PHASE TRANSITION IN PASSIVE AND ACTIVE

GRANULAR SUSPENSIONS

A THESIS

submitted by

AJINKYA MUKUND KULKARNI

for the award of the degree

of

MASTER OF SCIENCE(by Research)

DEPARTMENT OF APPLIED MECHANICSINDIAN INSTITUTE OF TECHNOLOGY MADRAS.

OCTOBER 2017

THESIS CERTIFICATE

This is to certify that the thesis titled PHASE TRANSITION IN PASSIVE AND

ACTIVE GRANULAR SUSPENSIONS, submitted by Ajinkya Mukund Kulkarni,

to the Indian Institute of Technology, Madras, for the award of the degree of Master of

Science, is a bona fide record of the research work done by him under our supervision.

The contents of this thesis, in full or in parts, have not been submitted to any other

Institute or University for the award of any degree or diploma.

Prof. Mahesh V. PanchagnulaResearch GuideProfessorDept. of Applied MechanicsIIT-Madras, 600036

Place: ChennaiDate: 13th October 2017

i

DEDICATION

I dedicate this work to my family who stand by me, no matter what.

ii

ACKNOWLEDGEMENTS

Life till now has left me filled with gratitude to a lot of people. I would like to utilize

this opportunity to thank them. First, I thank Dr. Mahesh Panchagnula for his support,

guidance and encouragement while pursuing my M.S research at IIT Madras. His be-

lief in my abilities, an optimistic and upbeat attitude towards research, patience with

students and willingness to allow people to explore different directions, played a huge

role in the progress of my research. I thank him for putting in time and effort in order

to aid my personal, academic and research growth. I thank Dr. Sumesh Thampi for his

valuable insights into the latter part of my research. Without his guidance and critical

thoughts, this work would have been incomplete. A special thanks for the patience and

support he displayed during his Hydrodynamics of Complex Fluids course. It turned

out to be the best course I had ever taken. Thank you both, for believing in me and

inspiring me every day. Without your help, the road would have been lonely. I also

thank Dr. Pallab Sihna Mahapatra and Dr. Srikanth Vedantam for their guidance and

support throughout.

I thank my GTC members Dr. Sarith Sathian, Dr. Sridharakumar Narasimhan and

Dr. Vengadesan S for their suggestions and feedback that helped improve my research

work. I thank the technical and Non-technical staff at Applied Mechanics for their coop-

eration. I thank the IIT Madras staff, especially the mess staff for holding their ground

when the 2015 Chennai floods struck. Thanks to the Ministry of Human Resource and

Development, Govt. of India for funding my M.S.

Thanks to Dr. Nachiketa, Dr. Vadivu, Vijay Anand, Harinadh, Siddharth, Karthiga

Madam, Dhivyaraja, Dr. Sri Vallabha, Dr. Satish Akella, Mithun, for being supportive

and also for being part of healthy discussions in the research group. I will miss your

presence in the lab. I am grateful to Dr. Vadivu and Vijay Anand for proof reading

the thesis. I thank Antony, Mrityu, Vishnu, Rahul, Pandeyji, for keeping the lab cheer-

ful and bright. I am thankful to Atul Kulkarni, Rohan Biwalkar, Kalpesh Benuskar

and Sujeet Gaikwad for making my daily mess meals palatable. Special thanks to Raj

Dandekar, Rajat Dandekar, Vivek Kulkarni, Bhargav Ghanekar, Vaibhav Tipnis, Sanket

iii

Barve, Saurabh Deshpande, Nikhil Gokhale and Aneesh Bhagwat for their encouraging

words throughout my stay at IIT Madras. Our conversations will be missed. The times I

spent with them are some of my most cherished memories at IIT Madras. I feel blessed

to have met them.

I am grateful to Radhika for her wisdom, support and timely words of encourage-

ment. Thank you for being with me through my ups and downs. I couldn’t ask for a

better brother than Chaitanya, and I thank God for him. I cannot thank my Aai and Baba

enough for bringing us up despite the unconventionally tough circumstances. Thank

you Baba, for being able to do what few people can. You inspired me to take up this

journey and helped me pursue the impossible. I dedicate this work to you.

Ajinkya Mukund Kulkarni

iv

ABSTRACT

KEYWORDS: Granular Suspensions; Active Matter; Phase Transition; Human

Crowds.

The mechanics of matter composed of individual independently moving particles is

quite complex. Amongst such granular material, distinction can be made between pas-

sive particles which are propelled by external forces or active particles which are able

to self propel. A Discrete Element Model, which incorporates a novel coupled fluid-

particle discrete element simulations for two dimensional flow of passive granular slur-

ries, has been validated to replicate the dynamics of dense granular suspensions. The

model is suitably modified to mimic active suspensions. The nature of the transition

from disorganized thermal-like motion to organized vortical motion has been examined

via calculating the Binder cumulants.

An occurrence of metastability is investigated in the hysteretic region. The switching

between the steady states of the system in the hysteretic region has been triggered via

artificial nucleation of randomly picked particles spanning the entire domain. In addi-

tion, the effect of domain size on the nature of the phase transition has been studied.

The motivation for these phase transition has been explained via thrust generation abil-

ity and the geometry of the confinement. Finally, insights from this model are utilized

in modeling human crowds and in obtaining possible routes towards the prevention of

stampedes.

v

TABLE OF CONTENTS

ACKNOWLEDGEMENTS iii

Acknowledgements iii

ABSTRACT v

Abstract v

1 Introduction 1

1.1 Review of research in passive granular suspensions . . . . . . . . . 1

1.2 Review of research in active granular suspensions . . . . . . . . . . 3

1.3 Review of research in dynamics of human crowds . . . . . . . . . . 6

1.4 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Modeling and simulation of passive granular systems 10

2.1 Force formulation for passive granular suspensions . . . . . . . . . 10

2.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Modeling and simulation of active granular systems 14

3.1 Force formulation for active granular suspensions . . . . . . . . . . 14

vi

4 Results - passive granular systems 17

4.1 Validation of the model for passive granular suspensions . . . . . . 17

4.2 Modeling landslide dynamics using DEM . . . . . . . . . . . . . . 20

4.2.1 Simulation Details . . . . . . . . . . . . . . . . . . . . . . 21

4.2.2 Effect of varying dam height (H) . . . . . . . . . . . . . . 21

5 Results - dynamics of human crowds 27

5.1 Phases observed in active granular suspensions . . . . . . . . . . . 27

5.2 Order parameter and nature of the transition . . . . . . . . . . . . . 32

5.3 Bulk oscillation and Mean Square Displacement . . . . . . . . . . . 40

5.4 Panic factor and artificial transition . . . . . . . . . . . . . . . . . . 42

5.5 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Conclusion and Future Work 63

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

vii

LIST OF FIGURES

4.1 Regimes for wet granular flow in a drum of diameter 0.2 m with 3000

particles of 5 mm each, for varying coordination coefficient (Cv). (a)

Slipping regime (b) Cataracting regime (c) Rolling regime and (d) Cen-

trifuging regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 Instantaneous particle positions from (a) the experiments of Liao et al.

(2010), (b) the current simulations on a rotating drum. The region from

the free surface to the dotted green line is the fluidized region in the

granular material. (c) Plot of the mean velocity profile in the granular

material from the current simulations and from the experiments of Liao

et al. (2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Free surface profiles of wet granular flow over a dam for varying H/h

and Cv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 a) Phase diagram for dry free surface granular flows over an obstacle b)

Effect of obstacle on Fr for wet free surface granular flows for varying

Cv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.5 Normalized Free Surface (NFS) as a function of the incline length for

low Cv values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.6 Normalized Free Surface (NFS) as a function of the incline length for

high Cv values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

viii

5.1 As the coordination coefficient (Cv) increases, the system exhibits a first

order phase transition from a disordered phase to an ordered structure.

The velocities have been scaled by the characteristic velocity√βR. (a),

(b) show the radial, tangential velocity profile before transition. Inset

shows instantaneous fluctuations (c) shows the radial, tangential veloc-

ity profile after transition. (d), (e) Velocity vectors averaged over parti-

cles before the phase transition. (f) Velocity vectors after transition. 28

5.2 As the coordination coefficient (Cv) increases, the system exhibits a first

order phase transition from a disordered phase to an ordered structure.

The velocities have been scaled by the characteristic velocity√βR. (a),

(b) show the radial, tangential velocity profile before transition. Inset

shows instantaneous fluctuations (c) shows the radial, tangential veloc-

ity profile after transition. (d), (e) Velocity vectors averaged over parti-

cles before the phase transition. (f) Velocity vectors after transition. 31

5.3 The jump in the order parameter as a function of the coordination coef-

ficient (Cv) and the hysteresis associated with this phase transition for

various values of R/d . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.4 Binder cumulant is plotted as a function of the control parameter Cv 34

5.5 Cmeanv,crit as a function of R. . . . . . . . . . . . . . . . . . . . . . . . 36

5.6 Histograms of azimuthal (Vθ) and radial (Vr) velocities for increasing

Cv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.7 Variation of scaled Azimuthal Velocity (V ∗θ ) and scaled Average Speed

(|V ∗|) for different R/d ratios for their respective values of Cforwv,crit . 38

5.8 Variation of V ∗θ and |V ∗| for various domain sizes for the respective

Cforwv,crit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.9 Mean Square Displacement as a function of t for various values of Cv 41

5.10 Zoomed in view of the Mean Square Displacement as a function of t

for Cv = 0.43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.11 Panic Factor (P ) variation at a) Cv = 0.21, b) Cv = 0.34, c) Cv = 0.36,

d) Cv = 5, e) Cv = 10, f) Cv = 50 . . . . . . . . . . . . . . . . . . 43

ix

5.12 Forcing a stampede crowd to uniform velocity. (a), (b), (c), (d) respec-

tively represent the distribution of particles to which a momentum im-

pulse was imparted. (a) distributed through out the domain, (b) a stripe

at various radial locations, (c) a pocket of agents at the center and (d) a

pocket of agents off-center. e) Panic factor distribution at the Jammed

state at Cv = 0.34, f) Panic factor distribution at the Rotary state at

Cv = 0.34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.13 (a) Growth or decay of the order parameter as a result of forcing a stam-

peded system with an impulse. (b) Critical number of particles required

to drive the system back to uniform velocity decreases with increase in

the magnitude of the momentum impulse. The results on both back-

ward and forward trials are also shown. The velocities are scaled with

Vavg in the initial state of the perturbation. . . . . . . . . . . . . . . 46

5.14 Different phases of the system at β = 0.1m/s2. a) Rotary state with no

core formation, b) and c) Rotary state with core formation . . . . . 49

5.15 Different phases of the system at β = 0.25m/s2. a) Thermal-like state,

b) Rotary state without core formation and c) Rotary state with core

formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.16 Different phases of the system at β = 0.5 m/s2. a) Thermal-like state,

b) Rotary state without core formation and c) Rotary state with core

formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.17 Different phases of the system at β = 0.75m/s2. a) Thermal-like state,

b) Rotary state without core formation and c) Rotary state with core

formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.18 Different phases of the system at kn2× 104 N/m. a) High speed rotary

state, b) and c) Rotary state with core formation . . . . . . . . . . . 53

5.19 Different phases of the system at kn = 5 × 104 N/m. a) High speed

rotary state, b) Rotary state without core formation and c) Rotary state

with core formation . . . . . . . . . . . . . . . . . . . . . . . . . . 54

x

5.20 Different phases of the system at kn = 106 N/m. a) Rotary state with-

out core formation, b) Rotary state without core formation and c) Rotary

state with core formation . . . . . . . . . . . . . . . . . . . . . . . 55

5.21 Different phases of the system at kn = 5 × 106 N/m. a) Thermal-like

state, b) Rotary state without core formation and c) Rotary state with

core formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.22 Different phases of the system at kn = 2 × 107 N/m. a) Thermal-like

state, b) Rotary state without core formation and c) Rotary state with

core formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.23 Different phases of the system at 0.1m. a) Thermal-like state, b) Rotary

state without core formation and c) Rotary state with core formation 58

5.24 Different phases of the system at 0.5m. a) and b) Thermal-like state, c)

Rotary state without core formation . . . . . . . . . . . . . . . . . 59

5.25 Different phases of the system at 2m. a) and b) Thermal-like state, c)

Rotary state without core formation . . . . . . . . . . . . . . . . . 60

5.26 Invariance of Cv,crit with varying mass (m) and Cv . . . . . . . . . 61

5.27 Invariance of Cv,crit with varying spring stiffness coefficient (kn) and

Cv. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.28 Linear monotonic increase of Cv,crit with increasing β and Cv . . . 62

xi

CHAPTER 1

Introduction

The mechanics of matter composed of individual independently moving particles is

quite complex. Amongst such granular materials, distinction can be made between

passive particles which are propelled by external forces or active particles which are

able to self propel. Granular flows have been studied for many decades due to their

importance in natural and industrial processes. Granular materials are aggregations of

discrete, macroscopic particles. Granular matter behaves differently from conventional

matter such as solids and liquids. Under the right amount of shear, granular materials

can flow as fluids, or can resist deformation similar to solids. Interstitial fluid can

alter the particle-particle interaction and can influence the resulting dissipative forces.

These systems are ubiquitous and are relevant to varied fields as mining, agricultural

industries and civil engineering. Understanding and mitigating environmental disasters

such as landslides, avalanches and mudslides is of paramount importance and requires

thorough knowledge of the macroscopic behavior of such materials. In the current

work, we focus upon validating the existing Discrete Element Method (DEM) model by

Bonkinpillewar et al. (2015), Mahapatra et al. (2016), Mahapatra et al. (2017) for wet,

dense granular slurries in a circular geometry. This model was employed for examining

interaction of avalanches with dams. Finally, the existing model was suitably modified

to mimic dynamics in human crowds.

1.1 Review of research in passive granular suspensions

The inter-granular fluids (particularly if they are liquids) may have a significant effect

on the overall dynamics of passive suspensions. Among wet granular materials, a dis-

tinction is again made between partially and completely wet materials. In the presence

of small amounts of the inter-granular liquid, wet granular materials are cohesive due to

the surface tension of the liquid, which forms a liquid bridge between the particles. The

physical phenomena and the mechanical properties of partially wet granular materials

have been reviewed by Mitarai and Nori (2006). When the intergranular liquid com-

pletely wets the particles, cohesiveness is lost and the material behaves like a slurry.

Wet granular slurries are perhaps the most important form of granular materials from

an industrial standpoint. Liao et al. (2010) studied dynamics of slurries in a rotating

drum geometry. Rheology of cohesion-less granular particles on an inclined plane was

studied by Silbert et al. (2001). A steady state was identified when gravity is balanced

by frictional forces and particle collisions. Finger and Stannarius (2007) investigated

effects of interstitial fluids on axial segregation of granular slurries in a rotating drum

geometry. Additionally, a scaling relation was obtained which was in good agreement

with the existing Bagnold velocity profile. Application of DEM method to particulate

flows was studied by Govender et al. (2013) using Positron Emission Particle Tracking

(PEPT). It was concluded that DEM methodology was indeed an accurate represen-

tation for wet granular flows. Additionally it was shown that further improvement in

the calculation of the friction coefficient further improves this agreement. Studies on

inclined plane flows have been used to develop constitutive models of such materials,

whereas, rotating drum flows have been used to study particle segregation and the de-

velopment of microstructure. From these studies, six different regimes have been iden-

tified in rotating drum flows by Ottino and Khakhar (2000) namely: slipping, slumping,

rolling, cascading, cataracting and centrifuging . As shown by Chou et al. (2011), the

transition between these regimes is complex and much work is still necessary to under-

stand the various aspects of flows of slurries. In light of many physical phenomena that

are manifested in a rotating drum geometry, it may be useful to study flows in simpler

geometries in order to gain further insight. Since the first numerical simulations by

Cundall and Strack (1979), there have been a wide variety of models of granular mate-

rials. Most numerical methods rely on discrete element methods (DEM), which track

individual particles subject to forces from the neighboring particles. DEM has gained

in popularity with the increased availability of computational resources. Various vari-

ants of the DEM methodology have recently been reviewed by Thornton et al. (2000),

Thornton (2009) as well as by Cleary et al. (2013).

DEM approach treats the material as a collection of particles interacting through

inter-particle forces. The inter-particle forces consist of a conservative component

which accounts for elastic interactions and a non-conservative component to model

dissipative interactions. Numerous models have been proposed for the two components

2

of the interaction forces. Elastic interactions are usually modeled based on simple lin-

ear springs or using Hertzian contact theory. In a two dimensional setting, the Hertzian

contact model reduces to a linear repulsive interaction force between the particles with

a finite cutoff. Dissipative interactions arise from frictional interaction between par-

ticles, inelastic contact between particles or interaction between the particles and the

surrounding fluid medium. Various DEM approaches have broadly simulated the dis-

sipative forces using viscous dashpot models, as studied extensively by Navarro and

de Souza Braun (2013), or using coefficients of restitution modeling inelastic contact.

Thornton et al. (2011) have compared the features of these models in detail. In wet

granular slurries, the fluid-particle interaction is significant and has been traditionally

modeled through one-way coupled DEM-CFD models. Recent advances in the DEM-

CFD approach have been reviewed extensively by Zhong et al. (2016)

Cleary et al. (2006) proposed a one way coupled DEM-SPH model in three di-

mensions to capture the dynamics of internal flows in a wet granular slurry in a ball

mill. Applications of two way coupled models such as DEM-CFD models have been

studied recently by Jayasundara et al. (2011), where a coupled DEM-CFD model was

developed to simulate effects of wet granular systems inside a high speed stirred mill.

Furthermore the differences between simulations and PEPT experiments were shown to

be negligible. However, such approaches are computationally expensive and may not be

necessary in some regimes of fluid-particle interactions, where the interaction between

the fluid and the particles can be simplified. Flows of dense granular media has been

reviewed by Forterre and Pouliquen (2008) and by Campbell (1990). Recent advances

in DEM methods are presented in Guo and Curtis (2015)

1.2 Review of research in active granular suspensions

Among granular materials, distinction can be made between passive particles which are

propelled by external forces and active particles which are able to self-propel. Swarms

of such self propelling objects moving collectively display very interesting dynamics.

Couzin et al. (2002) laid out the fundamental rules of flocking and collective behav-

ior as observed in schools of fishes or flocks of birds. According to these rules, i)

Individual self propelling agents must maintain a repulsion potential between them

3

and ii) They tend to align with their neighboring flock. In this approach, each agent

was assigned three zones, namely, zone of repulsion, zone of attraction and a zone of

alignment/orientation. Most importantly, they reported multiple transitions between the

possible dynamical states.

Models describing these phenomena have generally been of two types: (a) contin-

uum models obtained by considering field of orientation vectors representing particles

coupled to standard hydrodynamics equations and (b) discrete particle models. Toner

and Tu (1995) proposed a non-equilibrium continuum dynamical model for the collec-

tive motion of the biological organisms. This model reduced to the classical XY spin

model at the limiting case of zero velocity of the agents. The most common particle-

based model without any hydrodynamic interactions has been the Standard Vicsek

model by Vicsek et al. (1995) and its variants. In this model, the particles are assumed

to interact with their neighbors within a certain distance and align themselves in the

average direction of the velocities of the neighbors. This model predicted a novel phase

transition from a disordered phase to a state of organized motion. This method works

well for low-noise and high-particle density systems. Wang et al. (2013) proposed a

dynamic variant of the Vicsek model which incorporated a distance based influence

for the agents. This study also revealed the existence of a critical alignment required

to achieve a high degree of consensus amongst the agents. This collective behavior is

observed in a range of length scales spanning micro-organisms to human crowds. At

smaller length scales, Wolgemuth (2008) studied flows in bacterial colonies. Bacterial

turbulence was reported and large-scale coherence emerging from the hydrodynamic in-

teractions was observed. Riedel et al. (2005) investigated rotating waves and dynamic

vortices in swarms of highly dense sperm cells. The sperm cells showed two levels of

order namely: i) Organized vortical state and ii) Packing into an array. Furthermore, the

hydrodynamic interactions between the cell colonies were studied and quantified.

Deseigne et al. (2010) experimentally studied the collective motion of vibrated

disks. The disks achieved alignment in a confined domain from self propulsion and

repulsion with varying the amplitude of the vibrations. Additionally, Giant Number

Fluctuations (GNF) were observed, which agreed well with the theoretical model. This

was a first of it’s kind study which explored the differences between polar and nematic

order in collective motion. Theurkauff et al. (2012) observed clustering phase of self

propelled colloids at intermediate densities. Even in self-propelling active matter, the

4

organized collective motion arises from two reasons: (1) Individual particles are able

to observe neighbors and adjust the direction of motion and (2) hydrodynamic effects

of the surrounding medium could provide the coupling. In general, the effects of these

two mechanisms may not be equivalent and a combination of them could contribute

to the collective motion. Recently, interest has been generated in the latter case where

hydrodynamic interactions are thought to help reduce the total energy expenditure of

the particles by minimizing viscous dissipation.

Hydrodynamic interactions are important in the motion of bacteria and other mi-

croorganisms using beating of flagella or cilia in a viscous medium. Baskaran and

Marchetti (2009) derived a continuum model based on hydrodynamic interactions us-

ing a neutral buoyant model swimmer. This model was an attempt to unify the theories

of active suspensions. Furthermore it was emphasized that fluctuations lead to pattern

formation at the boundaries and showed that large-scale nonequilibrium phenomena

can be observed due to hydrodynamic interactions between the active particles. Lu

et al. (2013) used an improved self-propelled model for identifying the transition from

a collective state to random motion of Bacillus subtilis when subjected to photoactiva-

tion. Laskar and Adhikari (2015) described the spontaneous motion of active filaments

in three dimensions by considering a minimal active filament model with the hydrody-

namic interaction. This transition was inferred to have had originated from the destabi-

lizing nature of the hydrodynamic interactions competing with the stabilizing nature of

elasticity. Finally it was concluded that hydrodynamic instabilities are sufficient for the

onset of motion.

Collective behavior has also been observed in absence of any explicit alignment

mechanism in the self-propelled suspension as shown by Bialké et al. (2012) using

Brownian dynamics simulations. Experiments as well as theoretical studies confirm the

presence of different kinds of global patterns of the self-propelled systems. Yeo et al.

(2016) and Tsang and Kanso (2015) reported that geometric confinement and flagellar

activity of micro-swimmers resulted in hydrodynamically triggered phase transitions.

Three phases were subsequently identified: chaotic swirling, stable circulation, and

boundary aggregation. In the high viscous regime, different low-energy states of the

self-propelled particles, namely the coherent flock, the rigid rotation, and the random

droplet have been identified by Al Sayegh et al. (2016) using simulations with Morse

potential interactions. Solon et al. (2015) demonstrated the microphase separation us-

5

ing Viscek model in the coexistence region. They observed a hysteretic transition from

the gas, microphase and liquid states. Wioland et al. (2013) studied confinement effects

on small droplets containing Bacillus subtilis. A novel phase was discovered formed by

spiraling cells, surrounded by a counter-rotating boundary layer. A continuum model

was also developed to explain these flows which was in good agreement with the exper-

iments. Jamming in systems of self-propelled particles is of recent interest. Hinz et al.

(2014) studied mixtures of active and passive agents and their corresponding dynam-

ical states. A force based model was developed using Dissipative Particle Dynamics

(DPD) and an existence of a mesoturbulent state was reported for low Peclet (Pe) num-

bers. This also shed light on inducing the transitions between such states by altering the

number fraction of the active and passive agents. Angelini et al. (2011) reported a glassy

behavior in epithelial cell sheets. The glassy phase was observed when the density of

the particles is greater than the glass transition density. Henkes et al. (2011) numeri-

cally studied a system of dense soft spheres having repulsive potentials in a confined

domain. A possible route towards a glassy regime was investigated and the phases were

characterized via Mean Square Displacement (MSD) of the individual particles. Self

organization in cytoskeletal elements was studied by Kohler et al. (2011). A unifying

dynamical framework was provided by Caussin et al. (2014) which explained patterns

in polar active materials.

Synthetic particles such as Janus particles driven by temperature, light or chemical

composition gradients across the particle surface are some examples of synthetic active

systems which were studied by Sobrino Fernandez et al. (2014) and Liu et al. (2016).

Bechinger et al. (2016) and Elgeti et al. (2015) have reviewed recent advances in active

matter in their seminal works.

1.3 Review of research in dynamics of human crowds

Human crowds have been shown to be well-modeled using the DEM approach. A

key focus of the recent literature pertaining to human crowd dynamics has been on

evacuation and escape dynamics. Helbing et al. (2000) proposed a simple dynamic

model based on desired speed and relaxation times. A force based model was formu-

lated which assumed repulsive interactions between the agents. Escape dynamics of

6

the crowd from a single exit was studied and the following conclusions were drawn. i)

Clogging at the exits is similar to formation of arches and jamming in granular flows

through hoppers and ii) ’Faster’ is slower effect: Increasing speeds at the clogging point

does not aid escape, but delays it. Furthermore, a measure of the panic in the crowd,

namely, Panic Factor was proposed which was a measure of individual behavior com-

peting with the crowd behavior. Tajima and Nagatani (2001) studied exit dynamics of

crowds using a lattice-gas model, and found a phase transition emerging from distinct

dynamical phases such as arching. Consequently, a phase transition was observed from

choking to a decaying flow. Helbing et al. (2006) applied continuity equation towards

formulating an analytical approach for bottleneck flows. A shock wave approach was

presented to model density in escape situations and bottleneck flows. Helbing et al.

(2007) gathered data of crowd panic in large gatherings. The disaster at Mecca on 12

January 2006 was analyzed by extracting the position and velocities of individual per-

sons. The variation of local densities within the crowd was explored, which explained

the phase transition from laminar phase to stop-and-go waves. Recently, insights from

granular flows have been applied to understand collective behavior of human crowds.

Baglietto and Parisi (2011) presented an off-lattice model for pedestrian dynamics. The

agents were modeled as soft repulsive disks with variable radius and a racetrack domain

was utilized to study this model. Additionally the fundamental diagrams obtained were

validated with experimental data. Recently, Silverberg et al. (2013) identified a phase

transition from a disorganized to a vortical state in social gatherings.

Hajj in Arabia studied by Curtis et al. (2011), Ahmad et al. (2014) and Kumbh

Mela in India studied by Baranwal et al. (2015) are the two biggest periodic human

gathering events on earth. Estimates have shown that ∼ 106 to 107 people gather into a

confined space during these events. The dynamics of such large crowds and its possible

spontaneous transition to a stampede has perplexed researchers for over thirty years.

Such transitions are, of course, a matter of grave importance to both law enforcement

and public safety. Evacuation dynamics and crowd behavior has also been studied and

reviewed by Barnett et al. (2016), Hughes (2003) and Buzna et al. (2007)

7

1.4 Motivation and Objective

1.4.1 Motivation

The dynamics of granular suspensions have interested researchers for several years.

One key difference between non-particulate and particulate suspensions is the change in

flow viscosity as a function of shear rate and the resulting effect on the dynamics. Most

of the studies in the literature have focused on dry granular flows and have neglected

the effect of the inter-granular fluid (usually air). Recognizing that the inter-granular

fluids (particularly if they are liquids) may have significant effect in the dynamics of

granular suspensions, there have been a number of recent studies on wet granular flows.

The transition between the regimes of granular flows is complex and much work is still

necessary to understand the various aspects of flows of slurries. Recently more atten-

tion has been devoted towards understanding the collective behavior of active granular

matter. The organized collective motion in such systems arises from two reasons: (i)

Individual particles are able to observe neighbors and adjust the direction of motion,

and (ii) Hydrodynamic effects of the surrounding medium could provide the mecha-

nism behind the coupling. In general, the effects of these two mechanisms may not be

equivalent and a combination could contribute to the collective motion.

1.4.2 Objective

The objective of this study is two fold:

1. Validation of a model incorporating a novel coupled fluid-particle discrete ele-ment simulations for two-dimensional flow of a passive granular slurry.

2. Application of the model to study the dynamics of human crowds and possibleprevention of stampedes.

1.5 Outline of the thesis

This thesis contains 6 chapters. The introduction and the Literature survey forms the

contents of chapter 1. Chapter 2 deals with the numerical modeling of dense passive

granular suspensions using DEM method. Chapter 3 explains the modifications made

8

to the existing model to mimic active granular suspensions. The results of the method-

ology for passive granular suspensions are described in Chapter 4. Chapter 5 deals with

the results of the application of the DEM model for human crowds. The Conclusion

and future work is presented in Chapter 6.

9

CHAPTER 2

Modeling and simulation of passive granular systems

In this chapter, the Discrete Element Method (DEM) methodology presented by Bonk-

inpillewar et al. (2015), Mahapatra et al. (2016) and Mahapatra et al. (2017) is dis-

cussed. This body of previous work forms the basis for the formulation of the current

model. The granular slurry is modeled as granular particles comprising of monodisperse

soft discs having a circular cross-section which are densely packed in a fluid medium.

The situation of a dense packing is attained when ρfvf � ρsvs, where ρf and ρs corre-

spond to the density of the interstitial fluid and the density of the particles respectively.

vf and vs represent the volume fraction for the fluid and the particle respectively. This

inequality implicitly assumes ρf < ρs, which is generally true. In the next section, we

describe in detail the forces on the individual particles.

2.1 Force formulation for passive granular suspensions

The temporal evolution of the particles is given according to the Newton’s second law

of motion. Collisions between the particles (and with the boundary wall) are modeled

using (i) the interaction force (Fpp) - a contact force and (ii) the drag force (FD) - a

non-contact force. The formulation is outlined as follows which follows the work of

Bonkinpillewar et al. (2015)

mdv(i)

dt= F(i)

pp + F(i)D +mg(i) (2.1)

where F(i)pp and F

(i)D represent, respectively, the particle-particle interaction force and the

drag force acting on the passive granular particles. The inter-particle contact force F(i)pp

is modeled based on Hertzian Contact Theory. This force acts along the line joining the

centers of the particles and it of a conservative nature. Excessive overlaps between the

particles are avoided by imposing a finite cutoff for the deformations experienced by

the particles. These particle-particle interaction forces are modeled as

Fijpp =

−knδij if |ri − rj| < d

0 otherwise.

Here the extent of total deformation of two soft particles i and j involved in the colli-

sion event is calculated as δij = (|ri − rj| − d)ri−rj|ri−rj | , a vectorial distance along the

separation vector ri − rj . The position vector ri gives the location of the center of the

ithparticle. The strength of the elastic force arising from the deformation of the particle

is characterized using the coefficient kn. Herztian theory relates kn to known material

properties such as the elastic modulus in the limit of small deformations.

Next the modeling of the dissipative force is been discussed. In the low Re set-

ting, the dissipative force has been modeled as a Stokes’ drag. This assumption holds

true because the Reynolds number based on the interstitial gap between the particles

is low, since the case under focus is dense suspensions. Hence, the dissipative force is

formulated as

F(i)D = Cvd(v(i) − v) (2.2)

where Cv is the drag coefficient that controls the coupling strength between the fluid

and the particle. This coefficient is exactly equal to 3πµ (the Stokes’ drag) for a single

particle translating with a relative velocity v − v in a fluid of viscosity µ in inertia less

flow conditions. However, we do not simulate the dynamics of interstitial fluid, instead

v that represents the velocity of the fluid at the center of the particle, is calculated as

a weighted average of velocity of neighboring particles. Considering a neighborhood

radius of h around any particle, we can calculate v as

v =

∑nj=1 mjWij (|ri − rj|, h)vj∑nj=1mjWij (|ri − rj|, h)

. (2.3)

Here n is the number of neighbors in a radius of h and Wij is a Gaussian function

specifying the weight-age given to each particle in calculating the average velocity v.

11

This function is defined as

Wij =

exp(−η |ri−rj |

2

h2

)if |ri − rj| ≤ h

0 otherwise

where η controls the distribution of contributions of surrounding particles. In the current

study η is taken to be 1.

The choice of this weight function is based on the work of Drumm et al. (2008)

and arises from the requirement that velocity at a point in the fluid is most affected by

particles in its immediate neighborhood. In addition, the effect of a given particle de-

creases as the distance from the point under consideration increases. A natural function

that fits this requirement is a Gaussian function centered at the point of interest. The

fluid velocity at a point is thus calculated based on the velocities of the particles within

a distance h with an influence coefficient η. Furthermore, it is found out that the value

of h doesn’t have a great effect on the overall dynamics for 3d ≤ h ≤ 10d. Hence in

the current work, h is chosen to be 5d, implying that we allow five layers of particles

to influence the fluid velocity at a given point. When n is less than two, v is set to 0

implying that the surrounding fluid is at rest. This is consistent with the inertia less fluid

assumption. We note that this provides a means of obtaining the fluid velocity without

recourse to a solution of the hydrodynamics.

There are two advantages in using the Eq. (2.2) and Eq (2.3). Firstly, this formula-

tion avoids the need of a cumbersome solver for the fluid dynamics by approximating

the fluid velocity to v at the center of the particle. Linearity between the drag force FD

and the relative velocity v − v implies that the fluid inertia is neglected in the model

though particle inertia is considered relevant (see Eq. (2.1)). This is expected to be

true in dense granular suspensions where the fluid volume fraction is much smaller than

the particle volume fraction, as is the case with slurries. The advantage of this method

is that it couples the effect of the surrounding liquid phase without the need for the

solution of the complete Navier-Stokes equations. The simplification is attained by ne-

glecting the inertia of the fluid. The fluid is assumed to be carried by the neighboring

particles and only provides a secondary mechanism for exchanging momentum between

the particles. This simplification yields a great deal of computational efficiency while

retaining the essential physics.

12

2.2 Simulation details

In molecular dynamics, the most commonly used time integration algorithm is the

Velocity-Verlet algorithm as presented by Swope et al. (1982). Following are the steps

in the algorithm.

1. Position update is performed as: x(t+ ∆t) = x(t) + v(t)∆t+ 12a(t)∆t2

2. Updated acceleration: a(t+ ∆t) = Ftotal(t+∆t)m

, where Ftotal has been calculatedbased upon the updated x(t+ ∆t)

3. Velocity update is given by: v(t+ ∆t) = v(t) + 12[a(t) + a(t+ ∆t)]∆t

As evident from the above steps, calculating the net force (Ftotal) is the most com-

putationally expensive part. In order to reduce the computational time, a linked list

algorithm is applied. Linked-list algorithm was initially used in molecular dynamics

(MD) simulations as an optimization tool. The idea within the Linked-list algorithm is

to discretize the whole domain into cells and save the particle information with respec-

tive cells. This information is then utilized to optimize the simulation algorithm. The

model described in this chapter is exercised on a variety of simulation conditions and

the results discussed in the subsequent chapters.

13

CHAPTER 3

Modeling and simulation of active granular systems

In this section we explain the extension of the previously discussed model to active

systems. In this study, focus is set upon studying human crowd dynamics. Traditionally

the Standard Vicsek model developed by Vicsek et al. (1995) as well as its dynamical

equivalent by Toner and Tu (1995) have been used to explore the dynamics of active

matter systems. We use a dynamical version of the well-studied agent based Vicsek

model utilized by Helbing et al. (2000) and Vicsek et al. (1995) to simulate crowd

dynamics. In this study, each agent is modeled as having a mass m and occupying an

area of a soft disc of diameter d. N such agents are confined in a circular boundary.

3.1 Force formulation for active granular suspensions

The temporal evolution of the particles is given according to the Newton’s second law of

motion. The agent responds to three classes of ‘interaction forces’. These three classes

of forces are given by

mdv(i)

dt= F(i)

pp + F(i)sp + F

(i)D (3.1)

where F(i)pp , F(i)

sp and F(i)D represent, respectively, the sum of all repulsive forces on the

ith agent due to binary interactions between the agents, a self propelling force generated

by the agent i and an alignment force on the ith agent due to agent-neighboring crowd

interactions. In order to model human crowds where agents exhibit a finite relaxation

time associated with the changes in their momentum mv, agent inertia is explicitly

incorporated, which is not part of the classical Vicsek model formulated by Vicsek

et al. (1995). The repulsive interaction force between the agent i and a neighboring

agent j occurs due to space exclusion modeled as

F(i,j)pp =

−knδ(i,j) if |r(i) − r(j)| < d

0 otherwise.

and F(i)pp =

∑j F

(i,j)pp . Here the extent of compression between the two agents i and

j involved in the collision event is calculated as δ(i,j) = (|r(i) − r(j)| − d) r(i)−r(j)|r(i)−r(j)| , a

vectorial distance along the separation vector r(i) − r(j). The position vector r(i) gives

the location of the center of the ith agent. The coefficient kn determines the strength of

the space exclusion force similar to that in Herztian contact theory. The self propelling

force F(i)sp is a thrust force produced by the agent. This force is given by

F(i)sp = m

(β − α|v(i)|

)v(i) (3.2)

where β characterizes the strength of self propelling force and v(i) is a unit vector

aligned along the instantaneous velocity of the agent. α is a small positive coefficient

introduced to ensure that a single agent does not exhibit unbounded acceleration. One

can also rationalize α as a net momentum sink since the thrust force itself is being mod-

eled as a monopole force. For dense ensembles, this issue does not arise. Consequently,

this choice was exercised in this study.

The last in the list is the collective crowd influence force experienced by each agent,

and it is given by

F(i)D = Cvd(v(i) − vc) (3.3)

where Cv is a co-ordination coefficient that controls the coupling strength between the

ith agent and its neighborhood crowd. This neighborhood spans a radius of h around

any agent, consists of n neighboring agents. We calculate vc using

vc =

∑nj=1mjWij

(|r(i) − r(j)|

)v(j)∑n

j=1 mjWij (|r(i) − r(j)|). (3.4)

Here Wij is a Gaussian function specifying the weight-age given to each agent in cal-

culating the average velocity v. This function is defined as

Wij =

exp(−η |r

(i)−r(j)|2h2

)if |r(i) − r(j)| ≤ h

0 otherwise

Here, η controls the distribution of contributions of surrounding agents to the motion

of the ith agent, which has been presented by Mahapatra et al. (2016). This agent-

15

crowd interaction enforces an alignment between the v(i), the velocity of the ith agent

and the mean motion of its neighbors. The present model does not feature explicit

thermal noise. However, the collisions between the agents randomizes both position

and velocity distribution of the agents and such randomization is augmented by the

presence of self propelling agents which together act as a source of deterministic noise.

Furthermore, this drag force penalizes absence of local coordination and therefore gives

rise to coordinated motion. In the absence of the fluid (or this force), only thermal

motion will be observed. Hence, compared to the classical Vicsek model by Vicsek

et al. (1995) or its variations from Nagai et al. (2015) and Peruani et al. (2012), the

present model incorporates several modifications to the equation of motion of agents to

simulate a crowd.

16

CHAPTER 4

Results - passive granular systems

The dynamics of passive granular systems have been studied using the model described

in the previous chapters. The results from that study are presented in this chapter.

4.1 Validation of the model for passive granular suspen-

sions

The model described above was validated qualitatively against the experimental mea-

surements of granular slurry in a circular drum by Liao et al. (2010). Their measure-

ments were performed on a rotating drum system containing a mono-disperse wet gran-

ular slurry. The simulations are performed for a two dimensional circular cavity for

3000 mono-disperse particles with the drum diameter maintained constant at 0.2 m.

The diameter of the particles is chosen to be 5mm. The walls of the cavity are modeled

as fixed particles with the same diameter as the particles in the granular suspension, and

the drum is rotated at 2.3 rpm. The value for the influence co-efficient η and influence

radius h was set at a constant value of 1 and 5d respectively in our study. The algebraic

sum of the forces on each particle are solved numerically for the position and veloc-

ity as a function of time using the Velocity-Verlet algorithm, which uses a linked-list

algorithm to reduce the computation time. Their measurements were performed on a

rotating drum system containing a monodisperse wet granular slurry. The following

conclusions were drawn from the comparison.

Firstly, we are able to simulate the slipping, rolling/cascading, cataracting and cen-

trifuging regimes that have been observed in wet granular flows. These regimes, as

shown in Fig. 4.1, have been realized in the appropriate ranges of Froude (Fr) num-

bers. However, the model is more successful in capturing the dynamics when the par-

ticles are packed as a dense slurry, for example during slipping, rolling, cascading or

centrifuging. However, during cataracting, the motion of free particles is less accurately

captured.

(a) (b)

(c) (d)

Figure 4.1: Regimes for wet granular flow in a drum of diameter 0.2 m with 3000 par-ticles of 5 mm each, for varying coordination coefficient (Cv).(a) Slipping regime (b) Cataracting regime (c) Rolling regime and (d) Cen-trifuging regime

Secondly, qualitative comparison with their experiments was in the form of com-

paring the structure of the granular matter. Fig. 4.2(a) show a comparison of the instan-

taneous particle positions from our simulation with a photograph from the experiments

by Liao et al. (2010). Both these images have been obtained at similar instants of time

and for similar parameter values. The number of particles was chosen such that half the

drum was filled with particles, similar to the experiments. We note that the arrangement

of the granular matter in both the images is qualitatively similar. In particular, the shape

of the fluidized zone, indicated by the dotted line in Fig. 4.2(b) matches well with their

experimental measurements.

Thirdly, we show a quantitative comparison of the mean velocity in the granular

material as a function of the non dimensional depth from the free surface. Fig. 4.2(c)

shows this comparison of our simulation prediction with the experimental measure-

ments of Liao et al. (2010). In this plot, Vs is the mean velocity of the granular material

18

(a) (b)

0 0.25 0.50 0.75 1.00

0.25

0.50

0.75

1.0

y/h

V/Vs

Experiments Simulations

(c)

Figure 4.2: Instantaneous particle positions from (a) the experiments of Liao et al.(2010), (b) the current simulations on a rotating drum. The region fromthe free surface to the dotted green line is the fluidized region in the gran-ular material. (c) Plot of the mean velocity profile in the granular materialfrom the current simulations and from the experiments of Liao et al. (2010)

19

at the free surface and y/δ is the normalized depth from the free surface. δ is the depth of

the fluidized zone and is measured in both the experiments and in our simulations as the

point where the particle mean velocity is nearly zero. Physically, this is the maximum

depth (measured from the free surface) in the granular material at which the velocity

follows the direction of the rotating drum. δ = 27 mm in both the simulations and the

experiments. The value of Cv is chosen by trial and error to match the experimental

results. The value of Cv in our model was set equal to 20 mPa.s. The viscosity of the

interstitial fluid in the experiments by Liao et al. (2010). was 60.1 mPa.s. However, it

has to be borne in mind that our simulations are two-dimensional, while the experiments

were of course, three-dimensional. It was observed that the choice ofCv only influences

the quantitative nature of the velocity profile. Qualitatively, the form observed in Fig.

4.2(c) would still be observed. It can be observed from this figure that the simulations

are able to capture the mean velocity profile in the particle phase in good quantitative

measure. These observations indicate that the proposed model captures the physics of

wet granular slurry dynamics to good accord.

4.2 Modeling landslide dynamics using DEM

The work in this section has been carried out during my six month research internship

at Hong Kong University of Science and Technology, in the Department of Civil and

Environmental Engineering.

This work is based on modeling free surface debris flow impacting an obstacle

downstream. A brief introduction about landslide modeling and mitigation is discussed.

Prevention of landslides has been a central area of research for civil engineers.

Avalanches have been the cause of many deaths and understanding their dynamics with

defense structures remains poorly understood. Typically landslides/avalanches are high

speed free surface flows which can be either wet or dry. Amongst the defense structures

used to prevent it’s runout, retaining dams remain as a favored choice due to it’s easy

construction and ability to sustain modifications as shown by Faug et al. (2008); Chu

et al. (1995). Traditional methods to reinforce these retaining dams include attaching

catchment nets or providing braking mounds.

Hakonardottir et al. (2003) studied the interactions of supercritical granular flow

20

with obstacles. The resulting runout lengths, geometry of the airborne jet were quan-

tified. Their study concluded by showing that the airborne jet behaves like an inviscid

fluid. Faug et al. (2008) experimentally investigated the influence of obstacle on gran-

ular avalanches. Three regimes were discovered in the flow depending on the incoming

Fr number of the flow. For very low Froude (Fr) numbers, a dead zone is formed in

front of the obstacle. For higher Fr numbers, a granular bore if formed. A transition

from the dead zone to a granular bore is achieved if the height of the obstacle (h) is in-

creased. Faug et al. (2015) presented a depth averaged analytical solution for granular

flows, and constructed a phase diagram as shown in Fig. 4.4(a). This phase diagram

clearly marks the transition from airborne jets to standing jumps. Pudasaini et al. (2007)

and Pudasaini and Kröner (2008) also presented analytical solutions for granular flows

as they hit obstacles downstream. In this present study, we numerically investigate the

effect of viscosity (Cv) and height of the dam on the Froude number (Fr) for wet free

surface granular flows. We observe a continuous phase transition from the airborne jet

regime to the granular bore regime as the dam height is increased.

4.2.1 Simulation Details

Dense wet granular flows on a 35◦ incline were simulated. The number of particles

were chosen to be around 50, 000. The ratio of height of the dam to the incoming flow

(H/h) was varied from 0.75 to 2.5. The viscosity of the suspension (Cv) was varied

from 50 to 200 mPa.s

4.2.2 Effect of varying dam height (H)

Fig. 4.3 shows the effect of increasing dam height and increasing viscosity (Cv) the

incoming flow. For low H/h ratio, the dead zone does not form upstream of the dam

and the change in the slopes of the free surface at the dam location is smooth.

However, as H/h increases, the granular discontinuous jump becomes increasingly

prominent. This effect is similar when Cv is varied for a fixed H/h ratio. For higher

values of H/h and Cv, there is a formation of dead zone upstream of the dam as shown

in Figs. 4.3(f) and Figs. 4.3(h). At such high values of H/h, the incoming flow slides

over the dead zone and overflows.

21

(a) H/h = 0.1, Cv = 20 (b) H/h = 0.1, Cv = 50

(c) H/h = 0.15, Cv = 20 (d) H/h = 0.15, Cv = 50

(e) H/h = 0.2, Cv = 20 (f) H/h = 0.2, Cv = 50

(g) H/h = 0.25, Cv = 20 (h) H/h = 0.25, Cv = 50

Figure 4.3: Free surface profiles of wet granular flow over a dam for varying H/h andCv

22

(a) Phase diagram obtained by Faug (2015)

0.8 1.2 1.6 2 2.4

0

2

4

6

H/h

Fr

Cv = 50

Cv = 70

Cv = 90

Cv = 110

Cv = 150

Cv = 200

(b) Effect of obstacle (H/h) on Froude number Fr from the current study

Figure 4.4: a) Phase diagram for dry free surface granular flows over an obstacleb) Effect of obstacle on Fr for wet free surface granular flows for varyingCv

23

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

L∗

NFS

H∗= 0.64 H∗

= 0.97 H∗= 1.21 H∗

= 1.70

(a) Cv = 50

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

L∗

NFS

H∗= 0.64 H∗

= 0.97 H∗= 1.21 H∗

= 1.70

(b) Cv = 70

Figure 4.5: Normalized Free Surface (NFS) as a function of the incline length for lowCv values

24

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

L∗

NFS

H∗= 0.64 H∗

= 0.97 H∗= 1.21 H∗

= 1.70

(a) Cv = 90

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

L∗

NFS

H∗= 0.64 H∗

= 0.97 H∗= 1.21 H∗

= 1.70

(b) Cv = 110

Figure 4.6: Normalized Free Surface (NFS) as a function of the incline length for highCv values

25

Fig. 4.4(a) reflects the above points as observed by Faug (2015) There are several

points to note from this phase diagram.

1. For high incoming Fr flows, and low H/h ratio, the incoming flow flows overthe dam as an airborne jet. In this regime, the runout length is high.

2. For intermediate values of Fr and H/h, we have a steady state in which a deadzone is in co-existence with the overflowing flow over the obstacle.This typicallyhappens at Fr ≈ 1.

3. At high Fr and high H/h values, there is formation of a standing jump near thewall. This happens because the grains in the dead zone are able to interact withthe rapid incoming flow. This aids the formation of the so-called granular bore,which is a dead zone extending further up the dam.

Fig. 4.4(b) shows effect of H/h on Froude number Fr obtained from the current

study. As evident from the figure, the Froude number decreases as H/h ratio is in-

creased, for a fixed Cv. An increased H/h values leads to a gradual formation of a

dead zone. Similarly for a fixed H/h ratio, increasing Cv decreases the Froude number.

Fig. 4.5 and Fig. 4.6 show the variation of Normalized Free Surface (NFS), which is

the ratio of the height of the free surface at a particular section of the incline and the

incoming flow height, with the incline (L∗) for fixed values of Cv. Hence we propose

that NFS is a good measure of the surface roughness of the flow profile, which is useful

in qualitatively understanding the extent of dead zone formation.

The following conclusions can be drawn from the above figures.

1. At low values ofCv the variation inNFS is insignificant except near the overflowregion. The NFS value rises monotonically with increasing H/h, indicating theincrease in the particles participating in the dead zone formation.

2. For intermediate and high values of Cv, there is a sudden increase in NFS athigher H/h ratios, indicating that the dead zone formation process is a rapid onefor high viscosity flows.

It can be concluded from this study that the transition from a granular bore regime

from an airborne jet regime is sudden and probably discontinuous at high viscosities of

the incoming flow.

26

CHAPTER 5

Results - dynamics of human crowds

Understanding the dynamics associated with human crowds is the focus for this study.

Each human being is modeled as a discrete self-propelling agent. We consider 6120

mono-dispersed active agents each of diameter d = 0.5 m in a domain of circular

shape whose radius R may vary between 14 m and 26 m. Particle density is chosen

as 4 persons/m2 as observed by Baglietto and Parisi (2011) and m = 60 kg to sim-

ulate the typical crowd conditions observed. We select kn = 2.8 × 106 N/m, α = 0,

β = 0.01 m/s2 and vary Cv in the range 0.2 − 1 Pa.s. The walls of the domain are

modeled as fixed particles with the same diameter as the particles. In order to describe

the neighborhood, the value for the influence coefficient η and influence radius h are set

at a constant value of 1 and 5d respectively in the present study. Simulations are initial-

ized by placing the agents in the domain at arbitrary locations and assigning non-zero

velocities in random directions.

5.1 Phases observed in active granular suspensions

We first discuss the differences in the states and dynamics exhibited by the system by

varying the value of the coordination coefficient Cv. Fig. 5.1 illustrates the three differ-

ent states observed at various values of Cv. Radial variation of the radial and tangential

components of the velocity field are also shown. Fig. 5.1(a) corresponds to a small

value of Cv, where active particles execute a disordered motion. This state resembles a

situation of crush in human crowds. Both the radial and angular components of veloc-

ity field fluctuate around the mean value of zero. As Cv is increased, the drag force on

each particle increases. Consequently the mean velocity of the particles decreases. At

Cv = 0.34, the particle velocities are so small that the system appears jammed as shown

in Fig. 5.1(c). However at this state, the motion of the particles is still disordered. On

increasing the value of Cv further, we observe the sudden onset of collective motion.

Due to the circular confinement, this collective motion results in an organized rotary

(a) Cv = 0.21

0 0.5 1

0

0.5

1

r/R

V ∗

r V ∗

θ

0 0.2−1

1

(b) Cv = 0.21

(c) Cv = 0.34

0 0.5 1

0

0.5

1

r/R

V ∗

r V ∗

θ

0 0.2−1

1

(d) Cv = 0.34

(e) Cv = 0.36

0 0.5 1

0

0.5

1

r/R

V ∗

r V ∗

θ

(f) Cv = 0.36

Figure 5.1: As the coordination coefficient (Cv) increases, the system exhibits a firstorder phase transition from a disordered phase to an ordered structure. Thevelocities have been scaled by the characteristic velocity

√βR. (a), (b)

show the radial, tangential velocity profile before transition. Inset showsinstantaneous fluctuations (c) shows the radial, tangential velocity profileafter transition. (d), (e) Velocity vectors averaged over particles before thephase transition. (f) Velocity vectors after transition.

28

state similar to a confined vortex as observed at Cv = 0.36 as shown in Fig. 5.1(e).

The tangential velocity profile describes this rotary state accurately, and the radial com-

ponent is small in magnitude and fluctuating. The linear variation of the tangential

velocity over a larger part of the domain indicates a core rotating like a rigid body. A

sheared fluid layer of particles is sandwiched between the core and the boundary since

the boundary exerts frictional forces on the particles.

In order to understand these observations better, we attempt to explain the underly-

ing mechanisms in each of these states. The competing motives come from the particle-

particle repulsive interactions introducing random noise in the system, which is further

amplified by the particular nature of the self-propelled force and the alignment mecha-

nism introduced through the drag forces arising from the neighboring particles. Since

thermal energy is ignored in the formulation, the disordered motion of the particles may

appear as a surprise. Indeed, in the absence of active forcing (β = 0), particles exhibit

no dynamics. A non-zero active forcing (β > 0) in the direction of their velocity vector

makes the particles to change their initial locations according to Eq. (3.1). During this

motion, they undergo collisions, which are expressed as the contact interactions (the

elastic repulsive force directed along the separation vector between the particles) and

the non-contact interactions (the drag force directed along vi − v). At small values

of Cv, contact interactions dominate the dynamics of the particles; they randomize the

motion of the particles and we obtain a disordered state similar to Brownian motion of

the particles. At higher values of Cv, drag forces start to dominate over elastic inter-

actions. This will dampen the motion of the particles and the average velocity in the

system decreases. It decreases to a point where the particles appear to be jammed in a

cage of neighboring particles without any substantial motion in the system.

In this model for human crowds, Cv manifests as a measure to induce local align-

ment of the individual agents. It not only provides the dissipative drag force on the par-

ticle but also causes the agent to align with its immediate neighbourhood flock. There-

fore, a further increase in the value of Cv changes the characteristics of the system

dramatically. Drag force from the surrounding particles dominate over the repulsive

elastic interactions, thus bringing in coordinated motion of the particles. The extent of

this coordination immediately spreads throughout the domain but limited by the confin-

ing circular domain. Particles exhibit circular motion, a preferred state under confine-

ment as observed in various experiments and simulations by Woodhouse and Goldstein

29

(2012) and Yeo et al. (2016).

At smaller values of Cv the collective motion disappears, the order in the velocity

field is lost and we obtain a state that resembles purely thermal (Brownian) motion of

particles as shown in Fig.5.1(a). These distinct states are also clearly visible in the ve-

locity profiles along the radius. In Fig. 5.1(f) the radial velocity is negligibly small and

the velocity field is dominated by its angular component which increases from zero at

the center to a maximum and reduces to zero again near the walls. On the other hand in

Figs. 5.1(b) and 5.1(d), the radial and azimuthal components of velocity are comparable

in magnitude. They fluctuate around a mean value of zero and are considerably small

compared to the state of collective motion. In the regime of collective motion agents

move ballistically while in disordered state their motion is diffusive.

Fig. 5.2 shows the spatially averaged velocity vectors for higher values of Cv. As

seen from Fig. 5.2(a), at Cv = 50, a hollow core is observed at the center of the domain.

The core appears as the coordination between the particles increases further leading to

density segregation. The size of the core is determined by the interplay between the

centrifugal forces and the dissipative force. The radial and azimuthal velocities are

shown in Fig. 5.2(b), Fig. 5.2(d) and Fig. 5.2(f). Furthermore, as Cv is increased

further (Cv = 10 Pa.s), the core disappears and the bulk displays signatures of both

an oscillatory bulk motion and the rotary motion spanning the entire domain as shown

in Fig. 5.2(c). The probability to form clockwise or anticlockwise rotations is equal.

Increased alignment at higher Cv manifests into increased drag. This leads to decrease

in Vavg, hence the centrifugal forces are unable to sustain the core. This is the reason

for the disappearance of the core. For very high values of Cv, we observe a novel

state, where all particles oscillate together horizontally like a viscous fluid with no bulk

motion of the particles as shown in Fig. 5.2(e)

30

(a) Cv = 5

0 0.5 1

0

0.5

1

r/R

V ∗

r V ∗

θ

(b) Cv = 5

(c) Cv = 10

0 0.5 1

0

0.5

1

r/R

V ∗

r V ∗

θ

(d) Cv = 10

(e) Cv = 50

0 0.5 1

0

0.5

1

r/R

V ∗

r V ∗

θ

(f) Cv = 50

Figure 5.2: As the coordination coefficient (Cv) increases, the system exhibits a firstorder phase transition from a disordered phase to an ordered structure. Thevelocities have been scaled by the characteristic velocity

√βR. (a), (b)

show the radial, tangential velocity profile before transition. Inset showsinstantaneous fluctuations (c) shows the radial, tangential velocity profileafter transition. (d), (e) Velocity vectors averaged over particles before thephase transition. (f) Velocity vectors after transition.

31

5.2 Order parameter and nature of the transition

In order to quantify these observations, the average speed of the particles is calculated

as

Vavg =1

T

∫ T

0

⟨1

N

N∑i=1

v(i)

⟩dt. (5.1)

which is the the magnitude of the average velocity of all the particles and use this quan-

tity as an order parameter to specify the state of the system. It is worth noting that Vavg

does not go to zero in the collective phase due to the addition of components of velocity

in polar coordinates. To systematically study the dynamics of the phases with varying

Cv, two sets of simulations were performed in the following manner.

In the first set, termed as forward sequential trials, Cv is varied from 0.20Pa.s to

0.50Pa.s through the transition point, in increments of 0.01. In each step, the steady

state end result of the previous state was taken as the initial condition. As evident from

the Fig. 5.3, the order parameter increases with an increase in Cv, and jumps sharply.

This point is named Cforwv,crit. Similarly, in the second set of simulations, termed as the

backward sequential trials, Cv is decreased through the transition point from 0.50Pa.s

to 0.20Pa.s Here the order parameter showed a sharp fall at the point appropriately

named as Cbackv,crit and the order parameter reached a value of zero. The critical value

of coordination coefficient of the forward trial Cforwv,crit was larger than the critical coor-

dination coefficient of the backward trial Cbackv,crit, indicating the presence of hysteresis,

which is a hallmark of first order transitions. Fig. 5.3 also shows the increase in the

width of the hysteresis loop as the domain size is increased. We confirm this, and iden-

tify the exact critical points, by calculating the Binder cumulant of the order parameter

through the transition. As reported earlier in the literature by Chaté et al. (2008), Selke

and Shchur (2009) and Durve and Sayeed (2016), we define the fourth order Binder

cumulant as,

G = 1−〈V 4

avg(t)〉3〈V 2

avg(t)〉2, (5.2)

where 〈V navg(t)〉 is the time averaged nth moment of the order parameter. Fig. 5.4 shows

the behavior of the Binder cumulant as a function of the control parameter Cv for var-

ious values of R. G varies from 2/3 to 1/3 as the system undergoes a phase transition

32

0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

Cv

Vavg

R/d = 29.4R/d = 37.3R/d = 41.1R/d = 53.0

Figure 5.3: The jump in the order parameter as a function of the coordination coefficient(Cv) and the hysteresis associated with this phase transition for various val-ues of R/d

33

0 4 8

−4

−2

−1

0

1/3

2/3

1

Cv

G

R/d = 24.0R/d = 29.4R/d = 37.3R/d = 53.0

Figure 5.4: Binder cumulant is plotted as a function of the control parameter Cv

34

from an ordered state to a disordered state as known for two dimensional systems. At

large values of Cv, 〈V 4avg(t)〉 ≈ 〈V 2

avg(t)〉2 due to the local organization of the velocity

of the agents and consequentlyG is 2/3. On the other hand it is known that a disordered

state will result into a G = 1/3. During the transition, Binder cumulant undergoes a

sharp dip due to the co-existence of two phases at the transition point as described by

Selke and Shchur (2009) proving that crowd stampede is indeed a first order phase tran-

sition from a uniformly moving crowd. Moreover the size and the sharpness of the dip

also increases with increase in the system size which is another feature of the first order

phase transition, as shown by Durve and Sayeed (2016).

Fig. 5.3 shows the variation of Vavg as a function of Cv for a fixed active forcing (con-

stant β). Different curves correspond to different domain sizes. It may be seen that

the average speed of the particles decreases with increase in drag force. As mentioned

earlier, beyond a critical coordination coefficient Cv,crit, drag force will dominate in

generating collective motion. This results in a sharp increase in the magnitude of the

order parameter. The system is in ordered state after this transition. Further increase

in drag force does not change the value of the order parameter considerably. When

particles are in random motion, the frictional forces exerted by the wall are signifi-

cantly small so that the size of the boundary does not affect the average speed of the

particles. However, in the rotary state boundary walls influence the motion the parti-

cles significantly. Therefore the transition from the thermal-like state to rotary state is

also dependent on the size of the domain. Since the perimeter to area ratio decreases

on increasing the size of the domain, the mean critical coordination coefficient Cmeanv,crit

which is the mean of Cforwv,crit and Cback

v,crit, decreases with increase in R. Therefore Cmeanv,crit

depends on the size of the domain as R−1, as depicted in Fig. 5.5. Fig. 5.6 shows the

instantaneous fluctuations for the azimuthal and the radial velocities as the Cv is varied

and two observations can be made. i) At Cv where the system is in a thermal-like state,

the azimuthal velocity Vθ and the radial velocity Vr fluctuate around a mean of zero, in-

dicating no bulk radial or azimuthal motion. ii) At higher values of Cv where the system

is in a rotary state, the radial velocity Vr fluctuates around a mean of zero, indicating no

bulk radial motion, but a non-zero shift in the mean value of the azimuthal velocity Vθ

is observed, indicating a bulk azimuthal motion.

Fig. 5.7(a) to Fig. 5.7(d) show the radial dependence of the scaled azimuthal veloc-

ity V ∗θ and scaled average speed |V ∗| for varying domain sizes. These velocities have

35

101.3

101.8

100.4

100.6

100.8

R/d

Cmean

v,crit

Figure 5.5: Cmeanv,crit as a function of R.

36

−1 0 10

150

300

Vθ

Vr

|V |

(a) Cv = 0.34

−1 0 10

150

300

Vθ

Vr

|V |

(b) Cv = 0.36

−0.2 0 0.20

200

400

Vθ

Vr

|V |

(c) Cv = 5

−0.1 0 0.10

150

300

Vθ

Vr

|V |

(d) Cv = 10

Figure 5.6: Histograms of azimuthal (Vθ) and radial (Vr) velocities for increasing Cv

37

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

r/R

V ∗

θ |V ∗|

(a) R/d = 29.4

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

r/R

V ∗

θ |V ∗|

(b) R/d = 37.3

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

r/R

V ∗

θ |V ∗|

(c) R/d = 45.1

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

r/R

V ∗

θ |V ∗|

(d) R/d = 53

Figure 5.7: Variation of scaled Azimuthal Velocity (V ∗θ ) and scaled Average Speed(|V ∗|) for different R/d ratios for their respective values of Cforw

v,crit

38

0 10 20 300

0.25

0.5

0.75

1

r (m)

|V∗|

R/d = 29.4R/d = 37.3R/d = 45.1R/d = 53

(a) Scaled average Speed (|V ∗|) variation for different R/d ratios

0 10 20 300

0.25

0.5

0.75

1

r (m)

V∗ θ

R/d = 29.4R/d = 37.3R/d = 45.1R/d = 53

(b) Average azimuthal speed (V ∗θ ) variation for different R/d ratios

Figure 5.8: Variation of V ∗θ and |V ∗| for various domain sizes for the respective Cforwv,crit

39

been scaled with the characteristic velocity√βR. These velocity profiles correspond to

the respective values of Cforwv,crit.

5.3 Bulk oscillation and Mean Square Displacement

Here, the existence of an oscillatory mode that arises in an active granular suspension is

investigated. This oscillatory mode is observed to be present in all four states discussed

in Fig. 5.1. In this mode, though there is no external field acting on the particles, the

bulk of the particles move as if the system is shaken by an external periodic forcing in

the plane of the simulation but in random directions. The amplitude of this oscillatory

motion is much smaller than the typical large length scale in the system (say, the domain

size (R)), that this mode is shadowed by other processes at small and large values of

Cv. However, just before the transition where the system appears to be in a jammed

state this oscillatory mode is distinctly present. Again, immediately after the transition

the system exhibits this oscillatory mode along with the rotary motion of the particles,

and this mode becomes less conspicuous with further increase in Cv.

We see signatures of this bulk oscillatory motion in the displacement trajectory of

the particles as well, as described below. The root mean square displacement analysis

of particles often give much insight into the particle dynamics, especially when the

processes are statistical in nature. The root mean square displacement of the particles

is calculated as a time averaged quantity of all the particles. The resulting data for each

Cv value has been averaged over 10 different sets of simulations. Fig. 5.9 shows the

root mean square displacement of particles for various values of Cv atR/d = 29.4. The

slopes of the curve lie between 12

and 1 indicating the diffusive and ballistic nature of

the motion of the particles at various values of Cv.

At very small values of Cv, the disordered motion is characterized by a smooth tran-

sition from ballistic to diffusive regime. This behavior is similar to that of a Brownian

particle. Though thermal noise is absent in the formulation, various forms of collisions

result into this particle dynamics. As the value of Cv increases, the velocity of the

particles go down initially (see the curve for Cv = 0.43). Correspondingly, the root

mean square displacement is also smaller. However, the slope the curve remains at 12

indicating the diffusive nature of the motion of the particles. Further increase in the

40

100

101

102

103

104

10−3

10−2

10−1

t

√

MSD

Cv = 0.2Cv = 0.43Cv = 0.44Cv = 0.48Cv = 0.55Cv = 0.8

1

1

1

0.5

Figure 5.9: Mean Square Displacement as a function of t for various values of Cv

0 20 40 60 80 1000

1

2

3

4

5

6x 10

−3

t

√

MSD

Figure 5.10: Zoomed in view of the Mean Square Displacement as a function of t forCv = 0.43

41

value of Cv retains this character in the particle dynamics till the critical coordination

coefficient Ccritv at which the motion changes to a ballistic regime. For example, see

the curve for Cv = 0.55 in Fig. 5.9. Recall that the particles in the present study are

confined in circular domain and therefore the root mean square displacement is cut off

by the system size due to the rotary motion of the particles. This leads to a saturation

length of the curves at very long times.

As discussed in the previous section, the presence of bulk oscillatory modes can be

detected in the Mean Square Displacement plots at early times. Fig. 5.10 shows the

early time behavior at Cv = 0.43. This is just below the critical coordination coefficient

Cv,crit where the oscillatory motion is pronounced.

5.4 Panic factor and artificial transition

To quantify the state of disorder in a human crowd, Helbing et al. (2000) introduced a

measure of selfishness or panic, and appropriately termed as panic factor. In this present

model, the panic factor takes a more generalized radially dependent form P (r) =

mβmβ+Cvdvc

, where vc is the averaged velocity of the neighbors according to the Gaus-

sian distribution mentioned earlier. In our model, the panic factor P is a measure of

the self interest (mβ) competing with the total resistance experienced by the individual

agent (mβ + Cvdvc). The radial variation of P can be seen from Fig. 5.11. For low

values of Cv, Fig. 5.11(a) and Fig. 5.11(b), which correspond to the thermal-like and

the jammed state respectively, the panic factor is isotropic and close to 1. This can be

explained as in the thermal-like state and the jammed state, vc is close to 0. Similarly at

higher values of Cv, Fig. 5.11(c), P decreases to a minimum value of 0.5 at r ≈ 0.7R.

A further increase in Cv leads to a uniform panic factor ≈ 0 throughout the system as

shown in Fig. 5.11(d) to Fig 5.11(f), as at higher Cv, the resistance from the crowd

dominates over self interest.

An interesting outcome of the hysteresis loop as shown in Fig. 5.3, is the existence

of two distinctly stable states of the system, namely the disordered state and the col-

lectively moving particles in the rotary state, for certain values of Cv between Cforwv,crit

and Cbackv,crit. This presents an opportunity to investigate the reason that triggers the phase

transition between the two states and utilize this knowledge for recovering order in the

42

0

0.2

0.4

0.6

0.8

1

(a) Cv = 0.21

0

0.2

0.4

0.6

0.8

1

(b) Cv = 0.34

0

0.2

0.4

0.6

0.8

1

(c) Cv = 0.36

0

0.2

0.4

0.6

0.8

1

(d) Cv = 5.0

0

0.2

0.4

0.6

0.8

1

(e) Cv = 10.0

0

0.2

0.4

0.6

0.8

1

(f) Cv = 50.0

Figure 5.11: Panic Factor (P ) variation at a) Cv = 0.21, b) Cv = 0.34, c) Cv = 0.36, d)Cv = 5, e) Cv = 10, f) Cv = 50

43

(a) (b)

(c) (d)

0 0.5 1

0

0.5

1

r/R

V ∗

r V ∗

θ P

0 0.2−1

1

(e) Cv = 0.34, Jammed state

0 0.5 1

0

0.5

1

r/R

V ∗

r V ∗

θ P

(f) Cv = 0.34, Rotary state

Figure 5.12: Forcing a stampede crowd to uniform velocity. (a), (b), (c), (d) respectivelyrepresent the distribution of particles to which a momentum impulse wasimparted. (a) distributed through out the domain, (b) a stripe at variousradial locations, (c) a pocket of agents at the center and (d) a pocket ofagents off-center. e) Panic factor distribution at the Jammed state at Cv =0.34, f) Panic factor distribution at the Rotary state at Cv = 0.34

44

system from a stampede. Therefore, the following set of simulations were conducted

for values of Cv inside the hysteresis loop. First, a set of simulations are conducted at

Cv = 0.34 Pa.s and R/d = 45.1. The state of the system is already disordered. In this

set of simulations, the goal is to transit the system from a disordered state as shown in

Fig. 5.12(e) to the state of collective motion as shown in Fig. 5.12(f). Considering that

the present model has circular agents whose only degree of freedom is expressed in the

form of velocity vectors we choose to perturb the velocity distribution of agents to drive

the system towards that of uniform velocity. The perturbation is in the form of assigning

a tangential velocity of a specific magnitude to a fraction of particles when the system

is at a statistically steady state of stampede. The distribution of fraction of agents were

selected in four different ways: (i) dispersed throughout the domain (Fig. 5.12(a)), (ii)

as a stripe at various radial locations(Fig. 5.12(b)), (iii) as a pocket located at the center

(Fig. 5.12(c)) and (iv) as a pocket located off center (Fig. 5.12(d)).

Similarly, a second set of simulations were performed at Cv = 0.31 Pa.s, when state

of the system is already in an organized rotary state. The goal in the second set of sim-

ulations is to explore the possibility of transition from an organized state to a state of

stampede. As the momentum impulse increases, critical number of selected particles

required for transition decreases. More importantly, amongst all the modes shown in

Fig. 5.12(a) to Fig. 5.12(d), the stripe at r = 0.7R requires the minimum number of

particles for the system to transit to the organized state. The reason for this is twofold:

• The panic factor attains a minimum at 0.7R as shown in Fig. 5.12(f). This min-imum value attained is close to 0.5, which can be the case only when mβ ≈Cvdvc. Hence, it can be deduced that the particles at the stripe at 0.7R experi-ence the least resistance from the crowd.

• As Fig. 5.12(f) shows, the azimuthal velocity (Vθ) is maximum at 0.7R, indicatingthat the average dissipation would be minimum at the stripe located at 0.7R

Hence it is beneficial to perturb the particles located at the region of 0.7R. For

the stripe selected at r = 0.7R, the order parameter Vavg increases and the system

transits into a uniformly moving state. However, in all other systems, the impulse slowly

decayed and the systems went back to the disordered state. The results of imparting a

scaled azimuthal velocity impulse of 0.557 to 18% of the total agents for all modes

of perturbation are indicated in Fig. 5.13(a) where the order parameter is plotted as a

function of time. Of course this behavior of the growth or decay of the impulse not

only depend on the distribution but also on the number of particles selected as well

45

100

101

102

103

0

0.1

0.2

Vavg

t

0.4R0.5R0.6R0.7R0.8R0.9RCenter PocketOffset P ocketDispersed

(a)

10 30 50 70 900

1.5

3

%

|Vavg|/|V

i|

Forward DispersedForward Stripe@0.5RForward Stripe@0.7R

10 30 50 70 900

1.5

3

%

|Vavg|/|V

i|

Backward DispersedBackward Stripe@0.5RBackward Stripe@0.7R

(b)

Figure 5.13: (a) Growth or decay of the order parameter as a result of forcing a stam-peded system with an impulse. (b) Critical number of particles required todrive the system back to uniform velocity decreases with increase in themagnitude of the momentum impulse. The results on both backward andforward trials are also shown. The velocities are scaled with Vavg in theinitial state of the perturbation.

46

as the strength of the momentum impulse. Therefore, we plot the critical number of

particles required for transition from a stampede to a uniformly ordered state for each

configuration in Fig. 5.13(b) as a function of strength of velocity impulse. Similarly we

plot the critical number of particles required for transitioning from a state of organized

motion to a stampede in Fig. 5.13(b). Consequently, it is clear that a nucleation-growth

kind of mechanism during the phase transition can be ruled out. Instead, the transition

between the disordered state and the rotary state is induced only when the perturbation

spans across a large region. Hence, it can be concluded that the transition is driven by a

long ranged order in the system.

5.5 Parametric study

In this section, the parametric dependence of the variables on the control parameter (Cv)

is presented. It is of interest to verify the existence of the aforementioned regimes as

the parameters in the system are altered.

Firstly, Fig. 5.14 to Fig. 5.17 show the phases obtained by increasing Cv for in-

creasing values of β, maintaining the other variables at their original values. As seen

from these figures, for lower values of β, a rotary state with core formation is observed.

As higher values of β are attained, the state with low Cv exhibits random thermal-like

motion, while states with higher Cv experience a core formation.

Secondly, Fig. 5.18 to Fig. 5.22 show the phases obtained by increasing the spring

constant (kn) for various values of β. These figures reveal that for lower values of kn, an

agitated rotary state with a high speed is observed. This is due to lower particle-particle

interaction forces, leading to higher accelerations, since the value for β remains the

same (= 0.01 m/s2). Finally, as the value of kn is increased, the three states (thermal-

like, jammed and rotary) are regained, and a solid core is observed at higher values of

Cv.

Thirdly, the variation of mass of the particles (m) is studied with increasing values

of Cv from Fig. 5.23 to Fig. 5.25. At low values of m, the system transits directly

from thermal-like motion to a rotary state with core. This can be due to the reduction

in mass which would give rise to a reduction in the contribution of inertia in generating

the noise in the system. Hence, for a higher alignment coefficient (Cv), the expectation

47

of a rotary core is reasonable. Fig. 5.26 shows the relative dependence of mass of the

particle (m) with increasing Cv. It can be inferred from this Fig. that in the range of

0.2m to 1m, Cv,crit remains fairly invariant. Similarly, Fig. 5.27 shows the dependence

of kn with increasing Cv. Again, it can be concluded that Cv,crit remains the same in

the range explored.

Finally, the dependence of β on Cv is shown in Fig. 5.28. This plot reveals the

following insights about the transition.

1. At very low values of Cv, a rotary state is observed for low values of β.

2. For a fixed β, there exists a finite value of Cv which transits the system into arotary state from the thermal-like phase.

3. As the value of β is increased, the value of Cv,crit necessary to achieve the transi-tion increases linearly. This is expected since higher values of β would generatemore thermal noise in the system, thus requiring a higher alignment strength (Cv)for transition.

48

(a) Cv = 0.2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(f) Cv = 2

Figure 5.14: Different phases of the system at β = 0.1 m/s2. a) Rotary state with nocore formation, b) and c) Rotary state with core formation

49

(a) Cv = 0.2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(f) Cv = 2

Figure 5.15: Different phases of the system at β = 0.25 m/s2. a) Thermal-like state, b)Rotary state without core formation and c) Rotary state with core forma-tion

50

(a) Cv = 0.2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(f) Cv = 2

Figure 5.16: Different phases of the system at β = 0.5 m/s2. a) Thermal-like state, b)Rotary state without core formation and c) Rotary state with core forma-tion

51

(a) Cv = 0.2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(f) Cv = 2

Figure 5.17: Different phases of the system at β = 0.75 m/s2. a) Thermal-like state, b)Rotary state without core formation and c) Rotary state with core forma-tion

52

(a) Cv = 0.2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(f) Cv = 2

Figure 5.18: Different phases of the system at kn2 × 104 N/m. a) High speed rotarystate, b) and c) Rotary state with core formation

53

(a) Cv = 0.2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(f) Cv = 2

Figure 5.19: Different phases of the system at kn = 5×104 N/m. a) High speed rotarystate, b) Rotary state without core formation and c) Rotary state with coreformation

54

(a) Cv = 0.2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(f) Cv = 2

Figure 5.20: Different phases of the system at kn = 106 N/m. a) Rotary state withoutcore formation, b) Rotary state without core formation and c) Rotary statewith core formation

55

(a) Cv = 0.2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(f) Cv = 2

Figure 5.21: Different phases of the system at kn = 5 × 106 N/m. a) Thermal-likestate, b) Rotary state without core formation and c) Rotary state with coreformation

56

(a) Cv = 0.2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 2

0 0.5 1

0

0.5

1

r/R

Vr Vθ

(f) Cv = 2

Figure 5.22: Different phases of the system at kn = 2 × 107 N/m. a) Thermal-likestate, b) Rotary state without core formation and c) Rotary state with coreformation

57

(a) Cv = 0.2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 1

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(f) Cv = 1

Figure 5.23: Different phases of the system at 0.1m. a) Thermal-like state, b) Rotarystate without core formation and c) Rotary state with core formation

58

(a) Cv = 0.2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 1

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(f) Cv = 1

Figure 5.24: Different phases of the system at 0.5m. a) and b) Thermal-like state, c)Rotary state without core formation

59

(a) Cv = 0.2

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(b) Cv = 0.2

(c) Cv = 0.5

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(d) Cv = 0.5

(e) Cv = 1

0 0.5 1

0

0.25

0.50

r/R

Vr Vθ

(f) Cv = 1

Figure 5.25: Different phases of the system at 2m. a) and b) Thermal-like state, c)Rotary state without core formation

60

Cv

m

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.5m

2m

1.5m

1m

0.9m

0.8m

0.7m

0.6m

0.5m

0.4m

0.3m

0.2m

0.1m

G

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Figure 5.26: Invariance of Cv,crit with varying mass (m) and Cv

Cv

kn

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.71.6e+06

1.8e+06

2e+06

2.2e+06

2.4e+06

G

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Figure 5.27: Invariance of Cv,crit with varying spring stiffness coefficient (kn) and Cv.

61

Cv

β

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70.001

0.003

0.005

0.007

0.009

0.012

0.014

0.016

0.018

0.02

G

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Figure 5.28: Linear monotonic increase of Cv,crit with increasing β and Cv

62

CHAPTER 6

Conclusion and Future Work

6.1 Conclusion

There are two areas of focus in the current work. The first involves validating a mini-

malistic, novel Discrete Element Method (DEM) model for dense granular suspensions

developed by Bonkinpillewar et al. (2015), which simplifies the fluid-particle interac-

tion dynamics, while still retaining the essential physics. This validation has been done

with the experiments conducted by Liao et al. (2010). Additionally, the regimes de-

scribed by Ottino and Khakhar (2000) namely: slipping, rolling/cascading, cataracting

and centrifuging regimes have been replicated by our model for varying viscosity. Fur-

ther, the model has been extended suitably to mimic the dynamical properties of dense

active suspensions. The specific objective studied here is modeling human crowd dy-

namics in confined spaces. A stampede-like behavior is observed at low values of the

strength of alignment (Cv). As Cv is gradually increased, a sudden jump in the order

parameter is seen. This phase transition is observed which transits into a state of col-

lective motion. This transition is shown to exhibit hysteresis and is deduced to be a

first-order transition by calculating the fourth-order Binder cumulants (G) of the order

parameter. An occurrence of metastability is investigated in the hysteretic region. The

switching between the steady states of the system in the hysteretic region has been trig-

gered via artificially perturbing randomly picked particles spanning the entire domain.

It is shown that the success of the perturbation is linked with the choice of nucleation.

Furthermore, a measure of disorder in the crowd, namely, the panic factor (P ) is in-

troduced. The spatial variation of P is studied and is linked with the success of the

artificial transition. A region of the domain has been identified which is most favorable

for the nucleation. This artificial transition can be utilized in avoiding stampedes in

large human gatherings.

6.2 Future Work

Several different directions open up for future work, based on the generic formulation

of the model developed by Bonkinpillewar et al. (2015). The following additions to the

model could prove valuable for future work.

• Incorporation of a term which mimics attractive potential between the particles,as studied by Iliass and Cambui (2016), which would eliminate the need for con-finement of the system.

• Modifying the model to replicate the dynamics in elliptical particles. These mod-ifications can be made via implementing the multi-sphere model for ellipticalparticles. Kruggel-Emden et al. (2008) and Hohner et al. (2011) have reviewedthe methods for the same.

• The effect of changing the confinement geometry, leading to possibly new dy-namical states is worth exploring. An interesting possibility is active matter ina racetrack geometry as explored by Theillard et al. (2017), Shelley (2013) andWioland et al. (2016)

• Activity induced fluidization of the system from the jammed state as reported byDerzsi et al. (2017) and Mandal et al. (2016) is of active interest in recent litera-ture. The same can be explored in the current model by altering the magnitude ofthe self-propelled force.

64

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72

LIST OF PAPERS BASED ON THESIS

Papers in Refereed Journals

1. Bonkinpillewar, P.D., Kulkarni, A., Panchagnula, M.V. et al. (2015) “A novelcoupled fluid-particle DEM for simulating dense granular slurry dynamics” Gran-ular Matter 17(4): 511-521.

2. Mahapatra, P. S., Kulkarni, A., Mathew, S., Panchagnula, M. V., and Vedantam,S. (2017). Transitions between multiple dynamical states in a confined denseactive particle system. Phys. Rev. E 95, 062610

Presentations in Conferences

1. Activity induced phase transition in mixtures of active and passive agentsPallab Sinha Mahapatra, Ajinkya Kulkarni70th Annual Meeting of the APS Division of Fluid Dynamics November 2017

2. Phase Transitions and Metastability in Self-Propelled Particle systemsSumesh Thampi, Ajinkya Kulkarni and Mahesh PanchagnulaCell and tissue motility, CECAM-HQ-EPFL May 2017

3. Phase transitions in Active Granular SuspensionsAjinkya Kulkarni, Sumesh Thampi and Mahesh PanchagnulaCompFlu 2016 December 2016

4. Phase Transitions and Metastability in Self-Propelled Particle systemsAjinkya Kulkarni, Sumesh Thampi and Mahesh Panchagnula69th Annual Meeting of the APS Division of Fluid Dynamics November 2016

5. Self Propelled particle systems: A study of the onset of organized motionAjinkya Kulkarni, Srikanth Vedantam and Mahesh Panchagnula68th Annual Meeting of the APS Division of Fluid Dynamics November 2015

Papers in preparation

1. Route towards collective order in panicked human crowdsAjinkya Kulkarni, Sumesh Thampi and Mahesh Panchagnulamanuscript under preparation.

2. Activity induced fluidization in active granular suspensionsAjinkya Kulkarni, Pallab Sinha Mahapatra, Mahesh Panchagnulamanuscript under preparation.

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