the outer radius to be sufficiently large so that the value of Ro
does not affect the subsequently reported results – Ro = 2R for R/d = 25, 51, and 101 and
5 Roughwalls consisting of glued grains of diameter 2d were also used in the annular shear flow simu-
lations of Kamrin and Koval [78]. We utilize this type of rough wall for our annular shear flow simulations
rather than the rough walls described in Section 3.2.2 in order to more easily construct the annular DEM
configuration.
49
(a) Pwall (b) 0.45
Ro R/d = 26
0.4 R/d = 51
vwall
µwall
R/d = 101
R 0.35
vθ (r − R)
0.3
10-4 10-3 10-2
v˜wall
(c) 120 (d) 100
1
100
µ/µwall
0.5
80
vθ /vwall
R/d
60 10-1 0
Flow 0 10 20
40 (r − R)/d
20 No flow
0 10-2
µs 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0 5 10 15 20 25 30
µwall (r − R)/d
Figure 3.4: (a) Configuration for two-dimensional DEM simulations of annular shear flow
for the case of R/d = 26. (b) The dependence of the stress ratio at the wall µwall on v˜wall for
inner wall radii R/d = 26, 51, and 101. (c) Flow threshold locus. (d) Normalized steady
velocity fields in the quasi-static regime (v˜wall ≈ 10−4 ) for the three loading length-scales
in (b). Inset: Steady stress ratio fields corresponding to each case. Throughout, symbols
represent the steady-state results of DEM simulations. In (b) and (d), solid lines are the
calculated steady results of the NGF model, (3.2) and (3.3). In (c), the solid line is the
analytical flow threshold predicted by the NGF model (3.16). In the inset of (d), the solid
lines are the anticipated µ-fields (3.13) used as input in calculations involving the NGF
model.
Ro = 4R for R/d = 11. The DEM configurations for R/d = 11, 26, 51, and 101 contain
4640, 5715, 23900, and 97696 flowing grains, respectively.
Analogous to planar shear with gravity, we may deduce the steady stress field from
quasi-static force and moment balances. The moment balance gives the shear stress field
to be τ(r) = |σrθ (r)| = |σθ r (r)| = τwall (R/r)2 , where r is the radial coordinate and τwall is
the inner wall shear stress, and the radial force balance gives that P(r) = −σrr (r) = Pwall
is spatially constant. Again, we assume that the normal stresses are equal – i.e., σθ θ ≈ σrr
– which is consistent with DEM simulation results. Therefore, for annular shear flow, the
50
stress ratio field varies as
2
τ(r) R
µ(r) = = µwall , (3.13)
P(r) r
where µwall = τwall /Pwall is the maximum value of µ, occurring at the inner wall (r = R).
Our discussion of simulation results for annular shear flow mirrors that of Section 3.4.1
for planar shear flow with gravity. We run DEM simulations for different values of the
inner wall speed vwall and radius R. Each simulation is first run to steady state through
an inner-wall tangential displacement of at least 48d,6 and the steady fields vθ (r), τ(r),
and P(r) are then extracted using 1000 system snapshots, which are evenly spaced over
an additional inner-wall tangential displacement of an equal amount as in the preceding
step. Further, τwall is measured by way of the average torque applied to the inner wall
at steady state. First, we explore the dependence of the inner wall stress ratio µwall on
v˜wall = (vwall /R) m/Pwall in DEM simulations for R/d = 26, 51, and 101, which is plot-
p
ted as symbols in Fig. 3.4(b). Again, a transition is observed from a rate-dependent but
size-independent regime at sufficiently high wall speed (v˜wall & 10−2 ) to a size-dependent
plateau regime as v˜wall is decreased. Here, we have restricted attention to a slightly lower
range of v˜wall than considered in Section 3.4.1 to ensure that centripetal acceleration plays
no role in our DEM simulations – a point which is verified by checking that the normal
stress σrr is spatially constant. A phase diagram of flowing and non-flowing states for
annular shear flow is shown in Fig. 3.4(c), in which pairs of R/d and the corresponding
plateau value of µwall are plotted as symbols and denote the DEM-calculated flow thresh-
old locus. Again, we observe strengthening as the system-size R/d is reduced.
Steady-state predictions of the NGF model are numerically calculated for given com-
binations of R/d and µwall as described in Section 3.4.1 except with µ-field given through
(3.13). The calculated relationships between µwall and v˜wall are plotted as solid lines in
6 Koval
and coworkers [41, 78] report that transients fully subside after an inner-wall tangential displace-
ment of approximately 50d in their simulations of annular shear flow and conservatively adopt an inner-wall
displacement of 100d as their steady-state condition. Our observation of transients is similar, and for effi-
ciency – since we simulate the full annular shear cell rather than an angular section – we adopt a steady-state
condition of 48d for the inner-wall displacement.
51
Fig. 3.4(b) for R/d = 26, 51, and 101, demonstrating good quantitative agreement with
DEM data and – most importantly – displaying a size-dependent plateau value of µwall . As
in Section 3.4.1, we calculate the theoretical flow threshold locus for annular shear flow via
the linear perturbation procedure described in Section 3.3. Substituting the µ-field (3.13)
into (3.6), defining a dimensionless transformed coordinate r˜ and a positive, dimensionless
constant α as
r √ R
λ + µs α= µwall
p
r˜ = and , (3.14)
Ad Ad
and simplifying, we obtain the following linear ODE for g(˜r):
d2g dg
r˜2 + r˜ − r˜2 − α 2 g = 0. (3.15)
d r˜2 d r˜
The solution of (3.15) is g = C1 Iıα (˜r) +C2 Kıα (˜r), where Iıα (˜r) and Kıα (˜r) are the modified
Bessel functions of the first and second kind of purely imaginary order, and C1 and C2
are constants. We consider the following homogeneous boundary conditions: a Neumann
condition at the inner wall, dg/dr|r=R = 0, and the far-field condition, limr→∞ g = 0. For r˜
and α positive, the function Iıα (˜r) is complex-valued, and its real part diverges as r˜ → ∞.
In contrast, Kıα (˜r) is real-valued for r˜ and α positive, and Kıα (˜r) → 0 as r˜ → ∞. Therefore,
the far-field boundary condition requires that C1 = 0. Then, applying the wall boundary
condition, setting λ = 0, and simplifying leads to the size-dependent flow threshold for
annular shear flow:
√ R √ R
0
Kıα µs =0 with α = µwall . (3.16)
Ad Ad
For a given value of the dimensionless inner wall radius R/d and the material parameters
µs and A, the smallest, positive value of µwall that satisfies (3.16) gives the flow threshold.
The theoretical flow threshold locus (3.16) is plotted as a solid line in Fig. 3.4(c), showing
that the NGF model captures the size-dependent flow threshold in this flow configuration.
52
(a) (b) 80
W
60
Pwall
Flow
W/d
40
G L vz (x)
x 20
z No flow
0
µs 0.3 0.35 0.4 0.45
µwall
Figure 3.5: (a) Configuration for two-dimensional DEM simulations of vertical chute flow
for the case of W /d = 60. (b) Flow threshold locus. Symbols represent upper and lower
bound estimates of the flow threshold based on the results of DEM simulations, and the
solid line is the analytical flow threshold predicted by the NGF model (3.19).
Lastly, we compare DEM data and NGF model predictions of steady velocity fields.
Steady normalized velocity fields vθ (r)/vwall in the plateau regime (v˜wall ≈ 10−4 ) for inner
wall radii R/d = 26, 51, and 101 are shown in Fig. 3.4(d) with symbols denoting DEM data
and solid lines representing NGF model predictions. Corresponding steady normalized
stress ratio fields µ(r)/µwall are shown in the inset of Fig. 3.4(d), confirming that the µ-
fields measured from DEM data are consistent with the anticipated stress ratio fields (3.13).
The NGF model quantitatively captures both steady flow fields and the size-dependence of
the flow threshold in annular shear flow – while using the same set of material parameters
as in planar shear flow with gravity.
3.4.3 Vertical chute flow
Finally, we consider vertical chute flow. The size-dependence of the flow threshold in this
configuration was first explored by Chaudhuri et al. [82] for a simulated, two-dimensional
system of soft, frictionless particles. Here, we perform an analogous analysis for our system
of stiff, frictional disks. Our DEM set-up is shown in Fig. 3.5(a), which is generated by
first creating a dense granular system between two parallel, rough walls as described for
planar shear in Section 3.2.2 and then rotating the system clockwise by 90◦ and applying
53
a gravitational body force along the z-direction. The rough walls – consisting of layers of
glued grains as in the planar shear flow simulations of Section 3.2.2 – are separated by a
distance denoted by W , which is varied in our simulations. The left vertical wall is fixed,
and the right vertical wall is fixed in the z-direction but can move slightly in the x-direction
so as to impose a compressive normal stress Pwall on the granular material, using the control
procedure described in Section 3.2.2. Periodic boundary conditions are prescribed along
the z-direction. In all cases, the length of the vertical chute L is taken to be 60d, which
is in a range that does not affect DEM results.7 We consider nominal chute widths of
W /d = 10, 20, 30, 40, and 60 – however, these values do vary slightly during flow – and the
DEM configurations contain 633, 1270, 1900, 2539, and 3806 flowing grains, respectively.
From a quasi-static force balance, we expect the shear stress field to be τ(x) = |σxz (x)| =
|σzx (x)| = φ ρs G|x|, where x is measured from the centerline of the chute, and the pressure
field to be P(x) = −σxx (x) = Pwall . Again, we assume that the normal stresses are equal –
i.e., σzz ≈ σxx – and verify this assumption against the DEM results. Therefore, for vertical
chute flow, the µ-field is
|x|
µ(x) = µwall , (3.17)
W /2
where µwall = φ ρs GW /2Pwall is the maximum value of µ, occurring at the walls (x =
±W /2).
Since vertical chute flow is gravity-driven – while planar shear flow with gravity and
annular shear flow are boundary-driven – our process for determining the flow threshold
from DEM simulations is different than previously described. In boundary-driven flow, we
specify arbitrarily-low wall velocities and extract the flow threshold from the steady-state
plateau forces applied to the wall. In contrast, for gravity-driven flow, we consider various
conditions – namely, combinations of W /d and µwall – and determine whether steady flow
7 If the vertical chute is taken to be too long, alternating dense and sparse regions will develop along the
chute. This is because the procedure employed to control the pressure cannot account for variations along the
length of the vertical chute due to the rigid nature of the walls. We have verified that our chute length L is
sufficiently short so that this issue does not arise while also being sufficiently long so that the reported results
do not depend upon L.
54
may be sustained, and in this way, the flow threshold is bounded. Our process is as follows.
Motivated by the methodology of Weinhart et al. [31] for assessing flow arrest in DEM
simulations of inclined plane flow, we utilize a criteria based on the kinetic energy. First,
for a given chute width W /d, a sufficiently large value of µwall is applied so that steady
flow is attained. Next, µwall is decreased to a target value – in practice, this is achieved
by decreasing the acceleration of gravity G – and the system is allowed to reach steady
state over a time period of 19470 m/Pwall . We confirm that the µ-fields measured from
p
steady-state DEM data are consistent with the intended µ-fields (3.17). Then, the mean ki-
netic energy per flowing grain as a function of time – denoted as Ekin (t) – is extracted from
5000 system snapshots distributed evenly over an additional time period of 19470 m/Pwall
p
after reaching steady state. For values of µwall in which the arithmetic average of the ki-
netic energy hEkin (t)i is greater than 10−2 Pwall d 2 ,8 flow is continuous, and Ekin (t) is nearly
time-independent with fluctuations – defined as (h(Ekin (t) − hEkin (t)i)2 i)1/2 – smaller than
hEkin (t)i. When µwall is decreased to a value for which hEkin (t)i = 10−2 Pwall d 2 , fluctua-
tions increase to be roughly equal to hEkin (t)i, and accordingly, we identify this condition
and the corresponding value of µwall as the upper bound of the flow threshold for a given
W /d. As µwall is further decreased, flow becomes intermittent, and kinetic energy fluctua-
tions further increase. In this intermittent regime, it is important to acknowledge the effect
√
that the wall damping parameter gp has on flow. If gp is too low (gp / mkn . 1), wall mo-
tion is underdamped, and the associated wall oscillations prevent flow from ceasing, even
√
when µwall is arbitrarily small. If gp is too high (gp / mkn & 104 ), wall motion is over-
damped, and the target wall pressure Pwall is not achieved. We find that for our choice of
√
gp / mkn = 100, static states may be achieved while maintaining the target wall pressure.
That said, using this value of gp , at low values of µwall , wall motion can still induce isolated,
infrequent grain rearrangements that are not associated with steady flow but do contribute
to the kinetic energy. To remove this effect, we median filter the measured kinetic energy
8 Since
kn = 104 Pwall in our simulations, the normalization factor Pwall d 2 is related to the elastic potential
energy scale, as in Weinhart et al. [31].
55
data Ekin (t) for values of µwall in the intermittent regime prior to arithmetically averaging
the kinetic energy data over time. When the average value of the filtered kinetic energy
data decreases to a very low value of 10−7 Pwall d 2 we deem flow to have ceased and denote
the corresponding value of µwall as the lower bound of the flow threshold.
Once upper and lower bounds of the flow threshold have been determined, we may
construct a phase diagram of flowing and non-flowing states for vertical chute flow, which
is shown in Fig. 3.5(b). For a given W /d, the ×-symbols denote the upper and lower
bounds determined as described in the preceding paragraph, and the range of µwall between
the symbols corresponds to the intermediate regime of intermittent flow and is denoted by a
dotted line. While the exact value of the chute width varies in our simulations, we find that
the differences between actual values of W near the flow threshold and the corresponding
nominal values are less than one grain diameter in all cases, so the values of W /d appearing
in Fig. 3.5(b) correspond to the nominal values. Due to the presence of the intermittent
regime, our determination of the flow threshold in vertical chute flow is less precise than
for the flow configurations considered in Sections 3.4.1 and 3.4.2; however, the increase of
the measured flow threshold with decreasing system size remains clear.
To calculate the theoretical flow threshold locus predicted by the NGF model for verti-
cal chute flow, we substitute the µ-field (3.17) into (3.6) to obtain
d2g λ + µs − µwall x/(W /2)
− xg
˜ =0 where x˜ = (3.18)
d x˜2 (2Adµwall /W )2/3
is a dimensionless transformed coordinate. The solution of (3.18) is g = C1 Ai(x)+C
˜ 2 Bi(x),
˜
where Ai(x)
˜ and Bi(x)
˜ are the Airy functions of the first and second kind, and C1 and C2
are constants. The homogeneous boundary conditions consist of the symmetry condition at
x = 0, dg/dx|x=0 = 0, and a homogeneous Neumann condition at the wall, dg/dx|x=W /2 =
0. Enforcing the boundary conditions and setting λ = 0 yields the size-dependent flow
56
threshold for vertical chute flow:
µs µs − µwall
0 0
Ai Bi
(2Adµwall /W )2/3 (2Adµwall /W )2/3
µs µs − µwall
0 0
− Bi Ai = 0. (3.19)
(2Adµwall /W )2/3 (2Adµwall /W )2/3
For a given value of the dimensionless chute width W /d and the material parameters µs
and A, the smallest, positive value of µwall that satisfies (3.19) gives the flow threshold.
The theoretical flow threshold locus (3.19) is plotted as a solid line in Fig. 3.5(b), using
the same material parameters µs and A as in Sections 3.4.1 and 3.4.2. The theoretical flow
threshold locus does a reasonably good job of quantitatively capturing the DEM data for
vertical chute flow.
3.5 Concluding remarks
In this chapter, we have studied the size-dependence of the flow threshold in three different
dense granular flow configurations – (1) planar shear flow with gravity, (2) annular shear
flow, and (3) vertical chute flow. Importantly, the flow threshold measured in DEM simu-
lations shows substantial size-dependence across all configurations – the details of which
are affected by the form of the stress field. We have also applied the NGF model to all
three flow configurations to obtain predictions of both the flow threshold and steady flow
fields. Notably, we have obtained analytical solutions for the predicted size-dependent flow
threshold in all three cases. The theory – using a single set of material parameters – pre-
dicts size-dependent flow thresholds that match DEM data rather well in all cases. Further,
the theory simultaneously predicts steady flow fields that are quantitatively consistent with
corresponding DEM data.
For illustrative purposes, a comparison of the analytical flow thresholds predicted by
the NGF model for all four flow configurations discussed in this chapter – inclined plane
57
Dimensionless length scale
200
Inclined plane – H/d
Planar shear with gravity – ℓ/d
150 Annular shear – R/d
Vertical chute – W/d
100
50
0
0.3 0.35 0.4 0.45 0.5
tan θ or µwall
Figure 3.6: Comparison of the analytical flow thresholds predicted by the NGF model for
inclined plane flow (3.8), planar shear flow with gravity (3.12), annular shear flow (3.16),
and vertical chute flow (3.19).
flow (3.8), planar shear flow with gravity (3.12), annular shear flow (3.16), and vertical
chute flow (3.19) – is plotted in Fig. 3.6 with the appropriate length-scale on the vertical
axis and the maximum value of µ associated with the flow configuration on the horizon-
tal axis. Notably, size-dependent strengthening in all three configurations considered in
Section 3.4 is significantly greater than the strengthening predicted for inclined plane flow.
Recall that the strengthening predicted by the theoretical flow threshold for inclined plane
flow is entirely due to boundary effects, since the µ-field is spatially constant, while the
additional strengthening associated with the other three theoretical flow thresholds is due
to the spatial-dependence of the stress field, rather than boundary effects. This observation
illustrates that the precise nature of the µ-field has a crucial effect on the resulting flow
threshold and must be accounted for when considering other size-sensitive flow stoppage
phenomena.
As a final comment, providing physical justification for fluidity boundary conditions
at walls remains an open issue. In the present work, our choice of a homogeneous Neu-
mann condition was based on pragmatic grounds – since a homogeneous Dirichlet fluidity
boundary condition was not directly observed in our DEM simulations – and past experi-
ence – which has shown that such a boundary condition enables an excellent description
58
of experiments of split-bottom flow [28] and chute flow [81]. In spite of the lack of a
physical underpinning for this choice, the favorable agreement between the flow thresholds
measured in DEM simulations and the corresponding analytical flow thresholds predicted
by the NGF model provides support for this choice of fluidity boundary condition. From
a broader perspective, the issue of specifying non-standard boundary conditions arises in
virtually all nonlocal constitutive approaches [46, 54, 55, 82], and motivating the choice of
these boundary conditions from a physical perspective remains an open challenge. In the
context of nonlocal fluidity models, some recent progress has been made on this point for
dense emulsions [83], and future work of this type is needed to develop a clearer micro-
scopic understanding of granular fluidity boundary conditions.
Chapter 4
Modeling shear-rate-gradient-driven
and pressure-gradient-driven
size-segregation in dense, bidisperse
granular flows
Note: A version of this chapter is being prepared for submission to the Journal of Fluid
Mechanics. Data and figures have been used with all co-authors’ consent.
Daren Liu and David L. Henann. “Modeling shear-rate-gradient-driven and pressure-
gradient-driven size-segregation in dense, bidisperse granular flows.”
4.1 Introduction
The preceding chapters focused on quasi-monodisperse granular systems, in which all
grains were approximately the same size. However, most dense granular systems occur-
ring in nature and industry are non-monodisperse – i.e., mixtures consisting of particles
of disparate sizes. In non-monodisperse granular systems, it is widely appreciated that the
59
60
constituent grains segregate based on size and form complex patterns during flow [18, 84–
92]. In geophysics, granular size segregation can manifest in landslides and debris flows,
in which more damage is done by the larger grains that segregate to the top of the flow,
while in industry, size-segregation can be an undesirable effect when blending granular
constituents of various sizes is a necessary processing step.
Granular size-segregation has been repeatedly observed in experiments in the literature.
For example, in Thornton et al. [93] for an initially well-mixed system of grains of two
different sizes – which we will henceforth refer to as a bidisperse system – flowing down
an inclined plane, larger particles segregate to the surface of the flowing layer while smaller
ones sink to the bottom. When driving similar bidisperse mixtures in a rotating drum, Gray
and Thornton [90] observed small grains segregating to form stripes. As a final example, in
split-bottom cell experiments by Hill and Fan [91], not only do the larger particles segregate
to the top, but they also segregate towards more rapidly shearing regions.
In terms of the mechanisms of size-segregation in dense granular flows, the current
understanding is that there are two driving forces. The first is pressure-gradients – most
commonly induced by gravity. In pressure-gradient-driven size-segregation, small particles
move more readily through the interstitial space that opens and closes during flow through
a process often referred to as “kinetic sieving,” leading to a system stratified along the di-
rection of pressure gradients [90, 94–96]. While pressure-gradient-driven segregation has
been the focus of significant study, grains can also segregate in inhomogeneous flows along
directions orthogonal to gravitationally-induced pressure gradients [91, 92, 97], driven in-
stead by gradients in shear-strain-rate. As an example, this mechanism has been observed
in the aforementioned split-bottom cell experiments of Hill and Fan [91], in which large
grains segregate towards shear bands. Shear-strain-rate-gradient-driven segregation has re-
ceived comparatively less attention and is thus less well understood.
Due to the complexity of coupled flow and segregation patterns, developing a general,
predictive, continuum model for granular size-segregation is an open challenge. Although
61
much invaluable progress has been made over the recent decades [18, 90, 94–98], no ex-
isting models fully account for both segregation dynamics and flow rheology in a self-
contained manner. Instead, they all require some flow field quantity, such as velocity or
stress fluctuation field, to be measured first from experiments or DEM simulations and
then used as model input. A full mechanical continuum model capable of predicting both
the flow fields and the evolution of segregation, based solely on the geometry of the flow
configuration, applied loads, and boundary/initial conditions, is still missing.
The main reason for the incompleteness of current models is the lack of a dense granular
flow rheology theory, which has recently been addressed by the nonlocal granular fluidity
(NGF) model presented in [27, 28, 32, 57, 81] and discussed in the previous chapters.
Thus, our overarching aim is to formulate and validate a predictive continuum theory for
simultaneous flow and size-segregation in dense granular systems by integrating the NGF
model with a phenomenological size-segregation model. This is a broad goal, and in this
work, we narrow our focus to a set of simpler quasi-one-dimensional flow configurations.
In most real-world flows, both the pressure-gradient-driven and shear-strain-rate-gradient-
driven segregation mechanisms are present, making it difficult to disentangle them. There-
fore, our plan is to isolate and examine the shear-strain-rate-gradient-driven mechanism
first before integrating it with the pressure-gradient-driven mechanism and incorporating
both mechanisms into a general, continuum size-segregation model. Specifically, we first
study the shear-strain-rate-gradient-driven segregation by considering flow of dense, bidis-
perse two-dimensional disks in two flow configurations – vertical chute flow and annular
shear flow – in which pressure is constant throughout and no hydrostatic pressure gradients
are present. Therefore shear-strain-rate-gradients are the sole drivers of segregation. Once
a model for shear-rate-gradient-driven segregation is established, pressure-gradients will
be reintroduced using two more flow configurations – inclined plane flow and planar shear
flow with gravity – in which both segregation mechanisms act simultaneously.
62
In order to inform continuum model development, we perform two-dimensional dis-
crete element method (DEM) simulations using the open source software LAMMPS [79],
which function as “numerical experiments,” and continue to use the grain interaction model
described in Chapter 3.2.1. We then coarse-grain DEM simulation results to obtain con-
tinuum flow and segregation fields. The coarse-graining technique that we employ stems
from the one utilized in Zhang and Kamrin [64] and in Chapter 3 and is described in Ap-
pendix B.2. The coupled continuum model that we develop is then validated by comparing
its prediction of transient flow and segregation against additional DEM simulations results.
This chapter is organized as follows. In Section 4.2, we discuss the continuum model
that we use to describe flow and size-segregation in bidisperse, dense granular materi-
als. Specifically, Section 4.2.1 introduces the mixture theory framework used to describe
dense, bidisperse granular mixtures, and in Section 4.2.2, we briefly revisit the local iner-
tial rheology and the nonlocal granular fluidity model for monodisperse granular systems
and discuss their extension to bidisperse systems. Then in Section 4.2.3, we hypothesize
a model for shear-rate-gradient-driven size-segregation. In Section 4.3 and Section 4.4,
we independently determine the two dimensionless material parameters that appear in the
shear-strain-rate-driven size-segregation model. Then, in Section 4.5, the proposed segre-
gation model is coupled with the NGF model and applied to both vertical chute flow and
annular shear flow to predict the segregation dynamics in transient flow, and the predicted
continuum fields are compared against DEM measurements. With a good understanding
of the shear-rate-gradient-driven segregation mechanism in hand, in Section 4.6, we turn
attention to the pressure-gradient-driven mechanism, introducing an additional pressure-
gradient-driven segregation flux term to the continuum model. In Section 4.7, we verify
the effectiveness of the model in the inclined plane flow configuration. Finally, in Section
4.8, we validate the full continuum model that accounts for both size-segregation mech-
anisms in planar shear flow with gravity. In the end, our model demonstrates a level of
63
dl
AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KokIeix46bGi/YA2ls1m0i7dbMLuRiihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLUsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZIphi2WiER1A6pRcIktw43AbqqQxoHATjC+nfmdJ1SaJ/LBTFL0YzqUPOKMGivdh49iUKm6NXcOskq8glShQHNQ+eqHCctilIYJqnXPc1Pj51QZzgROy/1MY0rZmA6xZ6mkMWo/n586JedWCUmUKFvSkLn6eyKnsdaTOLCdMTUjvezNxP+8XmaiGz/nMs0MSrZYFGWCmITM/iYhV8iMmFhCmeL2VsJGVFFmbDplG4K3/PIqaV/WPLfm3V1V640ijhKcwhlcgAfXUIcGNKEFDIbwDK/w5gjnxXl3Phata04xcwJ/4Hz+AEbtjcw=
ds
AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KokIeix46bGi/YA2ls1m0i7dbMLuRiihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLUsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZIphi2WiER1A6pRcIktw43AbqqQxoHATjC+nfmdJ1SaJ/LBTFL0YzqUPOKMGivdh496UKm6NXcOskq8glShQHNQ+eqHCctilIYJqnXPc1Pj51QZzgROy/1MY0rZmA6xZ6mkMWo/n586JedWCUmUKFvSkLn6eyKnsdaTOLCdMTUjvezNxP+8XmaiGz/nMs0MSrZYFGWCmITM/iYhV8iMmFhCmeL2VsJGVFFmbDplG4K3/PIqaV/WPLfm3V1V640ijhKcwhlcgAfXUIcGNKEFDIbwDK/w5gjnxXl3Phata04xcwJ/4Hz+AFGJjdM=
Figure 4.1: A representative sub-domain of a dense, bidisperse granular system consisting
of two-dimensional disks.
fidelity in simultaneously predicting flow and segregation dynamics that has not been pre-
viously achieved. We close with a discussion relating our model to others’ work and future
steps in Section 4.9.
4.2 Continuum framework
In this section, we discuss the continuum framework used to describe dense, bidisperse
granular systems and then propose constitutive models for rheology and size-segregation.
Throughout, we utilize a mixture-theory-based approach, which is common in continuum
modeling of dense, bidisperse mixtures [e.g., 90, 95, 97].
4.2.1 Bidisperse systems
We consider two-dimensional granular mixtures consisting of disks with two sizes – large
grains with an average diameter of d l and small grains with an average diameter of d s ,
as shown in Fig. 4.1. In order to rule out the effect of density-based segregation [99], all
particles have the same area-density ρs . Throughout, we utilize the notational convention,
in which we denote large-grain quantities using a superscript l and small-grain quantities
64
Grain size Solid area Concentration Velocity Relative
fraction area flux
Large grains dl φl cl = φ l /φ vli wli = cl (vli − vi )
Small grains ds φs cs = φ s /φ vsi wsi = cs (vsi − vi )
Mixture d¯ = φ= 1= vi = 0i =
cl d l + cs d s φl +φs cl + cs cl vli + cs vsi wli + wsi
Table 4.1: Summary of the quantities introduced to describe dense, bidisperse granular
mixtures.
using a superscript s, and the important kinematic quantities are summarized in Table 4.1.
The species-specific solid area fractions – i.e., the areas occupied by each species per unit
total area – are φ l and φ s , respectively, and the total solid area fraction is φ = φ l + φ s .
The concentration of each species then follows as cl = φ l /φ and cs = φ s /φ , so that cl +
cs = 1. The average mixture grain size is defined as the sizes of both species weighted
by their concentrations, d¯ = cl d l + cs d s . We make the standard assumption that the total
area (or volume for spherical grains) of the dense flows that we study does not change
[44, 90, 95, 97], and therefore φ remains approximately constant at each point in space and
at each point in time during the segregation process. This assumption may be tested by
experiments and DEM simulations, in which it is observed that volume dilatation at flow
initiation occurs over a much shorter time scale than the process of segregation.
Regarding the kinematics of flow, each species has an associated partial velocity, vli and
vsi , and the mixture velocity is given by vi = cl vli +cs vsi . The mixture strain rate tensor is then
defined using the mixture velocity in the standard way: γ˙i j = (1/2)(∂ vi /∂ x j + ∂ v j /∂ xi ),
where γ˙kk = 0 since we have assumed that the mixture area (volume) does not change. The
equivalent shear strain-rate is defined as γ˙ = (2γ˙i j γ˙i j )1/2 .
65
Then, the relative area flux for each grain type α is defined as wαi = cα (vαi − vi ) , so that
wli + wsi = 0. Conservation of mass for each species requires that Dcα /Dt + ∂ wαi /∂ xi = 0.
Due to the fact that cl + cs = 1, only one of cl and cs is independent. Therefore, we will
utilize cl as the field variable that describes size-segregation in the following. Conservation
of mass in terms of large particles is
Dcl ∂ wli
+ = 0. (4.1)
Dt ∂ xi
In order to close the system of equations, we require (1) a rheology that relates the strain-
rate tensor to the stress and (2) a constitutive equation for the flux wli – each of which are
discussed in the following subsections.
4.2.2 Granular rheology – from monodispersity to bidispersity
In this section, we discuss the rheology of dense, bidisperse granular mixtures. As before,
the starting point of this discussion is the local inertial rheology function µloc (I). Regarding
stress quantities, the mixture Cauchy stress is denoted as σi j = σ ji , and we define the
pressure P = −(1/3)σkk , the stress deviator σi0j = σi j + Pδi j , the equivalent shear stress
τ = (σi0j σi0j /2)1/2 , and the stress ratio µ = τ/P – all for the mixture. As shown by Rognon
et al. [100] and Tripathi and Khakhar [101], the inertial rheology function µloc (I) may
be straightforwardly generalized from monodisperse to bidisperse systems by defining the
inertial number for a bidisperse system as I = γ˙ d¯2 ρs /P, where the average mixture grain
p
size for a bidisperse system d¯ has been used in place of d for a monodisperse system.
Then, the same local rheology function µloc (I) utilized for the monodisperse system may
be used for bidisperse systems without any change to the parameters appearing in the fitting
function. This approach neglects the potential effect of new dimensionless quantities that
arise in a bidisperse granular system, such as the grain size ratio d l /d s , but has been shown
to capture DEM data well.
66
(a)
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moving
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z
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flowing
grains
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fixed
wall
L = 60d¯0
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Figure 4.2: (a) Configuration for two-dimensional DEM simulations of bidisperse planar
shear flow. Upper and lower layers of black grains denote rough walls. Dark gray grains in
the flowing cell indicate large flowing grains, and light gray grains indicate small flowing
grains. A 10% polydispersity is utilizedp for each species to prevent crystallization. (b) The
local inertial rheology - µ versus I = γ˙ d¯2 ρs /P - for monodisperse as well as bidisperse
mixtures for grain-size ratios of d l /d s = 1.5, 2.0, 2.5, and 3.0 and cl = 0.5. The solid line
represents the best fit to the monodisperse DEM data with µs = 0.272 and b = 1.168.
To demonstrate this point, consider homogeneous, planar shear of a dense bidisperse
system of disks, illustrated in Fig. 4.2(a) for the case of d l /d s = 1.5 and a system-wide
large-grain concentration of cl = 0.5. In this case, no segregation will occur since the
flow is homogeneous and no pressure or strain-rate gradients are present. We utilize the
DEM procedures described in detail in Chapter 3.2 in order to extract the relationship
between µ and I for bidisperse mixtures with grain size ratios of d l /d s = 1.5, 2.0, 2.5,
and 3.0 and cl = 0.5. The simulated relationships are plotted in Fig. 4.2(b) using black
symbols, along with our monodisperse data from Fig. 3.2(b) plotted as gray symbols. The
relationship between µ and I is observed to be approximately independent of d l /d s . As for
the monodisperse case (Fig. 3.2(b)), the DEM data for bidisperse mixtures of disks may be
fit by a Bingham-like functional form:
µloc (I) = µs + bI, (4.2)
67
where µs = 0.272 and b = 1.168 are the dimensionless material parameters for monodis-
perse disks. In this way, we may capture the rheology of bidisperse mixtures in homoge-
neous planar shear without introducing additional fitting functions or adjustable parame-
ters.
Then, in order to consider inhomogeneous flows, it is necessary to generalize the non-
local granular fluidity model to the case of dense, bidisperse mixtures. As before, we
introduce the granular fluidity g – a positive, scalar field quantity – and recognize that g
represents the fluidity of the mixture. Then, we utilize the steady-state form of the NGF
model, which relates the stress state, the strain-rate, and the granular fluidity through two
constitutive equations: (1) the flow rule
γ˙ = gµ, (4.3)
and (2) the nonlocal rheology
∂ 2g
g = gloc (µ, P) + ξ 2 (µ) , (4.4)
∂ xi ∂ xi
where gloc (µ, P) is the local fluidity function and ξ (µ) is the stress-dependent cooperativity
length. Consistent with the new definition of the inertial number involving d¯ and the local
inertial rheology function (4.2), the local fluidity function is taken to be
P (µ − µs )
r
if µ > µs ,
ρs d¯2 bµ
gloc (µ, P) = (4.5)
0 if µ ≤ µs ,
with {µs = 0.272, b = 1.168}. We utilize an analogous approach to extend the cooperativity
length ξ (µ) to the bidisperse case, in which d for monodisperse grains in (3.5) is replaced
by d¯ for bidisperse grains:
Ad¯
ξ (µ) = p . (4.6)
|µ − µs |
68
Furthermore, we continue to use the numerical value of the nonlocal amplitude for monodis-
perse disks, A = 0.9, which was determined in Chapter 3. These choices will be tested in
later sections by comparing flow fields predicted by the NGF model to measured flow fields
in DEM simulations of bidisperse, inhomogeneous flows.
4.2.3 Segregation model
We take the flux wli to be comprised of two contributions – a diffusion flux wdiff
i and a
seg
shear-strain-rate-gradient-driven segregation flux wi :
seg
wli = wdiff
i + wi . (4.7)
Each contribution is discussed further below.
• The diffusion flux acts counter to segregation to mix the species and is taken to be
given in the standard form wdiff = −D ∂ cl /∂ xi , where D is the binary diffusion
i
coefficient [102]. From dimensional arguments, we expect
D = Cdiff d¯2 γ,
˙ (4.8)
in which Cdiff is a dimensionless material parameter which must be determined [96,
99]. Therefore, we have that the diffusion flux is
l
wdiff ¯2 ∂ c .
i = −Cdiff d γ˙ (4.9)
∂ xi
• Regarding segregation, a major question is what field quantity drives the segregation
flux. Gradients of a number of kinematic quantities are possible – e.g., strain-rate,
velocity fluctuations, or fluidity. For perspective, we note that recent works [97, 98,
103, 104] have shown that gradients in kinetic stress – which are defined by way of
69
the velocity fluctuation – correlate well with segregation flux. In the present work,
we adopt the simplest approach and hypothesize that the segregation flux is driven
by gradients in the shear strain-rate γ˙ and take the segregation flux to be given in the
following phenomenological form:
seg S ¯2 l ∂ γ˙
wi = Cseg d c (1 − cl ) . (4.10)
∂ xi
The factor cl (1 − cl ) ensures that segregation ceases when the bidisperse mixture
becomes either all large (cl = 1) or all small (cl = 0) grains, and the factor d¯2 is
S is a dimensionless material
present for dimensional consistency. The quantity Cseg
S to depend on the size ratio d l /d s , we will
property. While it is possible for Cseg
demonstrate that this effect is negligible in the DEM simulations of Section 4.4 and
S as a constant, dimensionless material parameter for disks, which
therefore treat Cseg
will be determined by fitting to DEM simulation results.
Combining (4.9), (4.10), and (4.7) with conservation of mass (4.1), we obtain the fol-
lowing governing equation for cl :
Dcl ∂ 2 ˙∂c
l
l ∂ γ˙
+ ¯
−Cdiff d γ S ¯2 l
+Cseg d c (1 − c ) = 0, (4.11)
Dt ∂ xi ∂ xi ∂ xi
S } represent two constant dimensionless material parameters that remain
where {Cdiff ,Cseg
to be determined.
4.3 Diffusion flux
First, we determine the value of Cdiff for our dense, bidisperse system of dry, frictional
disks. Consider homogeneous planar shear flow of such a bidisperse mixture, as shown in
Fig. 4.2(a). Again, no segregation occurs in this setting, since neither of the segregation
70
Figure 4.3: The binary diffusion coefficient (measured by way of the Mean Square Dis-
placement) versus γ˙d¯2 in homogeneous, steady planar shearing DEM simulations and
forpgrain-size ratios of d l /d s = 1.5, 2.0, 2.5, 3.0, and 4.0. Both axes are normalized by
d s Pw /ρs . Each symbol is measured from one DEM simulation of a specified size ratio at
one shearing rate. The solid line represents the best fit of a linear relation with Cdiff = 0.20.
driving forces – pressure-gradients or shear-strain-rate-gradients – are present [101]. Dur-
ing steady, planar shearing, the motion of individual grains in the direction transverse to
flow (the z-direction in Fig. 4.2(a)) approximates a random walk. Therefore, by measuring
the Mean Square Displacement (MSD) of the system of N particles as a function of time,
we may determine the binary diffusion coefficient D through [102, 105–108]
1 N
MSD(t) = ∑ (zn (t) − zn (0))2 = 2Dt. (4.12)
N n=1
We simulate homogeneous, steady planar shear flows for grain-size ratios of d l /d s =
1.5, 2.0, 2.5, 3.0, and 4.0 and at various shearing rates. Indeed, after a sufficiently long
time, the measured MSD is linear in time in all cases, allowing us to extract the diffusion
coefficient D for each case.
The diffusion coefficient D is plotted against γ˙d¯2 (with both quantities normalized by
d s Pw /ρs ) in Fig. 4.3. In general, the diffusion coefficient data collapses to a linear rela-
p
tion with D ∼ γ˙d¯2 across the range of size ratios considered. A best fit of the slope of the
71
linear relation – the line in Fig. 4.3 – yields:
D
Cdiff = = 0.20. (4.13)
γ˙d¯2
In order to further validate the fitted value of Cdiff , we have performed a consistency
test by considering planar shearing of an initially fully-segregated cell, which is described
in Appendix C. In this case, diffusion drives remixing of the two species. Using the fitted
value of Cdiff in (4.13), we are able to quantitatively capture the diffusive remixing process,
which provides confidence in our fitted value of Cdiff .
4.4 Shear-rate-gradient-driven segregation flux
Having independently determined the material parameter Cdiff , we next turn to testing the
constitutive equation for the shear-rate-gradient-driven segregation flux (4.10) and deter-
S by studying two representative flow configurations –
mining the material parameter Cseg
vertical chute flow and annular shear flow.
4.4.1 Vertical chute flow
Consider a dense, bidisperse granular mixture flowing down a long vertical chute with
rough parallel walls separated by a distance W under the action of gravity G, as described in
Chapter 3.4.3 for the monodisperse case. Our corresponding DEM set-up for the bidisperse
case is shown in Fig. 4.4(a) for W = 60d¯0 – where d¯0 is the system-wide average grain
size. In all cases, we take the chute length to be L = 60d¯0 and apply periodic boundary
conditions along the z-direction. The left vertical wall is fixed, and the right wall is fixed
in the z-direction but can move slightly in the x-direction so as to maintain a constant wall
pressure Pw , utilizing the same wall-position control method described in Section 3.2.2.
72
(a)
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sha1_base64="olQckvWpi9PQefRsoZ66tYDUCxg=">AAAB6nicbZC7SgNBFIbPxluMtxhLm8EgJE3YTRPLgI1lRHOBZAmzk9lkyOzsMnNWCEsewcZCEVufyM63cXIpNPGHgY//nMOc8weJFAZd99vJ7ezu7R/kDwtHxyenZ8XzUsfEqWa8zWIZ615ADZdC8TYKlLyXaE6jQPJuML1d1LtPXBsRq0ecJdyP6FiJUDCK1nqo0OqwWHZr7lJkG7w1lJulQRWsWsPi12AUszTiCpmkxvQ9N0E/oxoFk3xeGKSGJ5RN6Zj3LSoaceNny1Xn5No6IxLG2j6FZOn+nshoZMwsCmxnRHFiNmsL879aP8Xwxs+ESlLkiq0+ClNJMCaLu8lIaM5QzixQpoXdlbAJ1ZShTadgQ/A2T96GTr3mWb63adRhpTxcwhVUwIMGNOEOWtAGBmN4hld4c6Tz4rw7H6vWnLOeuYA/cj5/AKWrjg0=
sha1_base64="UOXgvcP94iudHKn4eCKm0HkqXuU=">AAAB6nicbZC7SgNBFIbPeo3xFmNpMxiE2ITdNFoGbCwjmgtkQzg7mU2GzM4uM7NCWPIIIlgoYuuT+Ah2Poi9k0uhiT8MfPznHOacP0gE18Z1v5y19Y3Nre3cTn53b//gsHBUbOo4VZQ1aCxi1Q5QM8ElaxhuBGsnimEUCNYKRlfTeuueKc1jeWfGCetGOJA85BSNtW7LeN4rlNyKOxNZBW8BpVrRL39/PPr1XuHT78c0jZg0VKDWHc9NTDdDZTgVbJL3U80SpCMcsI5FiRHT3Wy26oScWadPwljZJw2Zub8nMoy0HkeB7YzQDPVybWr+V+ukJrzsZlwmqWGSzj8KU0FMTKZ3kz5XjBoxtoBUcbsroUNUSI1NJ29D8JZPXoVmteJZvrFpVGGuHJzAKZTBgwuowTXUoQEUBvAAz/DiCOfJeXXe5q1rzmLmGP7Ief8BK0qQrg==
W = 60d¯0 (b) AAAB6nicbZA9TwJBEIbn8AvxC7W02UhMsCF3NFiS2FhilI8ELmRv2YMNe3uX3TkTcuEn2FhojK2/yM5/4wJXKPgmmzx5ZyY78waJFAZd99spbG3v7O4V90sHh0fHJ+XTs46JU814m8Uy1r2AGi6F4m0UKHkv0ZxGgeTdYHq7qHefuDYiVo84S7gf0bESoWAUrfVQDa6H5Ypbc5cim+DlUIFcrWH5azCKWRpxhUxSY/qem6CfUY2CST4vDVLDE8qmdMz7FhWNuPGz5apzcmWdEQljbZ9CsnR/T2Q0MmYWBbYzojgx67WF+V+tn2J442dCJSlyxVYfhakkGJPF3WQkNGcoZxYo08LuStiEasrQplOyIXjrJ29Cp17zLN+7lWY9j6MIF3AJVfCgAU24gxa0gcEYnuEV3hzpvDjvzseqteDkM+fwR87nD4QijTc=
sha1_base64="7+/fWWeKMsXdQfedjuUhfFe3mVE=">AAAB6nicbZC7SgNBFIbPxluMtxhLm8EgJE3YTRPLgI1lRHOBZAmzk9lkyOzsMnNWCEsewcZCEVufyM63cXIpNPGHgY//nMOc8weJFAZd99vJ7ezu7R/kDwtHxyenZ8XzUsfEqWa8zWIZ615ADZdC8TYKlLyXaE6jQPJuML1d1LtPXBsRq0ecJdyP6FiJUDCK1nqoBNVhsezW3KXINnhrKDdLgypYtYbFr8EoZmnEFTJJjel7boJ+RjUKJvm8MEgNTyib0jHvW1Q04sbPlqvOybV1RiSMtX0KydL9PZHRyJhZFNjOiOLEbNYW5n+1forhjZ8JlaTIFVt9FKaSYEwWd5OR0JyhnFmgTAu7K2ETqilDm07BhuBtnrwNnXrNs3xv06jDSnm4hCuogAcNaMIdtKANDMbwDK/w5kjnxXl3PlatOWc9cwF/5Hz+AKcwjg4=
sha1_base64="8ril912Pc8gSBXI7VO1m24KfyKs=">AAAB6nicbZC7SgNBFIbPeo3xFmNpMxiE2ITdNFoGbCwjmgtkQ5idzCZDZmeXmbNCWPIIIlgoYuuT+Ah2Poi9k0uhiT8MfPznHOacP0ikMOi6X87a+sbm1nZuJ7+7t39wWDgqNk2casYbLJaxbgfUcCkUb6BAyduJ5jQKJG8Fo6tpvXXPtRGxusNxwrsRHSgRCkbRWrfl4LxXKLkVdyayCt4CSrWiX/7+ePTrvcKn349ZGnGFTFJjOp6bYDejGgWTfJL3U8MTykZ0wDsWFY246WazVSfkzDp9EsbaPoVk5v6eyGhkzDgKbGdEcWiWa1Pzv1onxfCymwmVpMgVm38UppJgTKZ3k77QnKEcW6BMC7srYUOqKUObTt6G4C2fvArNasWzfGPTqMJcOTiBUyiDBxdQg2uoQwMYDOABnuHFkc6T8+q8zVvXnMXMMfyR8/4DLM+Qrw==
AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU8mKqBeh4KXHCvYD2qVks9k2NJvdJtlCWfo7vHhQxKs/xpv/xrTdg7Y+GHi8N8PMPD8RXBuMv53CxubW9k5xt7S3f3B4VD4+aek4VZQ1aSxi1fGJZoJL1jTcCNZJFCORL1jbHz3M/faEKc1j+WSmCfMiMpA85JQYK3nt+xvc84nKglkf98sVXMULoHXi5qQCORr98lcviGkaMWmoIFp3XZwYLyPKcCrYrNRLNUsIHZEB61oqScS0ly2OnqELqwQojJUtadBC/T2RkUjraeTbzoiYoV715uJ/Xjc14Z2XcZmkhkm6XBSmApkYzRNAAVeMGjG1hFDF7a2IDoki1NicSjYEd/XlddK6qrq46j5eV2r1PI4inME5XIILt1CDOjSgCRTG8Ayv8OZMnBfn3flYthacfOYU/sD5/AHJqZF4
Pw
AAAB8HicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9BjwkmME85BkCbOTTjJkZnaZmVXCkq/w4kERr36ON//GSbIHTSxoKKq66e6KEsGN9f1vb219Y3Nru7BT3N3bPzgsHR03TZxqhg0Wi1i3I2pQcIUNy63AdqKRykhgKxrfzvzWI2rDY3VvJwmGkg4VH3BGrZMe6r2sqyV5mvZKZb/iz0FWSZCTMuSo90pf3X7MUonKMkGN6QR+YsOMasuZwGmxmxpMKBvTIXYcVVSiCbP5wVNy7pQ+GcTalbJkrv6eyKg0ZiIj1ympHZllbyb+53VSO7gJM66S1KJii0WDVBAbk9n3pM81MismjlCmubuVsBHVlFmXUdGFECy/vEqal5XArwR3V+VqLY+jAKdwBhcQwDVUoQZ1aAADCc/wCm+e9l68d+9j0brm5TMn8Afe5w+qypBT
L = 60d¯0
vz (x) G
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x
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z
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sha1_base64="iQsejc10lqvV7elqr+Sfe0ARqtI=">AAAB6nicbZC7SgNBFIbPxluMtxhLm8EgJE3YTRPLgI1lRHOBZAmzk9lkyOzsMnNWCEsewcZCEVufyM63cXIpNPGHgY//nMOc8weJFAZd99vJ7ezu7R/kDwtHxyenZ8XzUsfEqWa8zWIZ615ADZdC8TYKlLyXaE6jQPJuML1d1LtPXBsRq0ecJdyP6FiJUDCK1nqosOqwWHZr7lJkG7w1lJulQRWsWsPi12AUszTiCpmkxvQ9N0E/oxoFk3xeGKSGJ5RN6Zj3LSoaceNny1Xn5No6IxLG2j6FZOn+nshoZMwsCmxnRHFiNmsL879aP8Xwxs+ESlLkiq0+ClNJMCaLu8lIaM5QzixQpoXdlbAJ1ZShTadgQ/A2T96GTr3mWb63adRhpTxcwhVUwIMGNOEOWtAGBmN4hld4c6Tz4rw7H6vWnLOeuYA/cj5/AKi1jg8=
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(d)
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sha1_base64="WUgomnvU4FaQ4JOGbHHwyg7Y4fs=">AAAB6nicbZC7SgNBFIbPeo3xFmNpMxiEpAm7abQM2FhGNBdIljA7O5sMmZ1dZs4KYckj2FgoYusT2fk2Ti6FJv4w8PGfc5hz/iCVwqDrfjtb2zu7e/uFg+Lh0fHJaems3DFJphlvs0QmuhdQw6VQvI0CJe+lmtM4kLwbTG7n9e4T10Yk6hGnKfdjOlIiEoyitR6qYW1Yqrh1dyGyCd4KKs3yoAZWrWHpaxAmLIu5QiapMX3PTdHPqUbBJJ8VB5nhKWUTOuJ9i4rG3Pj5YtUZubJOSKJE26eQLNzfEzmNjZnGge2MKY7Nem1u/lfrZxjd+LlQaYZcseVHUSYJJmR+NwmF5gzl1AJlWthdCRtTTRnadIo2BG/95E3oNOqe5XubRgOWKsAFXEIVPLiGJtxBC9rAYATP8ApvjnRenHfnY9m65axmzuGPnM8fqjqOEA==
sha1_base64="Hv7hFYbybvEwIdcXlRv5ceViIfw=">AAAB6nicbZDLSgMxFIbPeK31VuvSTbAIdVNmutFlwY3LivYCnaFkMpk2NJMMSUYoQx9BBBeKuPVJfAR3Poh708tCW38IfPznHHLOH6acaeO6X87a+sbm1nZhp7i7t39wWDoqt7XMFKEtIrlU3RBrypmgLcMMp91UUZyEnHbC0dW03rmnSjMp7sw4pUGCB4LFjGBjrdtqdN4vVdyaOxNaBW8BlUbZr35/PPrNfunTjyTJEioM4VjrnuemJsixMoxwOin6maYpJiM8oD2LAidUB/ls1Qk6s06EYqnsEwbN3N8TOU60Hieh7UywGerl2tT8r9bLTHwZ5EykmaGCzD+KM46MRNO7UcQUJYaPLWCimN0VkSFWmBibTtGG4C2fvArtes2zfGPTqMNcBTiBU6iCBxfQgGtoQgsIDOABnuHF4c6T8+q8zVvXnMXMMfyR8/4DL9mQsQ==
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Figure 4.4: (a) Initial well-mixed configuration for two-dimensional DEM simulation of
bidisperse vertical chute flow with 4327 flowing grains. The chute width is W = 60d¯0 .
As in Fig. 4.2, black grains on both sides represent rough walls (only large particles are
used as wall grains here). (b) Segregated configuration after flowing for a total time of
t˜ = t/ d s ρs /Pw = 4.3 × 105 . (c) & (d) Spatiotemporal evolution of the large grain
p
concentration cl and the normalized velocity (vcen − vz ) ρs /Pw , respectively. (e) & (f)
p
Spatial profiles of concentration and normalized velocity at three times – t˜ = 4 × 103 , 4 ×
104 and 4 × 105 – as indicated by the dashed lines in (c) & (d).
73
Recall that the stress field in vertical chute flow is statically-determinate and given by:
τ(x) = φ ρs G|x|
P(x) = Pw
(4.14)
µ(x) = µw |x|/(W /2)
where µw = φ ρs GW /(2Pw ).
We note that while flow is driven by gravity, pressure is constant throughout the chute and
no hydrostatic pressure gradients are present. Therefore, segregation occurs only due to
shear-strain-rate-gradients, enabling us to consider this effect in isolation.
There are four important dimensionless parameters that describe vertical chute flow
of dense bidisperse granular mixtures: (1) W /d¯0 , the dimensionless chute width; (2) µw ,
the maximum stress ratio, which occurs at the walls and controls the total flow rate; (3)
cl0 (x), the initial large-grain concentration, which is not necessarily constant and can be a
spatially-varying field; and (4) d l /d s , the bidisperse grain-size ratio. Note that the parame-
ters {W /d¯0 , µw , cl0 , d l /d s } are related to the geometry, loads, and initial conditions of the
problem and are not material parameters.
As a representative base case, we consider the parameter group {W /d¯0 = 60, µw =
0.45, cl0 = 0.5, d l /d s = 1.5}. We then run the corresponding DEM simulation starting from
the well-mixed initial configuration shown in Fig. 4.4(a) and observe that after a simulation
time of t˜ = t/(d s ρs /Pw ) = 4.3 × 105 the large, dark grains segregate towards the regions
p
near the walls where the shear-strain-rate is greatest, while the small, light (in color) grains
gather in bands just inside these regions, as shown in Fig. 4.4(b). A well-mixed core persists
along the center of the vertical chute where the shear strain-rate is nearly zero. To obtain a
more quantitative picture of the segregation process, we coarse-grain the concentration field
cl and the velocity field vz in both space and time and plot contours of the spatiotemporal
evolution of cl and vz in Figs. 4.4(c) and (d), respectively. Both the concentration and
velocity fields evolve quickly in time during flow initiation. Then, over longer times, the
74
Figure 4.5: Collapse of Cdiff d¯2 γ(∂
˙ cl /∂ x) versus d¯2 cl (1 − cl )(∂ γ/∂
˙ x) for several cases of
vertical chute flow of bidisperse disks. Symbols represent coarse-grained quasi-steady
DEM field data, and the solid line is the best linear fit using CsegS = 0.23.
evolution of these fields becomes slower and may eventually be regarded as quasi-steady.
For the example shown in Fig. 4.4, a true steady-state, corresponding to a fully-segregated
configuration, is not reached within the simulated time window. Spatial profiles of the
concentration and velocity fields at three snapshots in time – t˜ = t/(d s ρs /Pw ) = 4 ×
p
103 , 4 × 104 and 4 × 105 as indicated by the dashed lines in Figs. 4.4(c) and (d) – are
plotted in Figs. 4.4(e) and (f). These three snapshots correspond to early, medium, and late
times with respect to the segregation process. The spatial cl profiles shown in Fig. 4.4(e)
demonstrate the transition from a well-mixed state to a segregated state with large-grain-
rich and small-grain-rich regions. The normalized velocity fields (vcen − vz ) ρs /Pw –
p
relative to the velocity at the center of the chute, vcen = vz (x = 0) – show that the velocity
field rapidly develops into a steady flow field (Fig. 4.4(f)) – even while the segregation
process is still occurring, and the cl field continues to evolve.
At long times, near the end of the simulated time window (t˜ = t/(d s ρs /Pw ) & 3 ×
p
105 ), the concentration field evolves very slowly – Dcl /Dt ≈ 0 – and therefore, according
seg
to equations (4.1) and (4.7), the total flux is approximately zero – wli = wdiff
i + wi ≈ 0i
– meaning that the segregation flux is approximately balanced by the diffusion flux in this
quasi-steady flow regime. Then, using the expressions for the two fluxes, equations (4.9)
75
and (4.10), this observation implies that
∂ cl ∂ γ˙
Cdiff d¯2 γ˙ S ¯2 l
≈ Cseg d c (1 − cl ) . (4.15)
∂x ∂x
The field quantities appearing in this expression may be obtained by coarse-graining the
DEM data in the quasi-steady regime. Therefore, since Cdiff has been previously de-
S as follows. First, we ac-
termined, (4.15) may be used to determine the parameter Cseg
¯ and vz by spatially coarse-graining 152 evenly-
quire the field quantities cl (and hence d)
distributed snapshots in time in the quasi-steady regime (t˜ > 3 × 105 ).1 Then, we aver-
age these fields in time, yielding fields that only depend on the spatial coordinate x, and
take spatial gradients to obtain ∂ cl /∂ x, γ˙ = ∂ vz /∂ x, and ∂ γ/∂
˙ x = ∂ 2 vz /∂ x2 .2 Next, as
suggested by (4.15), we plot Cdiff d¯2 γ(∂
˙ cl /∂ x) versus d¯2 cl (1 − cl )(∂ γ/∂
˙ x) in Fig. 4.5, in
which each point represents a unique x-position. A linear relation is observed, support-
ing our choice for the form of the constitutive equation for the segregation flux (4.10). In
order to obtain further evidence for this choice, we consider four additional cases: (1) a
lower flow rate {W /d¯0 = 60, µw = 0.375, cl0 = 0.5, d l /d s = 1.5}; (2) a narrower channel
{W /d¯0 = 40, µw = 0.45, cl0 = 0.5, d l /d s = 1.5}; (3) more large grains {W /d¯0 = 60, µw =
0.45, cl0 = 0.75, d l /d s = 1.5}; and (4) a larger size ratio {W /d¯0 = 60, µw = 0.45, cl0 =
0.5, d l /d s = 3.0}. Coarse-graining the quasi-steady fields for each case and including the
field data in Fig. 4.5, we observe a strong linear collapse. Finally, the dimensionless mate-
S may be obtained from the slope of the linear relation in Fig. 4.5 (indi-
rial parameter Cseg
S = 0.23 and
cated by the solid line). We determine the numerical value for disks to be Cseg
note that this value indeed appears to be independent of the grain-size ratio d l /d s .
1 Results are not particularly sensitive to the quasi-steady regime criterion, and we only provide this value
as a guideline.
2 A detailed description of our coarse-graining technique may be found in Appendix B.2.
76
4.4.2 Annular shear flow
The constitutive equation for the segregation flux (4.10) and the fitted value of the material
S = 0.23 should be general across different flow configurations. To
parameter for disks Cseg
test this, we apply the same process described in the preceding section for vertical chute
flow to a different flow configuration – annular shear flow. In this configuration, flow
is boundary-driven, but like the vertical chute, it is a constant-pressure configuration that
eliminates hydrostatic pressure gradients and displays only shear-rate-gradient-driven size-
segregation.
Our DEM simulations of annular shear flow of a dense, bidisperse granular mixture
follow the procedure described in Chapter 3.4.2 for the monodisperse case. Consider a
dense, bidisperse granular mixture in a two-dimensional annular shear cell with rough cir-
cular walls of inner radius R and outer radius Ro , shown in Fig. 4.6(a). The circumferential
velocity of the inner wall is prescribed to be vw , and its radial position is fixed. The outer
wall does not rotate, and its radial position fluctuates slightly so as to maintain a constant
imposed wall pressure Pw , utilizing a wall-position control method analogous to that de-
scribed in Section 3.2.2. In this configuration, flow tends to localize near the inner wall,
with the velocity field rapidly decaying with radial position. We choose the outer radius Ro
to be sufficiently large so that it does not affect the resulting flow and segregation fields,
and based on our experience, we take Ro = 2R. Therefore, the role of the outer wall is sim-
ply to apply a far-field pressure, and otherwise, it does not affect the flow and segregation
fields. Also, as in Chapter 3.4.2, we simulate the full shear cell instead of applying periodic
boundary conditions on a slice [27, 41, 78].
77
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Figure 4.6: (a) Initial well-mixed configuration for two-dimensional DEM simulation of
bidisperse annular shear flow with 40108 flowing grains. The inner wall radius is R = 60d¯0 ,
and the outer wall radius is Ro = 2R. The circular walls consist of rings of large particles.
(b) Segregated configuration after flowing for a total time of t˜ = t/ (R/vw ) = 584. (c) & (d)
Spatiotemporal evolution of the large grain concentration cl and the normalized rotational
velocity vθ /vw respectively. (e) & (f) Spatial profiles of concentration and normalized
velocity at three times – t˜ = 5, 50 and 500 – as indicated by the dashed lines in (c) & (d).
78
Recall that the stress field in annular shear flow is statically-determinate and given by:
τ(r) = τw (R/r)2
P(r) = Pw
(4.16)
2
µ(r) = µw (R/r)
with µw = τw /Pw ,
where τw is the inner-wall shear stress. It is important to note that τw is not directly pre-
scribed in our DEM simulations. Instead, the inner-wall velocity vw is prescribed, and
τw and hence µw arise as a result, as illustrated in Fig. 3.4(b) for the monodisperse case.
Therefore, the four important dimensionless parameters that describe annular shear flow
of dense bidisperse granular mixtures are (1) R/d¯0 , the dimensionless inner-wall radius;
q
2
(2) v˜w = (vw /R) πρs d¯0 /(4Pw ), the dimensionless inner-wall velocity, which determines
τw and µw ; (3) cl0 (r), the initial large-grain concentration; and (4) d l /d s , the bidisperse
grain-size ratio.
We choose a representative base case of quasi-static annular shear flow identified by the
parameter set {R/d¯0 = 60, v˜w = 0.01, cl0 = 0.5, d l /d s = 1.5}. The well-mixed initial config-
uration for the base-case DEM simulation is shown in Fig. 4.6(a), and the segregated con-
figuration after driving flow for a total time of t˜ = t/ (R/vw ) = 584 is shown in Fig. 4.6(b).
The large, dark grains segregate into a ring near the inner wall, while the small, light grains
form a band just outside this region. Outside of these bands, where the shear strain-rate
is very small, the grains remain well-mixed. Contours of the coarse-grained concentration
field cl and the tangential velocity field vθ /vw are shown in Figs. 4.6(c) and (d), illustrating
the time-evolution of these fields. Snapshots of the spatial profiles of the concentration and
velocity fields at selected stages of the segregation dynamics – t˜ = t/ (R/vw ) = 5, 50, and
500 as indicated by the dashed lines in Figs. 4.6(c) and (d) – are shown in Figs. 4.6(e) and
(f). The radial cl profiles demonstrate the formation of large-grain-rich and small-grain-
rich regions with a persistent well-mixed far-field. As in the case of vertical chute flow,
79
Figure 4.7: Collapse of Cdiff d¯2 γ(∂
˙ cl /∂ r) versus d¯2 cl (1 − cl )(∂ γ/∂
˙ r) for several cases of
annular shear flow of bidisperse disks. Symbols represent coarse-grained, quasi-steady
DEM field data, and the solid line is the best linear fit using CsegS = 0.23.
the velocity field quickly develops into a quasi-steady flow field – while the large grain
concentration field cl evolves over a longer time-scale before approaching a quasi-steady
state near the end of our simulated time window. The flowing zone is localized near the
inner wall with slow creeping flow observed far from the wall.
Again, at long times, near the end of the simulated time window (t˜ = t/ (R/vw ) & 500),
the concentration field evolves very slowly, which we identify as the quasi-steady regime.
In this regime, the segregation and diffusion fluxes approximately balance, implying that
∂ cl ∂ γ˙
Cdiff d¯2 γ˙ S ¯2 l
≈ Cseg d c (1 − cl ) . (4.17)
∂r ∂r
As for the case of vertical chute flow, we spatially coarse-grain the DEM data for 144
evenly-distributed snapshots in time in the quasi-steady regime (t˜ & 500) to obtain field
quantities in (4.17) and then time-average these fields (before taking spatial derivatives).
Next, we plot Cdiff d¯2 γ(∂
˙ cl /∂ r) versus d¯2 cl (1 − cl )(∂ γ/∂
˙ r) in Fig. 4.7, with each point rep-
resenting a unique r-position. Finally, this process is repeated for four additional cases: (1)
a lower inner-wall velocity {R/d¯0 = 60, vw = 0.001, cl0 = 0.5, d l /d s = 1.5}; (2) a smaller
inner-wall radius {R/d¯0 = 40, vw = 0.01, cl0 = 0.5, d l /d s = 1.5}; (3) more large grains
80
{R/d¯0 = 60, vw = 0.01, cl0 = 0.75, d l /d s = 1.5}; and (4) a larger size ratio {R/d¯0 = 60, vw =
0.01, cl0 = 0.5, d l /d s = 3.0}, and the coarse-grained, quasi-steady fields are included in the
data plotted in Fig. 4.7. Collectively, we observe a strong collapse to a linear relation. Cru-
cially, the slope of the linear relation in Fig. 4.7 (indicated by the solid line) gives the same
value for the dimensionless material parameter obtained by fitting to vertical chute flow
S = 0.23. This observation of agreement in the best fit value of CS obtained via
data, Cseg seg
two independent routes provides support for our choice of the constitutive equation for the
S for disks.
segregation flux (4.10) and the fitted value of Cseg
4.5 Validation of shear-strain-rate-gradient-driven segre-
gation flux in transient flow
With a calibrated model for shear-strain-rate-gradient-driven size-segregation in hand (4.11),
we may now couple (4.11) with the NGF model, (4.3) and (4.4), in order to obtain predic-
tions of the simultaneous, transient evolution of segregation and flow fields, using a fixed
set of material parameters for disks:
S
{µs = 0.272, b = 1.168, A = 0.9,Cdiff = 0.20,Cseg = 0.23}. (4.18)
In the preceding section, we only used DEM data from the quasi-steady regime to inform
S . In this section, we compare model predictions
the model and determine the parameter Cseg
of the transient evolution of segregation and flow fields to the DEM data for both vertical
chute flow and annular shear flow without further parameter adjustment as a validation test
of the model.
81
4.5.1 Vertical chute flow
Continuum model predictions are obtained by solving the governing partial differential
equations using finite-differences. Summarizing the coupled boundary/initial-value prob-
lem for flow and segregation in the context of vertical chute flow, the unknown fields are
the strain-rate γ˙ = ∂ vz /∂ x, the granular fluidity g, and the large-grain concentration cl . The
governing field equations are (1) the flow rule,
γ˙ = gµ, (4.19)
(2) the nonlocal rheology,
∂ 2g
g = gloc (µ(x), Pw ) + ξ 2 (µ(x)) (4.20)
∂ x2
with gloc and ξ given through (4.5) and (4.6), respectively, and (3) conservation of mass for
the large grains,
∂ cl ∂ 2 ˙∂c
l
l ∂ γ˙
+ ¯
−Cdiff d γ S ¯2 l
+Cseg d c (1 − c ) = 0, (4.21)
∂t ∂x ∂x ∂x
where d¯ = cl d l +(1−cl )d s . Since the problem is statically-determinate, the stress ratio field
µ(x) is given through (4.16), and the balance of momentum does not enter the governing
equations.
Regarding boundary conditions, we impose Dirichlet fluidity boundary conditions at the
walls – i.e., g = gloc (µw , Pw ) at x = 0 and x = W – as well as no flux boundary conditions
at the walls – i.e., wl = −Cdiff d¯2 γ(∂ S d¯2 cl (1 − cl )(∂ γ/∂
˙ cl /∂ x) + Cseg ˙ x) = 0 at x = 0 and
x = W . Due to the time derivative in (4.21), an initial condition for the concentration
field cl0 (x) = cl (x,t = 0) is required. In order to account for the concentration fluctuations
inherent in the initial state, we obtain the coarse-grained cl -field from the initial DEM
configuration for each case and utilize this field as the initial condition field cl0 (x) in the
82
respective continuum simulation.
Then, for a given case identified through a set of input parameters {W /d¯0 , µw , cl0 (x),
d l /d s }, we obtain numerical predictions of the coupled model utilizing finite differences as
follows. First, at a given point in time, the concentration field cl (x) is known, allowing the
average grain-size field to be calculated through d¯ = cl d l + (1 − cl )d s . Using the stress ratio
field µ(x) for vertical chute flow (4.14), the local fluidity gloc (µ, P) and the cooperativity
length ξ (µ) – equations (4.5) and (4.6) – may be calculated at each spatial grid point. Then,
the nonlocal rheology (4.20) may be used to solve for the fluidity field g(x) at the current
step, using central differences in space. The strain-rate field follows using (4.19), which
may be integrated to obtain the velocity field vz (x). Next, (4.21) is used to determine the
concentration field at the next time step utilizing the forward Euler method and central dif-
ferences in space with one modification – the gradient of cl appearing in the diffusion flux
term in (4.21) is treated implicitly in order to improve numerical stability. This completes
one time step, and this process is repeated to step forward in time and calculate the transient
evolution of the concentration and flow fields – cl (x,t) and vz (x,t). In our finite-difference
calculations, we utilize a fine spatial resolution of ∆x d,¯ and we have verified that the
time-step is sufficiently small in order to ensure stable, accurate results.
We compare predictions of the coupled, continuum model to DEM data for all five
cases of vertical chute flow considered in Section 4.4.1 – (1) base case, (2) lower flow rate,
(3) narrower chute width, (4) more large grains, and (5) larger grain-size ratio – in order
to test the generality of the model. Figure 4.8 summarizes the comparisons for these five
cases. The first column of Fig. 4.8 shows the spatiotemporal contours of the evolution of cl
measured in the DEM simulations for each case, and the second column shows comparisons
of the DEM simulations (solid lines) and the continuum predictions (dashed lines) for the
p
c field at four time snapshots – t = t/ d ρs /Pw = 4 × 103 , 2 × 104 , 1 × 105 and 4 × 105
l ˜ s
– indicated by the horizontal dashed lines in the first column of Fig. 4.8. Based on Fig. 4.8,
the coupled model generally does a good job capturing the salient features of the evolution
83
(a)
t˜ = 4 ⇥ 103 t˜ = 2 ⇥ 104
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t˜ = 4 ⇥ 105
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t˜ = 1 ⇥ 105
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t˜ = 4 ⇥ 105
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(b)
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(c)
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Figure 4.8: Comparisons of the coupled continuum model predictions with corresponding
DEM simulation results during transient flow and for five vertical chute flow test cases:
(a) Base case {W /d¯0 = 60, µw = 0.45, cl0 = 0.5, d l /d s = 1.5}; (b) Lower flow rate case
{W /d¯0 = 60, µw = 0.375, cl0 = 0.5, d l /d s = 1.5}; (c) Narrower chute width case {W /d¯0 =
40, µw = 0.45, cl0 = 0.5, d l /d s = 1.5}. Continued on the next page.
of the cl field across all cases. For instance, for the narrower chute case shown in Fig. 4.8(c),
the segregation process nearly completes within the simulated time window with the mixed
core along the center of the chute nearly disappearing, and the continuum model prediction
captures this observation well.
Regarding flow fields, comparisons of the DEM-simulated and continuum-model-pre-
dicted quasi-steady, normalized velocity fields at t˜ = 4 × 105 are shown in the third column
of Fig. 4.8. Since the velocity field evolves minimally during the segregation process, only
the flow field at long time is shown. We note that the velocity field is very well-predicted
in all cases – including the creeping regions far from the wall. This favorable comparison
84
(d)
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(e)
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Figure 4.8: Continued: (d) More large grains case {W /d¯0 = 60, µw = 0.45, cl0 =
0.75, d l /d s = 1.5}; (e) Larger size ratio case {W /d¯0 = 60, µw = 0.45, cl0 = 0.5, d l /d s =
3.0}. For each case, the first column shows spatiotemporal contours of the evolution of cl
measured in the DEM simulations. The second column shows comparisons of the DEM
simulations (solid lines) and continuum model predictions (dashed lines) of the cl field
at four time snapshots representing different stages of segregation: t˜ = 4 × 103 , 2 × 104 ,
1 × 105 and 4 × 105 in the sequence of top left, top right, bottom left, bottom right. The
third column shows comparisons of the simulated and predicted quasi-steady, normalized
velocity profiles at t˜ = 4 × 105 .
provides support of our generalization of the NGF model to bidisperse granular systems
discussed in Section 4.2.2 – in particular, the choices to use the average grain size d¯ in the
expression for the cooperativity length (4.6) and to continue to use the numerical value for
the nonlocal amplitude determined for monodisperse systems, A = 0.9, without refitting.
Overall, the coupled continuum model is capable of quantitatively predicting both the flow
fields and the segregation dynamics in vertical chute flow.
4.5.2 Annular shear flow
We utilize an analogous approach to obtain continuum model predictions for the transient
evolution of concentration and flow fields in annular shear flow. The stress field in annular
shear is given by (4.16), and the governing equations (4.19), (4.20), and (4.21) are modified
85
(a)
t˜ = 5 t˜ = 50
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t˜ = 550
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t˜ = 200
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t˜ = 550
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(b)
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(c)
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Figure 4.9: Comparisons of the coupled continuum model predictions with corresponding
DEM simulation results during transient flow and for five annular shear flow test cases:
(a) Base case {R/d¯0 = 60, v˜w = 0.01, cl0 = 0.5, d l /d s = 1.5}; (b) Lower inner-wall velocity
case {R/d¯0 = 60, v˜w = 0.001, cl0 = 0.5, d l /d s = 1.5}; (c) Smaller annular shear cell case
{R/d¯0 = 40, v˜w = 0.01, cl0 = 0.5, d l /d s = 1.5}. Continued on the next page.
slightly to account for the divergence operator in cylindrical coordinates. Also, since vw ,
not µw , is specified in our DEM simulations of annular shear, while µw is specified in
our continuum simulations, we iteratively adjust the value of µw input into our continuum
simulations in order to achieve the target value of vw in the predicted quasi-steady flow
field. Otherwise, our approach is the same. Dirichlet fluidity boundary conditions and no
flux boundary conditions are imposed at the walls, and the initial concentration field is
extracted from the initial DEM configuration for each case.
Then, we validate continuum model predictions against DEM data in the five cases
86
(d)
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(e)
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Figure 4.9: Continued: (d) More large grains case {R/d¯0 = 60, v˜w = 0.01, cl0 =
0.75, d l /d s = 1.5}; (e) Larger size ratio case {R/d¯0 = 60, v˜w = 0.01, cl0 = 0.5, d l /d s = 3.0}.
For each case, the first column shows spatiotemporal contours of the evolution of cl mea-
sured in the DEM simulations. The second column shows comparisons of the DEM simu-
lations (solid lines) and continuum model predictions (dashed lines) of the cl field at four
time snapshots representing different stages of segregation: t˜ = 5, 50, 200 and 550 in the
sequence of top left, top right, bottom left, bottom right. The third column shows compar-
isons of the simulated and predicted quasi-steady, normalized velocity profiles at t˜ = 550.
of annular shear flow discussed in Section 4.4.2 – (1) base case, (2) lower inner-wall ve-
locity, (3) smaller inner-wall radius, (4) more large grains, and (5) larger grain-size ratio.
Figure 4.9 summarizes the comparisons for these fives cases and is organized in the same
manner as Fig. 4.8. Again, the coupled, continuum model does a good job capturing the
segregation dynamics and the quasi-steady flow field and the changes in these fields in-
duced by changing the input parameters.
87
4.6 Addition to the segregation model – Pressure-gradient-
driven segregation flux
To this point, we have solely considered shear-strain-rate-driven size-segregation, focusing
on two flow configurations that isolate this mechanism. Now that a predictive continuum
model for this mechanism has been established, we return to pressure-gradient-driven size-
segregation in order to incorporate this mechanism and obtain a more general model. As
in the preceding section, our approach is to first propose a constitutive equation for the
pressure-gradient-driven segregation flux based on previous works in the literature and di-
mensional arguments. This constitutive equation will then be added to the segregation
model and tested against DEM simulations of flow in two new configurations – inclined
plane flow and planar shear flow with gravity – in which both shear-strain-rate-gradient-
driven and pressure-gradient-driven size-segregation are present.
Following Hill and Tan [103], we posit that that the segregation flux may be additively
decomposed into a term associated with shear-strain-rate-gradient driven segregation, wSi ,
given through (4.10), and a term associated with pressure-gradient-driven segregation, wPi ,
seg
so that wi = wSi + wPi . Then, we expect that wPi acts along negative pressure gradients,
so that large grains will tend to rise to the top of a flowing granular media experiencing
hydrostatic pressure gradients. Additionally, the prefactor of wPi should involve the factor
cl (1 − cl ) in order to ensure that segregation ceases when the mixture becomes either all
large or all small grains as well as a factor γ,
˙ so that segregation ceases when there is no
flow. Therefore, our expected scaling for the pressure-gradient-driven segregation flux is
∂P
wPi ∝ −γc
˙ l (1 − cl ) .
∂ xi
The remaining prefactor must have units of [Length3 /Force], which using d¯ as the charac-
teristic length-scale and Pd¯ as the characteristic force-scale, we take to be d¯2 /P. Therefore,
88
we propose the following constitutive equation for the pressure-gradient-driven segregation
flux:
d¯2 γ˙ l ∂P
wPi = −Cseg
P
c (1 − cl ) , (4.22)
P ∂ xi
P is an additional dimensionless material parameter which scales the rate of
where Cseg
pressure-gradient-driven segregation. We emphasize that the hypothesized constitutive
equation (4.22) is based on a set of minimal requirements, and it is possible – if not likely
– that the flux will depend on other dimensionless quantities, such as the inertial number I
or the grain-size ratio d l /d s . Indeed, we will show that d l /d s does have an additional ef-
fect on the pressure-gradient-driven segregation flux that was not present in the shear-rate-
gradient-driven segregation flux. In the following sections, we will use a similar approach
P and test the constitutive equation
to determine the dimensionless material parameter Cseg
(4.22) as was utilized in Sections 4.4 and 4.5.
Adding the constitutive equation (4.22) to the segregation model in Section 4.2.3, the
total segregation flux now accounts for both mechanisms:
seg P d¯2 γ˙ l ∂P S ¯2 l ∂ γ˙
wi = −Cseg c (1 − cl ) +Cseg d c (1 − cl ) , (4.23)
P ∂ xi ∂ xi
and using (4.23) in conservation of mass (4.1), we obtain the following governing equation
for cl :
Dcl ∂ ¯2 l
P d γ˙ l l ∂P l ∂ γ˙ ∂ c
S ¯2 l
+ −Cseg c (1 − c ) +Cseg d c (1 − c ) −Cdiff d¯ γ˙
2
= 0, (4.24)
Dt ∂ xi P ∂ xi ∂ xi ∂ xi
S = 0.23,CP }.
with the dimensionless material parameter set {Cdiff = 0.20,Cseg seg
4.7 Inclined plane flow
Inclined plane flow is a broadly studied flow configuration [7, 11, 30, 58], which we will
P and test the coupled model
utilize to determine the value of the material parameter Cseg
89
(a) L= (b)
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z
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0
0
60d¯
H=
G
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vx (z) flowing
rough
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grains
bottom
✓
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Figure 4.10: (a) Initial well-mixed configuration for two-dimensional DEM simulation of
bidisperse inclined plane flow with 4311 flowing grains. The layer thickness is H = 60d¯0
with d¯0 the global mean particle size, and the inclination angle is θ = 20◦ . Black grains
represent the bottom, fixed surface. (b) Segregated configuration after flowing for a total
time of t˜ = t/ d s /G = 2.45 × 104 . (c) & (d) Spatiotemporal evolution of the large grain
p
l
√
concentration c and the normalized velocity down the incline, vx / Gds , respectively. (e)
& (f) Spatial profiles of concentration and normalized velocity at three times – t˜ = 2 × 102 ,
2 × 103 , and 2 × 104 – as indicated by the dashed lines in (c) & (d).
90
in the presence of hydrostatic pressure gradients. Consider a dense, bidisperse granular
mixture flowing down a long inclined surface under the action of gravity G with a surface
inclination angle θ and a layer thickness H. Figure 4.10(a) shows a representative DEM
setup for a layer thickness of H = 60d¯0 and an inclination angle of θ = 20◦ . The base of
the inclined plane (z = H) consists of a layer of fixed large grains, and the top of the layer
(z = 0) is a free surface. Periodic boundary conditions are employed along the direction of
the inclined plane – i.e., the x-direction. Throughout, we take the length along the periodic
direction to be 60d¯0 .
The stress distribution in the layer is statically-determinate and given by:
τ(z) = φ ρs Gz sin θ
P(z) = φ ρs Gz cos θ (4.25)
µ(z) = tan θ .
Note that the stress ratio µ = tan θ is spatially constant in this flow configuration and given
through the inclination angle θ . The four important dimensionless parameters that describe
inclined plane flow of dense bidisperse granular mixtures are (1) H/d¯0 , the dimensionless
layer thickness, (2) θ , the inclination angle, which controls the total flow rate, (3) cl0 (z), the
initial large-grain concentration field, and (4) d l /d s , the bidisperse grain-size ratio.
We choose a representative base case of inclined plane flow associated with the pa-
rameter set {H/d¯0 = 60, θ = 20◦ , cl0 = 0.5, d l /d s = 1.5}. The well-mixed initial configu-
ration corresponding to the base-case DEM simulation is shown in Fig. 4.10(a). As flow
and segregation progress, small grains move towards the bottom of the layer, while large
grains segregate towards the top free surface. The segregated configuration after a total
time of t˜ = t/ d s /G = 2.45 × 104 is shown Fig. 4.10(b). Contours of the coarse-grained
p
√
large-grain concentration and the normalized velocity field down the incline vx / Gds are
shown in Figs. 4.10(c) and (d), demonstrating the dynamics of the segregation and flow
fields. Snapshots of the spatial profiles of the concentration and velocity fields at three
91
selected points in time – t˜ = 2 × 102 , 2 × 103 , and 2 × 104 as indicated by the dashed lines
in Figs. 4.10(c) and (d) – are plotted in Figs. 4.10(e) and (f). The concave shape of the
flow field vx (z) is consistent with the so-called “Bagnold profile” for flow down an inclined
plane [7, 30] – vx (z) ∝ H 3/2 − z3/2 – which is obtained by integrating the strain-rate field
predicted by the local inertial rheology over the thickness of the layer. A quasi-steady state
in both the concentration and flow fields is attained at sufficiently long times.
Both hydrostatic pressure-gradients and shear-strain-rate-gradients are present in in-
clined plane flow. It is notable that due to the shape of the velocity profile (Fig. 4.10(f)), the
shear-strain-rate is greatest at the bottom of the layer. Therefore, shear-rate-gradients drive
smaller grains towards the surface (lower shear-rate region), while pressure-gradients drive
smaller grains to the bottom of the layer (higher pressure region) – i.e., shear-rate-gradient-
driven and pressure-gradient-driven segregation fluxes are opposed. The observation that
small grains segregate towards the bottom of the layer (Figs. 4.10(b) and (c)) suggests that
pressure-gradient-driven segregation dominates shear-rate-gradient-driven segregation in
this configuration. This feature makes inclined plane flow a good configuration for deter-
P .
mining the numerical value of the material parameter Cseg
For the base case, a quasi-steady state for both the concentration and flow fields is
reached after t˜ & 1.5 × 104 . In this regime, Dcl /Dt ≈ 0, and the diffusion flux approxi-
mately balances the total segregation flux, wdiff S P
i + wi + wi ≈ 0i , which implies that
∂ γ˙ ∂ cl ¯2
S ¯2 l P d γ˙ l ∂P
Cseg d c (1 − cl ) −Cdiff d¯2 γ˙ ≈ Cseg c (1 − cl ) . (4.26)
∂z ∂z P ∂z
We spatially coarse-grain the DEM data for 139 evenly-distributed snapshots in time in
the quasi-steady regime (t˜ & 1.5 × 104 ) to obtain the field quantities in (4.26) – i.e., cl ,
d,¯ and vx in addition to P and ∂ P/∂ z which are known and therefore not determined
by coarse-graining – and then time-average these fields, before taking spatial derivatives
to acquire ∂ cl /∂ z, γ˙ = ∂ vx /∂ z, and ∂ γ/∂ S
˙ z. Additionally, the values of Cdiff and Cseg
92
Figure 4.11: Collapse of Cseg S d¯2 cl (1 − cl )(∂ γ/∂
˙ z) −Cdiff d¯2 γ(∂
˙ cl /∂ z) versus (d¯2 γ/P)c
˙ l (1 −
cl )(∂ P/∂ z) for several cases of inclined plane flow of bidisperse disks with a grain-size
ratio of d l /d s = 1.5. Symbols represent coarse-grained, quasi-steady DEM field data, and
the solid line is the best linear fit using CsegP = 0.35.
S d¯2 cl (1 −
are known and not further adjusted. Next, as suggested by (4.26), we plot Cseg
˙ z) − Cdiff d¯2 γ(∂
cl )(∂ γ/∂ ˙ cl /∂ z) versus (d¯2 γ/P)c
˙ l (1 − cl )(∂ P/∂ z) in Fig. 4.11, in which
each point represents a unique z-position. This process is repeated for three additional
cases: (1) a lower inclination angle {H/d¯0 = 60, θ = 18◦ , cl0 = 0.5, d l /d s = 1.5}; (2)
a thinner layer {H/d¯0 = 40, θ = 20◦ , cl0 = 0.5, d l /d s = 1.5}; and (3) more large grains
{H/d¯0 = 60, θ = 20◦ , cl0 = 0.75, d l /d s = 1.5}, and the coarse-grained, quasi-steady fields
are included in the data plotted in Fig. 4.11. The collapse is comparatively noisier than that
observed in the case of shear-rate-gradient-driven segregation; however, a dependence is
evident, which may be fit by a linear relation. Then, the dimensionless material parameter
P is obtained from the slope of the best-fit linear relation in Fig. 4.11 (indicated by the
Cseg
P = 0.35 for a grain-size ratio of d l /d s = 1.5. We note
solid line) and is determined to be Cseg
P to depend upon the grain-size ratio. For exam-
that we observe the numerical value of Cseg
ple, for a case with a larger size ratio {H/d¯0 = 60, θ = 20◦ , cl0 = 0.5, d l /d s = 3.0}, we find
P = 0.65. Therefore, this data is not included in Fig. 4.11. More completely charac-
that Cseg
terizing the dependence of the pressure-gradient-driven segregation flux on the grain-size
ratio will be the subject of future work.
93
(a)
t˜ = 2 ⇥ 103
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t˜ = 2 ⇥ 102
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t˜ = 1 ⇥ 104
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t˜ = 2 ⇥ 104
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t˜ = 2 ⇥ 104
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(b)
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Figure 4.12: Comparisons of the coupled continuum model predictions with correspond-
ing DEM simulation results during transient flow and for five inclined plane flow test
cases: (a) Base case {H/d¯0 = 60, θ = 20◦ , cl0 = 0.5, d l /d s = 1.5}; (b) Lower flow-
rate case {H/d¯0 = 60, θ = 18◦ , cl0 = 0.5, d l /d s = 1.5}; (c) Smaller layer thickness case
{H/d¯0 = 40, θ = 20◦ , cl0 = 0.5, d l /d s = 1.5}. Continued on the next page.
The calibrated model for both shear-rate-gradient-driven and pressure-gradient-driven
size-segregation (4.24) may now be coupled with the NGF model, (4.3) and (4.4), and used
to obtain predictions of the transient evolution of segregation and flow fields. We use a
fixed set of material parameters for disks with a size-ratio of d l /d s = 1.5 – {µs = 0.272, b =
S = 0.23,CP = 0.35}. Continuum model predictions are
1.168, A = 0.9,Cdiff = 0.20,Cseg seg
numerically obtained as described in Section 4.5.1 for vertical chute flow. We utilize a
Dirichlet fluidity boundary condition at the rough bottom – i.e., g = gloc (µ = tan θ , P =
φ ρs GH) at z = H – and a homogeneous Neumann fluidity boundary condition at the free
surface – ∂ g/∂ z = 0 at z = 0 – in addition to no flux boundary conditions at both the bottom
94
(d)
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Figure 4.12: Continued: (d) More large grains case {H/d¯0 = 60, θ = 20◦ , cl0 =
0.75, d l /d s = 1.5}; (e) Larger size ratio case {H/d¯0 = 60, θ = 20◦ , cl0 = 0.5, d l /d s = 3.0}
with specifically fitted CsegP = 0.65. For each case, the first column shows spatiotempo-
ral contours of the evolution of cl measured in the DEM simulations. The second column
shows comparisons of the DEM simulations (solid lines) and continuum model predictions
(dashed lines) of the cl field at four time snapshots representing different stages of segrega-
tion: t˜ = 2 × 102 , 2 × 103 , 1 × 104 and 2 × 104 in the sequence of top left, top right, bottom
left, bottom right. The third column shows comparisons of the simulated and predicted
quasi-steady, normalized velocity profiles at t˜ = 2 × 104 .
and top. The initial concentration field is extracted from the initial DEM configuration for
each case.
We then compare continuum model predictions against DEM data in the four cases of
inclined plane flow discussed above – (1) base case, (2) lower inclination angle, (3) thinner
layer, and (4) more large grains – as a partial validation of the model. Figures 4.12(a)-
(d) summarize the comparisons and are organized in the same manner as Figs. 4.8 and
4.9. The predictions of the continuum model with the addition of the pressure-gradient-
driven segregation flux term match well with DEM measurements for inclined plane flow
for both the segregation dynamics and the quasi-steady flow field3 in all cases. Notably,
no slip boundary condition vx = 0 is enforced at the rough bottom in the continuum model predictions. 3A
However, it is unavoidable to have some slip at the base in the DEM simulations. In order to compare velocity
profiles, the DEM velocity profile is shifted to have zero velocity at z = H.
95
the model gives gives excellent predictions of the evolution of the cl field in spite of the
comparatively noisier data of Fig. 4.11. We also note that for the layer thicknesses and
inclination angles considered here, the NGF model plays a smaller role than in vertical
chute flow or annular shear flow. Indeed, the predicted velocity fields in Fig. 4.12 are quite
close to the Bagnold-type profile predicted by the local inertial rheology. For completeness,
we have also included continuum model predictions and comparisons to DEM data for a
higher grain-size-ratio of d l /d s = 3.0 using Cseg
P = 0.65 in Fig. 4.12(e), illustrating that the
pressure-gradient-driven segregation flux involves additional dependence on the size-ratio.
4.8 Planar shear flow with gravity
The continuum model is now equipped to predict size-segregation phenomena involv-
ing both pressure-gradient-driven and shear-rate-gradient-driven segregation mechanisms.
As a final test of the model, we consider another widely-studied quasi-one-dimensional
flow configuration – planar shear flow with gravity, as discussed in Chapter 3.4.1 for the
monodisperse case. Consider a dense, bidisperse granular mixture between two rough
parallel walls, separated by a distance H. The top wall moves in the x-direction with a
velocity vw , while the bottom wall remains fixed. The top wall imposes a compressive nor-
mal stress Pw , and the gravitational body force φ ρs G acts along the z-direction. Periodic
boundary conditions are employed along the x-direction, and the length of the region in the
x-direction is L = 60d¯0 , where d¯0 is the system-wide average grain size. The DEM set-up
is shown in Fig. 4.13(a) for H = 120d¯0 , which is sufficiently large so that it does not affect
flow and segregation fields and is used in all subsequently-considered cases.
96
The statically-determinate stress field in planar shear flow with gravity is
τ(z) = τw
P(z) = Pw + φ ρs Gz (4.27)
τ(z) µw
µ(z) = = .
P(z) 1 + z/`
where µw = τw /Pw is the maximum value of µ, occurring at the wall (z = 0), and ` =
Pw /φ ρs G is the loading length-scale, which is defined as the ratio of the wall pressure to the
gravitational body force and is distinct from the dimensions H and L. As in annular shear
flow, τw and hence µw are not directly prescribed in our DEM simulations. The top-wall ve-
locity vw is prescribed with τw and µw arising as a result. The four important dimensionless
parameters that describe planar shear flow of dense bidisperse granular mixtures with grav-
q
ity are (1) `/d¯0 , the dimensionless loading length-scale; (2) v˜w = (vw /`) πρs d¯02 /(4Pw ),
the dimensionless top-wall velocity, which determines τw and µw ; (3) cl0 (z), the initial
large-grain concentration field; and (4) d l /d s , the bidisperse grain-size ratio.
The remainder of the analysis follows the process of the preceding sections. We con-
sider a base case of quasi-static planar shear flow with gravity identified by the following
parameter set: {`/d¯0 = 60, v˜w = 0.02, cl0 = 0.5, d l /d s = 1.5}. The well-mixed initial con-
figuration for the base-case DEM simulation is shown in Fig. 4.13(a), and the segregated
configuration after a total time of t˜ = t/(`/vw ) = 2289 is shown in Fig. 4.13(b). Contours of
the coarse-grained concentration field cl and the normalized velocity field vx /vw are shown
in Figs. 4.13(c) and (d), illustrating the time-evolution of these fields. Finally, snapshots
of spatial profiles of these fields at selected times – t˜ = 20, 200, and 2000 – are shown in
Figs. 4.13(e) and (f). We observe that large grains segregate in a band just underneath the
top wall with a band of small grains beneath and a persistently well-mixed base of grains.
Note that since the shear-rate decreases with z and the pressure increases with z, both segre-
gation fluxes act along the same direction, and at this point, the respective contributions of
97
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wall x
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z
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flowing
vx (z)
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grains G
fixed
wall
L = 60d¯0
(c) (d)
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Figure 4.13: (a) Initial well-mixed configuration for two-dimensional DEM simulation of
bidisperse planar shear flow with gravity with 8701 flowing grains. Black grains represent
rough walls. (b) Segregated configuration after flowing for a total time of t˜ = t/(`/vw ) =
2289. (c) & (d) Spatiotemporal evolution of the large grain concentration cl and normalized
velocity field vx /vw , respectively. (e) & (f) Spatial profiles of concentration and normalized
velocity at three times – t˜ = 20, 200 and 2000 – as indicated by the dashed lines in (c) &
(d).
98
the two mechanisms is unclear. For this reason, planar shear flow with gravity would have
been a poor configuration for determining the forms of constitutive equations and fitting
material parameters but is an excellent configuration for model validation.
Again, at sufficiently long time, a quasi-steady-state is reached, implying that the fluxes
approximately balance as given in (4.26). We spatially coarse-grain the DEM data for 148
evenly-distributed snapshots in the quasi-steady regime (t˜ & 1600) to obtain all field quan-
tities in (4.26) and then time-average these fields (before taking spatial derivatives). As pre-
viously noted, unlike in inclined plane flow, both segregation mechanisms act in the same
direction. We compare the magnitudes of the two segregation fluxes (4.23) at each point in
space, revealing that the shear-rate-gradient-driven segregation flux dominates the pressure-
gradient-driven flux. That is to say, the right-hand-side of the flux balance equation (4.26)
for planar shear flow with gravity is negligible compared to the magnitudes of the terms on
the left-hand-side. This is because flow decays rapidly in planar shear flow with gravity,
giving rise to strong shear-strain-rate-gradients – a feature not present in inclined plane
flow. With this in mind, we plot Cdiff d¯2 γ(∂ P (d¯2 γ/P)c
˙ cl /∂ z) +Cseg ˙ l (1 − cl )(∂ P/∂ z) versus
d¯2 cl (1 − cl )(∂ γ/∂
˙ z) in Fig. 4.14 with each point representing a unique z-position, so that
S . The process is repeated for three ad-
the relation is expected to be linear with a slope of Cseg
ditional cases: (1) a lower top-wall velocity {`/d¯0 = 60, v˜w = 0.002, cl0 = 0.5, d l /d s = 1.5};
(2) a smaller loading length-scale {`/d¯0 = 40, v˜w = 0.02, cl0 = 0.5, d l /d s = 1.5}; and (3)
more large grains {`/d¯0 = 60, v˜w = 0.02, cl0 = 0.75, d l /d s = 1.5}, and the coarse-grained,
quasi-steady fields are included in the data plotted in Fig. 4.14. Indeed, we observe a col-
lapse to a linear relation with the slope of the linear relation (indicated by the solid line in
Fig. 4.14) consistent with the previously-fitted value of the dimensionless material param-
S = 0.23.
eter, Cseg
We then obtain predictions of the transient evolution of the segregation and flow fields
using the coupled, continuum model and a fixed set of material parameters for disks with a
size ratio of d l /d s = 1.5 – {µs = 0.272, b = 1.168, A = 0.9,Cdiff = 0.20,Cseg
S = 0.23,CP =
seg
99
Figure 4.14: Collapse of Cdiff d¯2 γ(∂
˙ cl /∂ z)+CsegP (d¯2 γ/P)c
˙ l (1 −cl )(∂ P/∂ z) versus d¯2 cl (1 −
cl )(∂ γ/∂
˙ z) for several cases of planar shear flow of bidisperse disks with gravity with a
grain-size ratio of d l /d s = 1.5. Symbols represent coarse-grained, quasi-steady DEM field
S = 0.23.
data, and the solid line is the best linear fit using Cseg
0.35}. As described in Section 4.5.1 for vertical chute flow, we utilize Dirichlet fluidity
boundary conditions and no flux boundary conditions at the rough walls – z = 0 and z = H.
The initial concentration field is extracted from the initial DEM configuration for each case.
In continuum simulations, the input value of µw is iteratively adjusted to achieve the target
value of vw in the predicted quasi-steady flow field.
Comparing transient model predictions of segregation dynamics and flow with DEM
measurements for the four cases of planar shear flow with gravity – (1) base case, (2) lower
top-wall velocity, (3) smaller loading length-scale, and (4) more large grains – good agree-
ment is observed, as shown in Figs. 4.15(a) through (d). The coupled continuum model
prediction for a larger size-ratio of d l /d s = 3.0 is shown in Fig. 4.15(e), obtained using
P = 0.65, which also displays good agreement. We emphasize that the continuum model
Cseg
gives good predictions of which segregation mechanism is dominant without supposing
any information beforehand. This highlights the importance of studying size-segregation
in a diversity of flow configurations – rather than focusing on a single configuration as is
often done in the literature [96, 98, 103].
100
(a)
t˜ = 20 t˜ = 200
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t˜ = 500
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t˜ = 2000
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t˜ = 2000
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(b)
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t˜ = 1600
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t˜ = 1600
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(c)
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Figure 4.15: Comparisons of the coupled continuum model predictions with corresponding
DEM simulation results during transient flow and for five planar shear flow with gravity
test cases: (a) Base case {`/d¯0 = 60, v˜w = 0.02, cl0 = 0.5, d l /d s = 1.5}; (b) Lower top-wall
velocity case {`/d¯0 = 60, v˜w = 0.002, cl0 = 0.5, d l /d s = 1.5}; (c) Smaller loading length-
scale case {`/d¯0 = 40, v˜w = 0.02, cl0 = 0.5, d l /d s = 1.5}. Continued on the next page.
4.9 Discussion and Conclusion
In this chapter, we have studied coupled size-segregation and flow in two-dimensional,
bidisperse, dense granular systems in four different configurations and developed a phe-
nomenological continuum model that captures the simultaneous evolution of both segre-
gation and flow fields. In our approach, we first focused on the shear-rate-gradient-driven
size-segregation mechanism in two configurations in which the pressure field is uniform –
vertical chute flow and annular shear flow – and based on observations from DEM simu-
lations, we proposed a phenomenological constitutive equation for the shear-rate-gradient-
driven flux. Then, we re-incorporated the pressure-gradient-driven segregation mechanism
101
(d)
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Figure 4.15: Continued: (d) More large grains case {`/d¯0 = 60, v˜w = 0.02, cl0 =
0.75, d l /d s = 1.5}; (e) Larger size ratio case {`/d¯0 = 60, v˜w = 0.02, cl0 = 0.5, d l /d s = 3.0}.
For each case, the first column shows spatiotemporal contours of the evolution of cl mea-
sured in the DEM simulations. The second column shows comparisons of the DEM simu-
lations (solid lines) and continuum model predictions (dashed lines) of the cl field at four
time snapshots representing different stages of segregation: t˜ = 1 × 103 , 1 × 104 , 2.5 × 104 ,
and 1 × 105 in the sequence of top left, top right, bottom left, bottom right. The third col-
umn shows comparisons of the simulated and predicted quasi-steady, normalized velocity
profiles at t˜ = 2 × 104 .
using two additional configurations – inclined plane flow and planar shear flow with grav-
ity – and similarly proposed a phenomenological constitutive equation for the segrega-
tion flux driven by this mechanism. Importantly, the coupled model correctly predicts the
segregation direction due to disparate particle sizes in dense granular flows: small grains
tend to segregate towards regions of lower shear-strain-rate and higher pressure and vice
versa for large grains. The segregation model involves three dimensionless parameters
S ,CP } which multiply the three fluxes appearing in the model – diffusion, shear-
{Cdiff ,Cseg seg
rate-gradient-driven, and pressure-gradient-driven fluxes, respectively. By coupling the
segregation model with the NGF model adapted to bidisperse systems, we may quantita-
tively predict both the flow fields and the segregation dynamics for dense flows of bidis-
perse disks, without requiring any flow field quantity input. Notably, the coupled model is
102
(a)
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Figure 4.16: Collapse of Cdiff d¯2 γ(∂
˙ cl /∂ xi ) versus d¯2 cl (1 − cl )(∂ g/∂ xi ) for several cases
of (a) vertical chute flowpand (b) annular shear flow of bidisperse disks. Collapse of
Cdiff d¯2 γ(∂ ¯ l (1 − cl )(∂ T /∂ xi ) for several cases of (c) vertical chute
˙ cl /∂ xi ) versus ρs /Pdc
flow and (d) annular shear flow of bidisperse disks. Symbols represent coarse-grained,
quasi-steady DEM field data, and the solid lines represent the best linear fit using Cseg S =
0.08 for (a) and (b) and CsegS = 0.7 for (c) and (d).
able to automatically capture when one segregation mechanism dominates over the other
(e.g., planar shear flow with gravity) or when both are important (e.g., inclined plane flow),
without any a priori information.
Size-segregation in granular materials is a complex and rich problem, so there remain
many avenues for model improvement and unresolved research questions to be answered.
One important outstanding question relates to the constitutive equation for the shear-rate-
gradient-driven segregation flux (4.10). Although our use of a constitutive equation driven
by gradients in γ˙ has worked well, there are other theories in the literature based on gra-
dients of other field quantities. In particular, Hill and coworkers [97, 103] have proposed
that gradients in the kinetic stress – which is related to velocity fluctuations and hence the
granular temperature – drive segregation. Since Zhang and Kamrin [64] have established
103
a connection between velocity fluctuations and the granular fluidity g, it is possible to pro-
pose other forms for the constitutive equation for wSi based on gradients in g. For example,
instead of (4.10), consider the following form for the segregation flux:
S ¯2 l ∂g
wSi = Cseg d c (1 − cl ) . (4.28)
∂ xi
Then, applying the quasi-steady-state condition,
∂ cl ∂g
Cdiff d¯2 γ˙ S ¯2 l
≈ Cseg d c (1 − cl ) , (4.29)
∂ xi ∂ xi
to the long-time DEM data for vertical chute flow and annular shear flow, we obtain the
collapses shown in Fig. 4.16(a) and (b), respectively.4 The solid lines represent the best
S = 0.08. The collapses are reasonable but not as strong as those shown
linear fit using Cseg
in Figs. 4.5 and 4.7 for a segregation flux based on gradients in the shear-strain-rate, leading
to our choice to work with the constitutive equation (4.10).
Additionally, we have also tested a possible constitutive equation for the segregation
flux driven by gradients in the granular temperature, defined as T = (δ v)2 , where δ v is the
velocity fluctuation.5 Then, consider the following form for the segregation flux:
wSi = Cseg
S ¯ l (1 − cl ) ∂ T ,
ρs /Pdc
p
(4.30)
∂ xi
where the inertial time d¯ ρs /P is included in the prefactor for dimensional reasons. As
p
4 The coarse-grained values of g and its gradient are obtained using the coarse-grained values of γ˙ and
its gradient along with µ and its gradient which are statically-determinate and therefore not determined by
coarse-graining.
5 The definitions of the granular temperature T and the velocity fluctuation δ v as well as the coarse-
graining method used to obtain these quantities follow Zhang and Kamrin [64]. They are treated as field
quantities just as γ.
˙ Following the description in (B.2), instantaneous T at a bin is the grain area weighted
summation of the square of the difference of the grain instantaneous velocity vector and the coarse-grained
instantaneous velocity field at that grain center, for all grains that are intersected by this bin.
104
above, upon applying the quasi-steady-state condition,
∂ cl ¯ l (1 − cl ) ∂ T ,
Cdiff d¯2 γ˙ S
ρs /Pdc
p
≈ Cseg (4.31)
∂ xi ∂ xi
Figs. 4.16(c) and (d) show the collapses for granular-temperature-gradient-driven segrega-
tion for the long-time DEM data for vertical chute flow and annular shear flow, respectively.
S = 0.7. The collapse is quite
In this case, the solid lines represent the best linear fit using Cseg
good; however, in order to utilize the constitutive equation (4.30) in practice, an additional
constitutive equation that gives the granular temperature field in terms of other continuum
quantities – such as strain-rate, stress, and fluidity – is needed. In summary, we acknowl-
edge that the possibility for an alternative form for the segregation flux wSi remains and
that future work involving additional, more-complex flow configurations – e.g., heap flow
or split-bottom flow – will be required to conclusively judge which constitutive equation is
the most predictive.
Another question regarding the constitutive equation for the segregation flux is the cl de-
pendence of the pre-factor. We simply use a symmetric dependence cl (1 − cl ), while other
researchers [98, 109] invoke more involved expressions that depend on both cl and d l /d s in
a more complex manner. Besides the constitutive equation for wSi , we have observed that
the material parameter in the constitutive equation for the pressure-gradient-driven flux,
P , is different for the two size ratios of 1.5 and 3.0 that we considered. In order to obtain
Cseg
a more general model, it will be necessary to determine the functional dependence of wPi
on d l /d s .
In order to apply the proposed continuum model in more complex flow configurations,
such as heap flows or split-bottom flow, it will be necessary to extend the theory to three-
dimensions. In developing a justification for a three-dimensional generalization based on
first-principles continuum thermo-mechanics [60], several important questions will need to
be answered: (1) What is the physical nature of the segregation flux? (2) Is the segregation
105
flux related to a dissipative or an energetic microforce? and (3) What restrictions does the
second law of thermodynamics place on the segregation flux? Such a general treatment is
invaluable as it will provide a firm foundation for further generalization to more complex
systems, such as continuously polydisperse systems [110, 111]. Finally, a robust numerical
implementation of the complex system of coupled equations that is capable of addressing
problems in general geometries is needed. Possible approaches may be based on the finite-
element method – as in previous work involving the NGF model [72] – or the material-point
method [35, 37], which is better equipped to model flows undergoing large distortions and
with evolving free surfaces, such as those arising in rotating drums.
106
Chapter 5
Conclusions and outlook
5.1 Concluding remarks
From both scientific and engineering perspectives, granular materials are an interesting
class of materials that pose important challenges to continuum modeling. The ability to
better understand and model these materials is desirable in multiple industries as well as in
geophysics in order to mitigate natural disasters such as landslides. The work in this thesis
addressed several critical questions related to flow, the flow threshold, and size-segregation
in dense granular materials. The main conclusions and contributions are summarized in the
following:
• The nonlocal granular fluidity (NGF) model was applied to steady, dense granular
heap flow in channels with frictional side walls, and the model was shown to quanti-
tatively predict all salient experimental observations, including (1) a rapidly-flowing
surface layer with rate-dependent thickness; (2) a creeping bed characterized by an
exponentially decaying velocity field with rate-independent decay length; and (3) the
relationships between volume flow-rate, maximum velocity, and surface inclination
angle.
• The flow threshold in dense granular materials demonstrates size-dependence across
107
108
three investigated configurations – planar shear flow with gravity, annular shear flow,
and vertical chute flow – in which the flow threshold increases as the size of a granu-
lar medium decreases. The dynamical form of the NGF model quantitatively captures
both the size-dependence of the flow threshold and steady flow fields in all three flow
configurations.
• A phenomenological continuum model for size-segregation in dense flows of bidis-
perse disks was proposed, involving (1) a diffusion flux, (2) a shear-strain-rate-
gradient-driven segregation flux, and (3) a pressure-gradient-driven segregation flux.
By coupling this model with an adapted NGF model for bidisperse granular materi-
als, we are able to quantitatively predict both flow fields and segregation dynamics
simultaneously, which no previous continuum model has achieved. The continuum
model was tested by comparing model predictions against DEM simulation results of
dense bidisperse flow in four configurations – vertical chute flow, annular shear flow,
inclined plane flow, and planar shear flow with gravity – and the model was shown
to be capable of capturing the different flow and segregation fields that arise as the
configuration and various input parameters are changed.
5.2 Future directions
While progress has been made, there are numerous directions for future research work.
Several directions are highlighted below.
• The NGF model remains phenomenological in nature, and further work is required to
elucidate its physical underpinning. Progress on this point has been made by Zhang
and Kamrin [64] and Bhateja and Khakhar [65] in establishing that the fluidity g has
a microscopic kinematic definition that holds across a wide diversity of flow geome-
tries. Furthermore, Zhang and Kamrin [64] showed that the constitutive equation for
the flow rule – i.e., γ˙ = gµ – is consistent with both granular kinetic theory as well as
109
an Eyring-like transition-state-theory picture. That said, the nonlocal rheology PDE
remains phenomenological, and while the NGF model has demonstrated accuracy
in flow predictions and versatility across flow geometries, obtaining the constitutive
equations from a bottom-up derivation is necessary to complete the picture. Such a
bottom-up derivation will help to provide physical meanings for the nonlocal ampli-
tude A and the cooperativity length ξ (µ) (2.8).
• As a corollary to the previous point, additional work is needed to obtain a clearer
microscopic understanding of the fluidity boundary conditions accompanying the
NGF model. In this thesis, fluidity boundary conditions were chosen primarily for
pragmatic reasons, and motivating the choice of fluidity boundary conditions from a
physical perspective remains an open challenge. As pointed out in Chapter 3, some
progress has been made on this point for dense emulsions by Mansard et al. [83], and
future work of this type is needed for dense granular materials.
As a specific class of boundary condition, the boundary between a dense, flowing
granular material and a smooth frictional wall remains challenging – yet very impor-
tant from a practical perspective. In the heap flow work of Chapter 2, the friction
coefficient between the granular material and the side walls was taken to be a con-
stant, which is a point worth further investigation. The work of Artoni and Richard
[75] has demonstrated a dependence of the wall shear stress on the slip velocity and
the velocity fluctuation – suggesting the possibility of a boundary condition in which
the wall friction coefficient is dependent on both the slip velocity and the granular
fluidity.
• In the quasi-static flow regime, the notion of Reynolds dilatancy – in which tran-
sient volumetric dilatation or compaction can occur during flow initiation – is well-
appreciated. Volume change in the context of Reynolds dilatancy depends upon the
initial preparation, and constitutive equations capable of quantitatively capturing this
110
phenomenology have been proposed in the context of local constitutive equations
[e.g., 112, 113]. In order to obtain a more general nonlocal model, future work is
needed to incorporate constitutive equations that account for Reynolds dilatancy, the
associated shear strengthening/weakening, and the effect of the initial state into the
NGF model.
• Concerning the continuum model for size-segregation, we have taken the segregation
flux to be driven by gradients in the shear-strain-rate γ,
˙ but potentially, gradients
in other shear-related quantities could drive this flux. In our DEM tests, the form
based on γ-gradients
˙ captures DEM data best, but other candidates remain possible
– such as the granular fluidity or the granular temperature. Since other works in
the literature are based on the closely-related notion of kinetic stress [97, 98, 103],
these forms are worth further investigation in additional flow geometries. Another
issue is the cl -dependence of the prefactors in the segregation fluxes. In this thesis,
we used a simple symmetric form, cl (1 − cl ), in both equations (4.11) and (4.22), but
P
more complex forms are possible [98, 109]. Furthermore, we have observed that Cseg
depends on the grain-size-ratio d l /d s , so it is necessary to determine the functional
P (d l /d s ) as a next step in order to obtain a more general size-segregation
form of Cseg
model.
• We have focused our segregation modeling work on quasi-one-dimensional flow
configurations. The eventual goal of our continuum modeling of flow and size-
segregation relates to general, three-dimensional flow settings. To achieve this, a
justification based on first-principles continuum thermo-mechanics – as was done
for the NGF model [60] – for the coupled model for segregation and flow is neces-
sary. Then, implementing the three-dimensional model, which involves a system of
highly coupled PDEs, in a finite-element-method or material-point-method frame-
work presents an additional, numerical challenge.
111
• In this thesis, we have focused on segregation due to differences in grain size. Segre-
gation also occurs due to mismatches in other quantities, such as density differences
[99] or differences in the surface friction coefficient [114], and additional modeling
work will be necessary to develop constitutive equations for these mechanisms.
• Finally, granular materials exist in vast and diverse combinations of forms. In this
thesis, we have focused attention on monodisperse and bidisperse mixtures of dry,
stiff, spherical particles. Extending continuum models for dense granular materials
to account for polydispersity [115], soft particles [61–63], various grain shapes and
orientations [116–118], and pore fluid effects [4, 119] remains a long-term goal in
granular materials modeling.
112
Appendix A
Technical details of DEM in LAMMPS
The DEM simulations described in this thesis were carried out using the open-source
molecular dynamics (MD) code LAMMPS [79]. Molecular dynamics is a computational
method widely used to study the physical movements of atoms and molecules, and with
properly defined particle shapes and interaction models, it can be used to model a granular
system. This appendix provides some technical details of the DEM simulations executed
in this thesis.
• The built-in pair style “gran/hooke/history” – a standard Hookean spring-dashpot
grain interaction model – is utilized. Interactions between grains occur only when
grains overlap. The normal force consists of two terms, a repulsive elastic contact
force and a damping force. The tangential force can consist of elastic and damping
terms, but damping in the tangential force is turned off in our simulations. The
tangential force is limited by Coulomb friction with a friction coefficient µsurf , and
typically, the tangential force is dominated by friction.
• The microcanonical ensemble NVE is utilized, in which the total number of grains
N, the system volume V , and the total energy E are conserved.
• A slight polydispersity (10% or 20%) in grain-size is always included for each species
113
114
of flowing grains. This is common practice in DEM simulations to prevent a collec-
tion of particles of the exact same size from crystallizing.
• In LAMMPS, granular particles are modeled as spheres in three-dimensions. For
our purposes of modeling two-dimensional granular systems, one layer of spherical
grains, all lying in the same plane, is utilized. All displacements and velocities in
the out-of-plane direction are constrained, and particles only rotate about the out-of-
plane direction. Thus, all contacts and motion only occur within the plane, and we
discuss all simulation results in this two-dimensional plane. For example, in bidis-
perse mixtures, the three-dimensional densities of the two species actually need to be
different in order for them to have the same area density – i.e., ρ2D = (2/3)dρ3D . One
drawback to this approach is that the mass moment of inertia for in-plane rotation of
spheres and disks is different. However, this effect appears to be negligible, since the
inertial rheology for disks and spheres confined to a plane are indistinguishable (see
Fig. 3.2(b)).
Appendix B
Averaging method – coarse graining
Coarse-graining refers to the micro-to-macro mapping method [104] utilized to extract
continuum fields from discrete DEM data. In this appendix, we briefly summarize the
coarse-graining techniques utilized in this thesis.
B.1 Slices technique
First, we introduce the spatial averaging method used to obtain steady, continuum veloc-
ity and stress fields from the monodisperse DEM simulations of Chapter 3. The method
follows the work of Koval and coworkers [41, 78] and is described for the cases of planar
shear flow and planar shear flow with gravity, in which quantities are averaged over the
x-coordinate shown in Figs. 3.2(a) and 3.3(a). First, for a snapshot at time t, we draw a hor-
izontal line at a given z-position, and assign each intersected grain i a weight Li defined as
the length of the horizontal line passing through grain i. Then, with the instantaneous veloc-
ity of grain i denoted as vi (t), the instantaneous velocity field is v(z,t) = ∑i Li vi (t)/ ∑i Li .
Regarding the stress field, the instantaneous stress tensor associated with grain i is σi (t) =
(∑ j6=i ri j ⊗ fi j )/ai , where ri j is the position vector from the center of grain i to the center
of grain j, fi j is the contact force applied on grain i by grain j, and ai = πdi2 /4 is the area
of grain i. The instantaneous stress field follows as σ(z,t) = ∑i Li σi (t)/L, where L is the
115
116
total length of the domain in the x-direction. The instantaneous velocity and stress fields
are then arithmetically averaged in time over many snapshots to obtain steady fields that
only depend upon the z-coordinate, such as those shown in Fig. 3.3(d). This process may
be adapted to spatial averaging over the angular coordinate in a polar coordinate system,
as described in Appendix B of Koval et al. [41], and used to obtain steady fields in annular
shear flow, such as those shown in Fig. 3.4(d).
B.2 Sliding bins technique
The slices technique works well for velocity and stress fields. However, as shown by Wein-
hart et al. [68], applying techniques of this type – i.e., ones with small or zero width in
the z-direction – to the solid area fraction field φ will lead to spatial fluctuations due to
particle-layering near walls. Since the concentration field cl is defined through the solid
area fraction, we utilize an alternate bin-based coarse-graining process to obtain contin-
uum quantities from DEM data in Chapter 4.
For specificity, we continue to consider two-dimensional flows that vary in the z-direc-
tion and are periodic in the x-direction – such as planar shear flow or planar shear flow with
gravity – and discuss a process that maybe adapted to spatial averaging over the angular
coordinate in a polar coordinate system. We draw a horizontal bin – i.e., a slender rectangle
that spans the simulation domain along the x-direction – centered at a given z-position and
with a finite width in the z-direction. Then, instead of defining the weight for a given
intersected grain i through a length, the weight is defined as the area of the intercepted
grain inside the bin Ai . Following Tunuguntla et al. [98] for a bidisperse system, denote the
sets of large and small grains intersected by a bin as F l and F s , respectively, so that the set
of all grains intersected by a bin is F = F l ∪ F s with F l ∩ F s = ∅. The instantaneous
solid area fraction field for species α is φ α (z,t) = (∑i∈F α Ai )/A, where A is the total area of
the bin, and the corresponding concentration field for species α is cα (z,t) = φ α (z,t)/φ (z,t)
117
with φ (z,t) = φ l (z,t) + φ s (z,t). Then, for consistency, the instantaneous velocity and stress
fields are defined as v(z,t) = (∑i∈F Ai vi (t))/(∑i∈F Ai ) and σ(z,t) = (∑i∈F Ai σi (t))/A.1
Depending on the desired spatial resolution for the coarse-grained fields and the se-
lected bin width, adjacent bins may overlap – hence the name “sliding bins.” The sliding
bins coarse-graining technique is utilized throughout Chapter 4, since our analysis of the
size-segregation process depends on accurately obtaining cl (and hence φ l and φ s ) fields.
Throughout, we take a bin width of 4d¯0 and a spatial resolution of roughly 0.1d¯0 . This
bin-width is sufficiently small so that the coarse-grained data is not over-smoothed but
sufficiently large so that layering effects are not observed in the solid area fraction and
concentration fields – i.e., we are in the plateau range [68] of bin-widths that produce
bin-width-independent continuum fields. We note that in the limit that the bin-width goes
to zero, we recover the slice-based coarse-graining process of Section B.1, and we have
tested that over a range of bin widths – between 0 and 9.6d¯0 – the coarse-grained velocity
and stress fields are very similar. For a given snapshot of a DEM configuration, we start
with a boundary bin centered 0.5d¯0 inward from the outermost flowing grain, which means
that the first few boundary bins contain empty area outside of the flowing region and are
thus less accurate. We remove this wall effect by deleting coarse-grained results near walls
and free surfaces. Specifically, when plotting the cl and v profiles and contours in Chap-
ter 4, coarse-grained data from bins centered within one-half of a bin-width (2d¯0 ) from the
outermost flowing grain are omitted at all walls and free surfaces. The DEM data collapses
S and CP (Figs. 4.5, 4.7, 4.11 and
used to best fit the dimensionless material parameters Cseg seg
4.14) are more sensitive to wall effects, and therefore in these figures, we do not include
spatial positions within 6d¯0 of walls or free surfaces.
Unlike for the steady flows of Chapter 3, which allowed for arithmetic time-averaging
as described in Section B.1, the flows of Chapter 4 involve transient, evolving fields due
1 We note that the definition for velocity is consistent with first defining species-level fields – i.e., vα (z,t) =
(∑i∈F α Ai vi (t))/(∑i∈F α Ai ) – and then calculating the mixture-level fields – i.e., v(z,t) = cl (z,t)vl (z,t)+(1−
cl (z,t))vs (z,t).
118
to the dynamics of the segregation process. In these cases, time-averaging is performed
by applying a normalized, cutoff Gaussian time filter to the DEM data at each z-position
– i.e., for each bin. Denote the Gaussian standard deviation as σt , so that the cutoff time-
width of the Gaussian kernel is 6σt . The time-smoothed field quantity at a given z-position
and time t is then the convolution of the DEM data over a time-period of 6σt , centered
at time t – i.e., for times from (t − 3σt ) to (t + 3σt ) – with the cutoff Gaussian kernel.
Concerning the choice of Gaussian time kernel width, we take σt ≈ 6156d s ρs /Pw (or for
p
our choice of normal contact stiffness kn and coefficient of restitution e, 2.8 × 105 τc , where
τc is the binary collision time) for vertical chute flow and σt ≈ 334 d s /G (or 6 × 104 τc )
p
for inclined plane flow.2 For the two boundary driven flows, σt varies across cases to
maintain the boundary shearing displacement inside the kernel time-width roughly equal
for each case for a given flow configuration. For the width of a Gaussian time filter, each
annular shear flow case has an inner-wall tangential displacement of ∼ 2000d¯0 , and the
top-wall shearing displacement for each planar shear flow with gravity case is ∼ 8000d¯0 .
(This distinction between the two configurations is due to the different wall displacements
needed to achieve a quasi-steady segregated state.) Finally, we note that our coarse-grained
continuum fields are relatively consistent for changes in Gaussian time kernel widths within
the tested ranges – i.e., σt ∈ (2638d s ρs /Pw , 9674d s ρs /Pw ) for vertical chute flow and
p p
(200 d s /G, 600 d s /G) for annular shear flow and kernel times corresponding to wall
p p
shear-displacements in (1000d¯0 , 2600d¯0 ) for annular shear flow and (3500d¯0 , 12500d¯0 )
for linear shear flow with gravity.
The collapses of Figs. 4.5, 4.7, 4.11, and 4.14 are obtained in the long-time regime, in
which the fields evolve slowly in time. Since the flow is quasi-steady, instead of subjecting
the DEM data to time filtering as described in the previous paragraph, the velocity and
concentration fields are simply arithmetically averaged in time to obtain fields that only
2 The binary collision time τq
c was defined for monodisperse systems in Chapter 3.2.1, and for bidisperse
2
mixtures, it is defined as τc = πρs d¯0 (π 2 + ln2 e)/16kn . The DEM time step is chosen as 0.1 of τc for
vertical chute flow and 0.03 of τc for annular shear flow, inclined plane flow, and planar shear flow with
gravity of bidisperse disks to ensure stable simulation results while balancing computation time.
119
depend on the spatial coordinate. Then, the necessary first and second-order spatial deriva-
tives of the field quantities – i.e., ∂ cl /∂ z, γ˙ = ∂ v/∂ z, and ∂ γ/∂
˙ z = ∂ 2 v/∂ z2 – are obtained
from these time-averaged fields. Instead of using a finite-difference-based approach, we
apply a Gaussian spatial derivative filter to the time-averaged data in order to obtain better
accuracy and more spatial smoothing. This approach is possible because the derivative of
the convolution of a signal and a kernel is equal to convolving the signal with the derivative
of the kernel. The standard deviation of the Gaussian spatial derivative filter σs = 0.5d¯0
throughout, which leads to a filter width of 30 bins.
As a final comment, other works in the literature [68, 104] have focused on choosing ap-
propriate filter functions – e.g., Lucy functions instead of Gaussian functions – and coarse-
graining widths. Although we have not exhaustively studied the effects of coarse-graining
widths, we have been careful to ensure that the reported results in Chapter 4 are indepen-
dent of these choices. Future work will focus on developing more systematic criteria for
selecting coarse-graining time and space scales for problems involving size-segregation.
120
Appendix C
Diffusion flux consistency check
S
Our process for determining the segregation flux – and hence the material parameters Cseg
P – is based on the assumption that the segregation and diffusion fluxes balance
and Cseg
in the quasi-steady regime. Therefore, it is essential that the dimensionless material pa-
rameter Cdiff appearing in the constitutive equation for the diffusion flux (4.9) has been
accurately determined, so that the coarse-grained diffusion flux is accurate. In Section 4.3,
we determined Cdiff for disks to be 0.20 using mean square displacement data from DEM
simulations of planar shear of a well-mixed bidisperse granular system. In this appendix,
we perform an independent consistency check that tests whether the constitutive equation
for the diffusion flux (4.9) using this fitted Cdiff value is capable of predicting the evolution
of the cl field in a diffusion-dominated problem.
Consider homogeneous planar shear flow of an initially-segregated system with large
grains on the bottom and small grains on the top, as shown in Fig. C.1(a). We run the DEM
simulation starting from the initially-segregated configuration, and the spatiotemporal evo-
lution of the coarse-grained cl -field is shown in Fig. C.1(b). We observe that the interface
between large and small grains, which is initially sharp, becomes diffuse with a transition
width that grows with time. We define a transition width at a given point in time as the dis-
tance between the points where cl equals 0.1 and 0.9 in snapshots of the spatial cl profile.
121
122
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wall x
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flowing
grains
fixed
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L = 60d¯0
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transition
width
Figure C.1: (a) Initially-segregated configuration for two-dimensional DEM simulation of
bidisperse planar shear flow with d l /d s = 1.5 and 8649 flowing grains. Black grains on
top and bottom represent rough walls. Dark gray grains in the flowing cell indicate large
flowing grains, and light gray grains indicate small flowing grains. A 10% polydispersity
is utilized for each species to prevent crystallization. (b) Spatiotemporal evolution of the
large grain concentration cl , illustrating the growing transition width due to diffusion. (c)
Normalized transition width versus square root of normalized time t˜ = t/(H/vw ). The
evolution of the transition width is dominated by the diffusion mechanism and is therefore
a good test of the material parameter Cdiff .
This transition width as a function of the square-root of time is plotted in Fig. C.1(c) as
solid lines for grain-size-ratios of d l /d s = 1.5 and 3.0, displaying roughly linear behavior
– typical of diffusive behavior – with a slight dependence on d l /d s .
Next, we apply the continuum model for the evolution of cl (4.11) to this problem. As
123
for planar shear flow of a monodisperse granular system (Fig. 3.2) or a well-mixed bidis-
perse system (Fig. 4.2), no pressure gradient is present, and the shear-strain-rate is approxi-
mately constant, leading to approximately zero shear-rate-gradient throughout.1 Therefore,
the diffusion flux is the dominant flux, which acts to remix the flowing grains. The equation
governing the evolution of cl (4.11) becomes
Dcl ∂ ∂ cl ∂ γ
˙
+ −Cdiff d¯ γ˙
2
+Cseg d¯ c (1 − c )
S 2 l l
= 0, (C.1)
Dt ∂z ∂z ∂z
where d¯ = cl d l + (1 − cl )d s . We note that due to the tiny effect of segregation and the
dependence of d¯ on cl , (C.1) is similar to but not exactly the same as the linear diffusion
equation in one dimension, so the solution is close to but not exactly an error function.
We obtain predictions for the evolution of the cl -field using the fully-segregated initial
condition for cl (z,t = 0), no-flux boundary conditions at z = 0 and z = H, a spatially-
constant value of inertial number I consistent with that prescribed in the DEM simulations,2
a given grain-size-ratio d l /d s , and Cdiff = 0.20. We extract the transition width as a function
of time from continuum simulation results for d l /d s = 1.5 and 3.0 and include these results
in Fig. C.1(c) as dashed lines. The continuum model predictions agree well with the DEM
data and are even capable of capturing the small difference due to the grain-size-ratio.
This result indicates that the expression for the diffusion flux (4.9) and the fitted material
parameter value Cdiff = 0.20 are indeed consistent with DEM data. We conclude by re-
emphasizing that this test is simply a consistency check for the constitutive equation for the
diffusion flux. We do not use this test to fit the value of Cdiff , nor do we consider this test
as validation of the model.
1 In the context of the local inertial rheology, since the stress ratio µ is spatially-constant in homogeneous
planar shear, the resulting inertial number field I is also spatially constant. Since the inertial number depends
on both d¯ and γ,˙ spatial variation in d¯ leads to spatial variation in γ.
˙ Even though this spatial variation in γ˙ is
slight, and we still include the shear-strain-rate-gradient-driven segregation flux when analyzing the problem
shown in Fig. C.1(a).
2 Note that since I is taken to be spatially-constant, γ˙ = (I/d) ¯ Pw /ρs varies slightly in space due to the
p
cl -dependence of d.¯
124
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