a viscoelastic model for dense granular flows

A viscoelastic model for dense granular flowsD. Z. Zhang and R. M. Rauenzahn

Citation: Journal of Rheology (1978-present) 41, 1275 (1997); doi: 10.1122/1.550844 View online: http://dx.doi.org/10.1122/1.550844 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/41/6?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Remarkable drag reduction in non-affine viscoelastic turbulent flows Phys. Fluids 25, 015106 (2013); 10.1063/1.4774239 Mesoscale hydrodynamic modeling of a colloid in shear-thinning viscoelastic fluids under shear flow J. Chem. Phys. 135, 134116 (2011); 10.1063/1.3646307 Effects of viscoelasticity on the probability density functions in turbulent channel flow Phys. Fluids 21, 115106 (2009); 10.1063/1.3258758 Comparison of measured centerplane stress and velocity fields with predictions of viscoelastic constitutivemodels J. Rheol. 47, 1331 (2003); 10.1122/1.1608951 Analysis of the Constant Rate Startup Flow of a Viscoelastic Fluid in Annular, Cylindrical, and Planar Conduits J. Rheol. 25, 193 (1981); 10.1122/1.549614

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A viscoelastic model for dense granular flows

D. Z. Zhanga),b) and R. M. Rauenzahn

Los Alamos National Laboratory, Theoretical Division, Fluid DynamicsGroup, T-3, B216, Los Alamos, New Mexico 87545

(Received 4 February 1997; final revision received 12 August 1997)

Synopsis

In traditional kinetic theory for a granular flow, it is usually assumed that particle interactions areinstantaneous and binary. For a dense granular system, these assumptions are usually invalid. In thispaper, we use an ensemble averaging technique to examine the effects of finite particle interactiontime and multiparticle collisions. The main objectives of this paper are to develop a method and toprovide a tool to study dense granular materials. As an example, we study flows of granular particlescoated with thin layers of resin. To model particle elasticity and resin viscosity, the force betweena pair of particles is approximated by a serial connection of a linear spring and a dashpot.Subsequently, a viscoelastic model is developed from the averaging method. In order to determinecoefficients in the constitutive model, direct numerical simulations are performed. When the particleconcentration is relatively low, the shear stress is quadratically proportional to the shear rate, inagreement with kinetic theories. At a high particle concentration, the shear stress depends linearlyon the rate of strain. The transition between this quadratic and linear dependence is similar to aphase transition. In a dense system, when the shear rate exceeds a critical value, shear bandformation is also observed. ©1997 The Society of Rheology.@S0148-6055~97!00506-3#

I. INTRODUCTION

Since the 1960s, kinetic theories have been used to model rapid granular flows. Inthose theories, it is always assumed that particle interactions are binary and can bemodeled as instantaneous collisions with a proper coefficient of restitution. The dynamicsduring the particle interactions are ignored. These assumptions are clearly valid in arelative dilute system where the particle contact time is negligible compared with themean-free flight time. In such systems, the duration of particle interaction is short, and apair of particles completes their interaction before another particle approaches the inter-acting pair. Based on these assumptions, many models have been successfully developed@Lun ~1991!; Campbell~1989!; Savage and Jeffrey~1981!#, and numerical simulationshave been performed to verify them@Walton and Braun~1986!; Campbell~1989!#. Goodagreement is usually found for systems with relative small particle volume fraction andwith the coefficient of restitution close to unity. In numerical simulation, both hard-sphere models@Campbell~1989!; Lun and Bent~1994!# and soft-sphere models@Waltonand Braun~1986!# have been used. As pointed out by Campbell~1989!, the soft-spheremodels introduce an effect of finite particle interaction time or contact time. This effect isnegligible in a dilute system but is important in a dense system. The objectives of this

a!Corresponding author.b!Electronic mail: [email protected]

© 1997 by The Society of Rheology, Inc.J. Rheol. 41~6!, November/December 1997 12750148-6055/97/41~6!/1275/24/$10.00


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paper are to understand the effects of the dynamics of multiparticle interaction during thefinite collision time and the related relaxation mechanism in a dense granular system.

In a dense system, the streaming part of the granular stress is at least one order smallerthan the collisional or contact part. Thus, in this paper, we shall focus on the collisionalstress. Because of finite contact time, the dynamics of multiparticle interactions are im-portant. For instance, the rotation of the line connecting the centers of the contactingparticles is significant because of the relative tangential velocity between them and finitecontact duration. This rotation causes a change in the force direction from that at initialcontact. The mean effect of this rotation transports the averaged contact force betweenthe two particles from one direction to the other according to the vorticity of the meanflow. Therefore, the stress tensor in such systems depends not only on the rate of strainbut also on the vorticity of the mean flow.

To consider the dynamics of multiparticle interaction, we employ the ensemble aver-aging technique recently developed by Zhang and Prosperetti~1994, 1997! in the study ofmultiphase flows. We shall derive the averaged equations for the granular system, fromwhich we obtain an expression relating collisional stress to the force system in thegranular material at the microscopic level. Then, the evolution equation of the collisionalstress is derived from the transport theorem proven by Zhang and Prosperetti~1994,1997!. The effect of pair rotation appears naturally in the constitutive relation as theJaumann time derivative of the collisional stress tensor.

As an example, we study a flow of particles coated with a thin layer of viscous resin.This type of granular flow is common in many industrial processes, especially in moldmaking processes in foundries. To derive the constitutive relation from the study of thecomplicated dynamics of multiparticle interaction, we assume that a thin layer of fluidalways exists around a particle and that solid–solid contact is unimportant. For thincoatings, according to the lubrication theory, the dominant force acts normally to thecolliding surface. In order to account for resin viscosity and particle elasticity, we modelthe force between a contacting pair by a serial connection of a spring and dashpot. Weassume that the spring is linear with constantK and the dashpot has a constant resistanceR. In this system, the time scale of energy dissipation during a particle contact is char-acterized by the relaxation timeTr 5 R/K. For a dense system, particle contact time is,on average, much longer than this relaxation time. Therefore, this relaxation is the domi-nant time scale in the system. Based on this, a viscoelastic model for the collisional stressis obtained from its evolution equation.

To study the model, we have numerically simulated a simple shear flow. At lowparticle volume fractions, our numerical results are in general agreement with kinetictheory @Lun ~1991!#. At volume fractions approaching that of random loose packing, atransition in rheological behavior occurs. Shear stress is proportional to the product ofcollision frequency and the force during a collision, both of which are proportional to theshear rate at low particle concentrations. However, at high volume fraction, particles areconstantly in contact, and the collision frequency saturates. In this case, this reasoningand our simulations predict a linear relationship between the shear stress and shear rate.

When the particle volume fraction is greater than 0.3, shear band formation is ob-served at high shear rates in our simulation. When a shear band forms, the particles aredivided into two layers sliding against each other. Inside each layer, little relative particlemotion is observed.

II. AVERAGED TRANSPORT EQUATIONS AND GRANULAR STRESS

The transport equations for rapid granular flow are usually derived from a kinetictheory @Lun and Savage~1987!; Lun ~1991!#. In a kinetic theory, it is usually assumed

1276 ZHANG AND RAUENZAHN


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that the flow is relatively dilute, particle–particle interactions are binary, and the durationof contact~collision! is negligible. With these assumptions, a set of averaged equations isderived from the Boltzmann equation. In a slow and dense granular flow, these conditionsare not usually satisfied. A general method of studying the force system and of obtainingthe averaged transport equations for multiphase flows has been developed in recent pa-pers by Zhang and Prosperetti~1994, 1997!. In granular flow, where interstitial fluid isnot present, this method becomes similar to one used by Irving and Kirkwood~1950! ina molecular system. In this paper, we shall briefly introduce the main steps and results ofthe method. Major attention will be focused on the derivation of the evolution equationfor the collisional stress.

We consider an ensemble ofN identical spherical particles with radiusa. Each flow inthis ensemble can be uniquely described by its configurationC N consisting of particlepositionsy(a), a 5 1,2,...,N, velocitiesw(a), and other parameters, sayH; necessary tocompletely determine the system. LetP(C N;t) be the probability distribution of havingconfigurationC N at time t. Since all particles in the system are identical, it is moreconvenient to normalize this probability as@Batchelor~1972!#

E P~C N;t !dCN 5 N!. ~1!

The number density of particles is then defined by

n~x,t ! 51

N!E dP~x,C N!P~C

N;t !dCN, ~2!

wheredP is the point indicator function

dP~x,C N! 5 (a 5 1

N

d~x2y~a!!. ~3!

The particle volume fraction at pointx is the probability of finding this point occupied bya particle and can be calculated as@Zhang and Prosperetti~1994!#

bD~x,t ! 5 Eux2zu , a

n~z,t !dz3 5 n~x,t !v11

10va2¹2n~x,t !1oS a2

L2 nv D . ~4!

wherev 5 43pa3 is the particle volume andL is the length scale over which the averaged

macroscopic quantities vary significantly.Let g(a)(C N;t) be a quantity associated with particlea, and define a function

g~x,C N,t ! 5 H g~a!~CN,t !, if x 5 y~a!, ;a 5 1,2,...,N,

0, otherwise.~5!

The average of the particle quantity is defined as

g~x,t ! 51

n~x,t !N!E dP~x,C N!g~x,C N,t !P~C

N;t !dCN

51

n~x,t !N!E (

a 5 1

N

d~x2y~a!!g~a!~CN;t !P~C

N;t !dCN. ~6!

The average defined this way is especially convenient to study quantities pertaining toentire particles, such as particle velocities and angular velocities.

1277DENSE GRANULAR FLOWS


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Following Zhang and Prosperetti~1994!, one can derive the transport equation for anaveraged particle quantity as

]ng

]t1“•~nwg! 5 n

dg

dt, ~7!

where the time derivative on the right is the total time derivative

dg

dt5

]g

]t1 (

a 5 1

N

w~a!•“y~a!g1w~a!

•“w~a!g1dH

dt•“H g . ~8!

Equation~7! is the fundamental equation in our following derivations of the averagedcontinuity and momentum equations for the granular system. After we obtain the expres-sion of the collisional stress later in this section, and introduce the force model in the nextsection, we use this transport equation again to derive the transport equation for thecollisional stress in Sec. IV.

The continuity equation for the granular flow can be derived by takingg 5 1:

]n

]t1“•~nw! 5 0. ~9!

If this is multiplied by the particle volumev and an error of ordera2/L2 is neglected asin Eq. ~4!, the continuity equation can be written in terms of particle volume fractionbD

]bD

]t1“•~bDw! 5 0. ~10!

The averaged particle momentum equation can be derived directly from the equationgoverning the motion of a particle, saya, in the system

mw~a! 5 fc~a!1mb, ~11!

wherem is a particle mass, andb is the external body force resulting from interactionswith objects outside the granular system, such as gravity. The forcefc

(a) is the forceapplied by other particles in the system. This force may be due to collision, electromag-netic interaction, etc. For simplicity, we will denote them as collisional forces. Themethod described in this paper is the outgrowth of the method developed by Zhang andProsperetti~1994, 1997!. In those papers, interactions between the surrounding fluid andthe particles are studied extensively. In this paper, we shall focus on the interactionamong the particles and neglect particle–fluid interactions.

Taking g(a) 5 mw(a) and then making use of Eqs.~7!, ~9!, and~11!, one finds

nmS]w

]t1w•“wD 5 “•~nmM !1nfc1nmb, ~12!

wherenmM is the streaming part of the granular stress,

M 5 ww2ww 5 2~w2w!~w2w!. ~13!

For the forcefc , we shall follow the same steps of Irving and Kirkwood~1950! toformally prove that it can be expressed in terms of a stress. The total forcefc

(a) acting onparticlea from other particles can be written as



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fc~a! 5 (

b Þ a

N

f~y~a!,y~b!,w~a!,w~b!,C N22;t !, ~14!

where superscriptb denotes that the force on particlea is applied by particleb. It maybe noted that this decomposition of the forcefc

(a) on a particle is always possible, sinceany internal force acting on the particle is always applied by another particle. Accordingto the definition of particle average, we can write

nfc 51

N! (@a,b#

E d~x2y~a!!f~y~a!,y~b!,w~a!,w~b!,C N22;t !P~CN;t !dC

N, ~15!

where@a,b# denotes summation over all possible pairs of particlesa andb, ~a Þ b!. Thesummation can be written in a symmetrized form as

nfc 51

2N! (@a,b#

E @d~x2y~a!!~ fP!~y~a!,y~b!,w~a!,w~b!,C N22;t !1d~x2y~b!!

3~ fP!~y~b!,y~a!,w~b!,w~a!,C N22;t !#dCN, ~16!

where fP stands for the product of the forcef and the probability densityP. Sinceparticlesa andb are indistinguishable, we have

P~y~a!,y~b!,w~a!,w~b!,C N22;t ! 5 P~y~b!,y~a!,w~b!,w~a!,C N22;t !. ~17!

Using Newton’s third law, one can write

f~y~a!,y~b!,w~a!,w~b!,C N22;t ! 5 2f~y~b!,y~a!,w~b!,w~a!,C N22;t !. ~18!

With these, we rewrite Eq.~16! as

nfc 51

2N! (@a,b#

E @d~x2y~a!!2d~x2y~b!!#

3~ fP!~y~a!,y~b!,w~a!,w~b!,C N22;t !dCN. ~19!

Note thaty(b) 5 y(a)1(y(b)2y(a)). The difference of thed function is expanded for-mally by Taylor series as

d~x2y~a!!2d~x2y~b!! 5 ~y~b!2y~a!!“xd~x2y~a!!1••• . ~20!

Substitution of this into Eq.~19! leads to

nfc 5 “•~bDs!, ~21!

wheres, the stress due to particle interaction~collision! is defined by

bDs 51

2N! (@a,b#

N E d~x2y~a!!f~CN;t !~y~b!2y~a!!P~C

N;t !dCN. ~22!

As found in a recent paper by Zhang and Prosperetti~1997!, terms similar to the higher-order terms neglected in Eq.~20! have contributions similar to correction of the effectiveviscosity and Faxe´n correction to the Stokes drag in particle suspensions. They can beneglected only when an error of orderO(a2/L2) is allowed. In this paper, we shall focuson the first-order term.



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With a similar method, Lunet al. ~1984! obtained an expression for the collisionalstress under the assumption of binary collision. The key step of this proof is the expan-sion of thed function, corresponding to translation of the center of particleb to the centerof particle a. This translation is legitimate only if it is conducted in such a way thatrelative positions among particles are unchanged and such that particle overlap does notoccur. This subtle point is not entirely trivial and is not clear in expression~20!. Theproof of Eq. ~21! given here is a formal proof only. A proof considering these factors,with a clearer physical picture and more mathematical rigor is provided in the Appendix.

With the collisional granular stress so defined, the averaged momentum equation canbe written as

bDrDS]w

]t1w•“wD 5 “•~bDrDM1bDs!1bDrDb, ~23!

whererD is the density of the particle material.In this equation, closure relations are needed for the Reynolds stress and the colli-

sional stress. The primary focus of this paper is on dense granular flows. In such flows,the collisional stress far exceeds the streaming part of the granular stress. In the remain-der of this paper, attention will be focused on the development of a constitutive relationfor the collisional stress. However, at this point we compare our collisional stress expres-sion with that of kinetic theories.

In order to do so, we write Eq.~22! in terms of the averaged forcef(2) conditional onthe specified velocities and positions of two contacting particles:

bDs 51

2E d3w~1!d3w~2!d3y~2!

3 f~2!~y~1!,y~2!,w~1!,w~2!;t !rP2~y~1!,y~2!,w~1!,w~2!;t !, ~24!

wherer 5 y(2)2y(1) is the relative distance between two particle centers, andP2 is thepair distribution function defined by

P2~y~1!,y~2!,w~1!,w~2!;t ! 5

1

N~N21!E P~y~1!,y~2!,w~1!,w~2!,C ~N22!;t !dC

~N22!,

~25!

anddC (N22) denotes that the integral is over all possible positions and velocities of theremainingN22 particles.

In a kinetic theory, the particle collision time is vanishingly small. Therefore, for afinite relative velocity between particles, solid deformation is negligible andr 5 2an.

bDs 5 4a3E dVnn

3E P2~y~1!,y~2!,w~1!,w~2!;t ! f ~y~1!,y~2!,w~1!,w~2!;t !drd3w~1!d3w~2!,

~26!

whereV is the solid angle. Here, we have assumed that the force between particles actsonly in the normal direction. We note that



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4a2dVEP2~y~1!,y~2!,w~1!,w~2!;t ! f ~y~1!,y~2!,w~1!,w~2!;t !d3w~1!d3w~2!dr, ~27!

is the averaged collisional force in the normal directionn, and is equal to the rate ofmomentum change of particles approaching the particle centered aty(1). In a dilutesystem, where particle interaction time is vanishingly small, this rate of momentumchange can be calculated as the product of the collision frequency and momentum ex-change during each collision@Savage and Jeffrey~1981!; Lun et al. ~1984!#,

2a2~11e!m~v•n!2P2~y~1!,y~2!,w~1!,w~2!;t !dV, v•n . 0, ~28!

wherev 5 w(2) 2 w(1), ande is the coefficient of restitution. In this case, Eq.~24! canbe written as

bDs 5 22~11e!a3mEv•n . 0

P2~y~1!,y~2!,w~1!,w~2!,t !~v•n!2nndVd3w~1!d3w~2!.

~29!

This is identical to Eq.~3.6! in Lun et al. ~1984! or Eq. ~2.19! in Savage and Jeffrey~1981!. In a simple shear flow, the relative velocityv between particles is proportional tothe imposed shear rate. Therefore, Eq.~29! predicts a quadratic dependency of the stresson the shear rate.

The arguments used to obtain Eq.~28! do not apply to a system in which particleshave finite interaction time during a collision, and multiparticle collisions must be con-sidered. Even in a binary collision with a finite contact duration, the relative tangentialvelocity causes the direction of the contact force to rotate during the collision, and thecollisional force is governed by a more complicated expression. To account for theseeffects, the dynamics of particle interaction need to be considered.

III. FORCE MODEL FOR RESIN COATED PARTICLES

Many industrial processes involve flow of granular particles coated with thin layers ofviscous resin. Typically, the averaged size of the particles is about a few hundred micronand the thickness of the fluid layer is about a few micron. One example of such a processis the sand flow in the process of making sand cores in foundries. In this process, theforce between contacting particles is determined by the flow of the liquid within the gapbetween particles. In principle, to obtain accurate forces between two particles, one mustsolve the flow field in the gap and air flow around the particles. To obtain detailedsolutions in the mixture requires computation of the detailed flow field around eachparticle. This itself is a topic of research in the modern theory of multiphase flows@Brady~1988!; Feng, Hu, Joseph~1994a,b!#, and is beyond the scope of this paper.

To approximate the force between particles, we assume that solid–solid contact be-tween the resin-coated particles is unimportant, and the force between the contactingparticle is always transmitted through resin. We note that the force transmitted from resincauses deformation of the particles and stores elastic energy in the particles. In order toaccount for both the viscosity of the fluid and elasticity of the particles, we model theinteraction between a pair of particles by a serial connection of a spring and a dashpot. Inthis way, we greatly simplify the detailed effect of the flow in the gaps, while retainingthe basic physics of the granular system. For simplicity, we also assume that the spring islinear with a constantK, and the dashpot has a constant resistanceR. In the real system,this is not necessary true. For instance, according to lubrication theory, the resistance ofthe dashpot is inversely proportional to the gap between the pair of particles. However,



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the assumptions contain the basic elements of the complex system, such as the finitecontact time of the particles in a collision, which is completely neglected by kinetictheories. The method put forth in this paper allows us to obtain a constitutive relationfrom a description of dynamics at the microscopic level, but the method is not restrictedto the force model used. We use this simple force model as an example to explore waysto achieve this goal and to understand the basic physics encountered in such systems.

In the force model, the strength of the dashpot characterizes energy dissipation. In adilute system, this energy dissipation is represented by the coefficient of restitution in abinary collision. In a dense system, with multiparticle interactions, energy dissipation ischaracterized by the relaxation timeTr 5 R/K. More general cases with nonspherical,polydisperse particles are considered in our recent paper@Zhang and Rauenzahn~1997!#,in which a similar constitutive relation is obtained.

According to lubrication theory, to leading order, the contribution of the normal forceis proportional toa/d, whered is the gap between particles, while the leading contribu-tion of the tangential force is proportional to ln(d/a). For a thin fluid coating around theparticle, the tangential force is negligible compared with the normal force. Therefore, wecan write the force between contacting particles as

f 5 Rxr 5 Kxe, r , r0, ~30!

wherexr is the deformation velocity of the dashpot,xe is the elastic deformation of thespring, andr 0 is the distance between two particles beyond which the particles cease tointeract.

The relative normal velocityvn of the particle centers is related to the deformationvelocity of the spring and dashpot

xr1xe 5 vn . ~31!

In order to understand this force model and to derive the evolution equation for thecollisional stress, we first study the properties of a binary collision in which equations ofmotion of the two particles can be solved analytically. We consider two particles ap-proaching each other with equal speedsv0 . At time t 5 0, the distance between thecenters isr 0 , and the interaction begins. The positions of the particle centers are denotedby x(t) and 2 x(t), implying that the origin of the coordinate system is at the middlepoint between them. The relative velocity,vn , between the centers is 2x(t). The equa-tion of motion for these two particles, by using Eqs.~30! and ~31!, can be written as

d3x

dt31

K

R

d2x

dt21

2K

m

dx

dt5 0. ~32!

The solution of this equation is

x~t! 5 Sr01v0Tn

2TrD2 v0

2TrexpS2

t

2TrDFTn

224Tr

2

2Trvsin~vt!1Tn

2 cos~vt!G, ~33!

where

v 5 A 2

Tn22

1

4Tr2, Tn 5 Am

K, Tr 5

R

K, ~34!

Tn is the time scale associated with the natural frequency of the spring-mass system, andTr is the relaxation time or the time scale associated with energy dissipation in thesystem. WhenTr < Tn /(2A2), v is imaginary, and the distance between two particles



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decreases monotonically. Otherwise, rebound is complete after timeTc when the distancebetween the particles becomesr 0 again. From Eq.~33!, we findTc satisfies

S124Tr

2

Tn2 D sin~vTc!

2vTr1cos~vTc! 5 expS Tc

2TrD. ~35!

The result of the numerical solution of Eq.~35! is shown in Fig. 1. The restitutioncoefficiente is the ratio of rebound velocity to incoming velocity

e 5 expS2Tc

2TrDFcos~vTc!1

Tn

A8Tr22Tn

2sin~vTc!G. ~36!

The relation between the coefficient of restitution and the time ratioTn /Tr is shown inFig. 2. In this simple model of particle interaction, the restitution coefficient depends onthe ratioTn /Tr only and is independent of the relative incoming velocity of the particles.This qualitative result is shown to be true for polymer spheres@Aidanpaa, Shen, andGupta~1996!# but is not true for metal particles@Goldsmith~1960!#. When the time ratioTn /Tr 5 AmK/R is greater than 0.542, the rebound time becomes infinite as shown inFig. 1. The two particles stick to each other and the corresponding restitution coefficientvanishes.

IV. TRANSPORT EQUATION FOR COLLISIONAL STRESS

In Sec. II, the collisional stress is expressed in terms of the microscopic forces be-tween contacting particles. In the limit of a small particle interaction time and binarycollisions, this stress can be modeled by kinetic theory. For a dense granular system, it isnecessary to consider the effect of finite particle interaction time and dynamics duringcontact. In order to do so, we derive the evolution equation for the collisional stress fromthe transport theorem~7!.

Let

FIG. 1. Contact time in a binary collision.



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gija 5

1

2 (b Þ a

N

f~y~a!,y~b!,w~a!,w~b!C

N22;t !~y~b!2y~a!!. ~37!

Noting that

bDs 5 ng, ~38!

and using Eq.~7! with g(a), one finds

bDS]s

]t1w•“sD 1“•~ngw8! 5 n

dg

dt, ~39!

wherew8 5 w2w is the fluctuating component of the velocity. The second term in Eq.~39! can be written as

ngijwk8~x,t ! 5 E r jdrE f i~2!wk8P2~x,r ,w8,v;t !d3vd3w8, ~40!

wheref (2) is the averaged force conditional on specifying velocities and positions of twocontacting particles. The velocity of the particle atx is w and the velocity of the particleat x 1 r is w 1 v. In Eq.~40!, the pair distribution function has been written as a functionof relative velocityv and the relative position vectorr between the centers of particles.The second integral in Eq.~40! is proportional to the correlation between the velocity ofthe particle atx and the force acted by the particle atx 1 r . The force is stronglycorrelated to the relative velocityv between the particles. When two particles are close toeach other in a dense system, it is expected that their velocitiesw andw 1 v are stronglycorrelated. In other words, the correlation betweenw andw 1 v is close to 1. Since thecorrelation of velocityw with itself is always 1, the correlation between velocityw andthe relative velocityv is weak, as is the correlation between thew ~or w8!, and the forcef between the two contacting particles. Indeed, this expected result is verified by direct

calculation ofngi j wk8 in the numerical simulation described in the next section. There-

FIG. 2. Restitution coefficient in a binary collision.



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fore, we will neglect this term from this point forward. Furthermore, in a simple shearflow, the divergence of this quantity vanishes identically due to the uniformity of theshear rate and constant particle concentration.

The last term in Eq.~39! can be written as

ndg

dt5

1

2N!E (

@a,b#

N F f~b!~w~b!2w~a!!1df~b!

dt~y~b!2y~a!!G

3d~x2y~a!!P~CN;t !dC

N. ~41!

For a system without tangential force on a particle surface, the force in Eq.~41! can bewritten as

f 5 f n, ~42!

wheren is the unit vector from the center of particlea to the center of particleb. Thetime derivativef can be expanded as

f 5d f

dtn1 f

dn

dt, ~43!

dn

dt5

1

r@~w~b!2w~a!!2~w~b!2w~a!!•nn#. ~44!

The derivative in the second term of Eq.~43! accounts for the change in forcef causedby the rotation of the line connecting the centers of the contacting particles. The rotationmentioned here happens in the finite duration of a particle contact. In kinetic theories ofgranular flows, such rotation of the force during a particle collision is neglected as is thefiniteness of collision time. As we shall see latter, the contribution of this rotationemerges as the Jaumann time derivative of the collisional stress.

The time derivative off in Eq. ~43! can be calculated by using Eqs.~30! and ~31!

df

dt5 2

f

Tr1Kv•n2 f v•nd~r 02r !, ~45!

where Tr 5 R/K is the relaxation time associated with energy dissipation at themicroscopic level.

The relative velocity can be written as

w~b!2w~a! 5 w~y~b!!2w~y~a!!1q 5 “w~y~a!!•~rn!1q, ~46!

wherer 5 u y(b) 2 y(a)u is the distance between the two particle centers, and velocityq 5 w8(2) 2 w8(1). Using Eqs.~42!–~46! in the right-hand side of Eq.~41!, andsubstituting into Eq.~39!, one finds

bD

ds

dt2bD@~“w!•s1s•~“w!T#1

bDs

Tr5 A, ~47!

where



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A 51

2N!E (

@a,b#~Krvnnn1fq1qf2 f q•nnn!d~y~a!2x!P~C

N;t !dCN

21

2N!e :E (

@a,b#f rnnnnd~y~a!2x!P~C

N;t !dCN

21

2N!E d~r 02r !rnnE (

@a,b#f vnd~y~a!2x!P~C

N;t !dCN. ~48!

Note that we have the upper convective time derivative of the stress tensor in the left-hand side of Eq.~47!. This is one of the objective time derivatives often encountered inrheology and is often introduced to establish objectivity of a constitutive relation. Here,this time derivative is obtained directly from the analytical derivation. The physicalorigin of this time derivative is the rotation and deformation of the line connecting thecenters of the contacting particles. The last term in Eq.~48! represents contributions fromthe formation and destruction of interacting particle pairs.

In order to close the transport equation~47! for the collisional stress, we now restrictourselves to the region of dense and slow granular flows. That is,bD is greater than therandom loose packing volume fraction about 0.52 andueu ! 1/Tr , where e is rate ofstrain of the mean flow field defined by

e 5 12@“w1~“w!T#. ~49!

In such systems, the particle contact time is much longer than the binary collision time,and the relation between stress and rate of strain differs from kinetic theory. Indeed, asthe numerical simulations described in the next section illustrate, in a dense system insimple shear flow, the particle contact time is about two orders larger than the binarycollision time.

We note thatA is a frame indifferent quantity. Therefore, it can only depend on theframe indifferent quantities of the flow field. For a given system, these quantities can bechosen as the rate of straine, and the Jaumann time derivative of the strain rate

D e

D t5

] e

]t1w•“ e2V• e1 e•V, ~50!

whereV is the vorticity tensor of the averaged velocity field,

V 5 12@“w2~“w!T#. ~51!

We assume that quantityA depends linearly on the Jaumann derivative with a scalecoefficientc1 . Furthermore, we assumeA is an isotropic function of the strain ratee.With these assumptions, representation theorems@Gurtin ~1981!# of an isotropic functionlead to a unique form of the closure relation:

A 5 c1

D e

D t1c2e1c3e• e1c4e : eI , ~52!

whereci , (i 5 1,2,3,4) are functions of the invariants of the strain rate tensor, volumefraction of particles, and material properties of the particles. In principle,A can alsodepend on spatial gradients of the rate of strain, but these are of order (a/L) e and areneglected for flows in which the change of the strain rate is unimportant at the particlescale~i.e. u“ eu/ueu ! 1/a).



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From substitution of Eq.~52! into Eq. ~47!, we have

ds

dt2@~“w!•s1s•~“w!T#1

s

Tr5

AmK

aFC1S de

dt2V• e1 e•VD 1C2AK

me

1C3e• e1C4e : eI G , ~53!

where now the constantsCi are dimensionless and are functions of the nondimensionalgroups:

Ci 5 CiSbD ,AmK

R,uTr eu D , ~ i 5 1,2,3,4!, ~54!

and I is the identity tensor. Here, we denote the invariants of the dimensionless tensorTr e by uTr eu.

Some possible variants of Eq.~53! need to be explored. For instance, one might followGordon and Schowalter~1972! and Phan-Thien and Tanner~1977! to rewriteq in A asje•r1q8. This would introduce a term proportional toe•s1s•e in the left side of Eq.~53!. However, with our assumption thatueu ! 1/Tr , explicit inclusion of this term isunnecessary. One can also consider other stress relaxation processes by adding terms likes/T to the right of Eq.~52!. Possible candidates for other relaxation time scales are theparticle contact time and the time scale associated with the rate of strain. However, in adense system, the contact time is of orderueu21, and additional stress relaxation termsare also unnecessary. Further, by the same argument, the terme•s1s•e contained in theupper convected time derivative of stresss can be ignored and Eq.~53! becomes

ds

dt2V•s1s•V1

s

Tr5

AmK

aFC1S de

dt2V• e1 e•VD 1C2AK

me

1C3e• e1C4e : eI G . ~55!

The constitutive model proposed in Eq.~55! is a viscoelastic model. This model for denseand slow granular flow contains four constants that must be determined.

V. NUMERICAL SIMULATION

In order to study the behavior of the closure relation, and to understand how granularstress departs from prediction of kinetic theory with increasing particle volume fraction,direct numerical simulations are necessary. Many numerical simulations of granularflows with various flow conditions have been carried out, but recently, more simulationsare based on a ‘‘soft-sphere’’ model, in which the dynamics of particle collisions andmultiparticle interactions are resolved. However, the choice of interparticle force modeland flow geometry distinguishes one simulation from another. For example, Walton andBraun ~1986! used a partially latching spring in their simulation with a reduced springconstant during unloading to account for energy dissipation. Potapov and Campbell~1996! used a parallel connection of spring and dashpot as the microscopic force modelto examine granular flows in a hopper. For our purposes, as mentioned earlier, a serialconnection of spring and dashpot allows us to consider particles lubricated by viscousresin.



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The numerical simulations performed in this study are simple shear flows, in whichthe vorticity magnitude is of the same order as the magnitude of the strain rate, which issmall compared with 1/Tr . Therefore, at steady state, the Jaumann derivative of thestress can be neglected, and the proposed constitutive relation~55! becomes

s 5R

aAm

KFC1~ e•V2V• e!1C2AK

me1C3e• e1C4e : eI G . ~56!

From this form, in principle, the four constantsCi , (i 5 1,2,3,4) can be determined fromthe four components of the stress tensors. Unfortunately, in the dense granular flowsimulated, the normal stress is about one or two orders smaller than the shear stress.Furthermore, because we simulate a rather small number~108! of particles, fluctuationsin the normal stress are rather large, and we were unable to obtain accurate values ofcoefficientC1 , C3 , and C4 . More efficient algorithms capable of handling about tenthousand or more particles are currently under construction. Nevertheless, we are able todemonstrate a transition in the stress–strain relation from kinetic theory predictions rel-evant to dilute systems to that appropriate for dense flows. We suspect that this transitionis similar to a phase transition in a molecular system.

In order to simulate simple shear flow in infinite space, we employ periodic boundaryconditions. The generation of the initial particle configuration with specified particleconcentration is described in Sanganiet al. ~1991!. Particles are first placed in a funda-mental cubic cell in a face-centered cubic~fcc! array. Then, each particle is subjected to30 000–50 000 random motions. The mean distance of each movement is proportional tothe mean-free path at the specified particle concentration. After each movement, wecheck for overlap, and only those movements that do not cause an overlap are accepted.Therefore, there are no initial particle interactions in the system. Particle interactions aredeveloped, as a consequence, by the evolution of the shear flow. Thus, we avoid intro-ducing artificial effects of the initial particle interaction.

The initial particle velocity is set to zero iny andz directions, and the initial velocityin the x direction is specified as proportional to itsz coordinate according to the shearrateg 5 2exz. Again, to avoid artificial effects on the initial particle velocity distribu-tion, no initial velocity fluctuations are given. The randomness of the particle velocity,related to the streaming part of the granular stress, is developed as a consequence ofinteractions among particles. With the initial particle positions and velocities specified,the entire space is filled with images of the fundamental cell. At a later time, the particlepositions in the fundamental cell are determined by solving the simultaneous equations ofmotion. For a particle in an image cell, the velocity is given by the sum of the velocity ofthe corresponding particle in the fundamental cell and a velocity offset equal to theproduct of the shear rate and their distance in thez direction. The position of any imageor fundamental particle is calculated by time integration of its velocity.

In these simulations, all particles have the same size. The radius of the particles ischosen to be one, and a length is nondimensionalized by the particle radius. The mass ofthe system is nondimensionalized by a particle mass. The time scale is determined by thenatural periodtN of the spring-mass system. One simulation time unit equals 200tN .The equations of motion are solved with a second-order explicit method similar to theone used by Walton and Braun~1986!. In order to resolve the dynamics of multiparticleinteractions, the time step used in this simulation is restricted to about 1/100–1/50 of abinary collision time, or about 0.01tN .

In Figures 3–10, we compare the four nonzero total stress components with kinetictheory and Walton and Braun’s numerical results. For low particle concentrations, kinetic



19:24:59

theory agrees reasonably well with both our results and those of Walton and Braun~1986!. At higher particle concentrations, Walton and Braun’s simulations predict highernormal stresses than kinetic theory while our results indicate lower normal stresses. Forthe shear stress, both our and Walton and Braun’s simulations yield a result larger thanthat of kinetic theory, though our shear stresses are usually larger than Walton and

FIG. 3. Dimensionless normal stresssxx (e 5 0.8). The solid line is the result of kinetic theory, Lun~1991!.The solid dots, squares, and triangles are our numerical results for shear rates of 0.25, 0.50, and 0.75, respec-tively.

FIG. 4. Dimensionless normal stresssyy (e 5 0.8). The solid line is the result of kinetic theory, Lun~1991!.The solid dots, squares, and triangles are our numerical results for shear rates of 0.25, 0.50, and 0.75, respec-tively.



19:24:59

Braun’s. The good agreement among both numerical simulations and kinetic theory atlow particle concentrations indicates that, in that region, the detailed particle interactionmodel is unimportant. The effect of interaction between particles can be characterized bya binary collision with a coefficient of restitution, provided that the restitution coefficient

FIG. 5. Dimensionless normal stressszz (e 5 0.8). The solid line is the result of kinetic theory, Lun~1991!.The solid dots, squares, and triangles are our numerical results for shear rates of 0.25, 0.50, and 0.75, respec-tively. The hollow circles, squares, and triangles are results of Walton and Braun~1986! with shear rates of10.0, 5.0, and 1.0, respectively.

FIG. 6. Dimensionless shear stresssxz (e 5 0.8). The solid line is the result of kinetic theory, Lun~1991!.The solid dots, squares, and triangles are our numerical results for shear rates of 0.25, 0.50, and 0.75, respec-tively. The hollow circles, squares, and triangles are results of Walton and Braun~1986! with shear rates of10.0, 5.0, and 1.0, respectively.



19:24:59

is sufficiently close to unity. As collisions become more inelastic, the details of particleinteractions gain importance because the ratio of particle contact time to mean-free flighttime increases. Consequently, the results depart from kinetic theory at lower particleconcentrations as shown in Figs. 7–10. In the high-density region (bD > 0.5), thedetailed dynamics of the particle interaction are important, and the different force models

FIG. 7. Dimensionless normal stresssxx (e 5 0.6). The solid line is the result of kinetic theory, Lun~1991!.The solid dots, squares, and triangles are our numerical results for shear rates of 0.25, 0.50, and 0.75, respec-tively.

FIG. 8. Dimensionless normal stresssyy (e 5 0.6). The solid line is the result of kinetic theory, Lun~1991!.The solid dots, squares and triangles are our numerical results for shear rates of 0.25, 0.50, and 0.75, respec-tively.



19:24:59

yield different stress behaviors. When the force between contacting particles is modeledas a serial connection of a linear spring and a dashpot, particles can easily stick togetherand interact for a period much longer than the binary collision time. Then, the particleDeborah number, defined as the ratio of the relaxation timeTr 5 R/K to the particle

FIG. 9. Dimensionless normal stressszz (e 5 0.6). The solid line is the result of kinetic theory, Lun~1991!.The solid dots, squares, and triangles are our numerical results for shear rates of 0.25, 0.50, and 0.75, respec-tively. The circles are results of Walton and Braun~1986! with a shear rate of 10.0.

FIG. 10. Dimensionless shear stresssxz (e 5 0.6). The solid line is the result of kinetic theory, Lun~1991!.The solid dots, squares, and triangles are our numerical results for shear rates of 0.25, 0.50, and 0.75, respec-tively. The circles are results of Walton and Braun~1986! with a shear rate of 10.0.



19:24:59

interaction time, is small, and the force between particles is largely determined by theviscosity of the resin coating on the particles. The granular system behaves more like aviscous fluid. Thus, the shear stress is much larger than the associated normal stresses.

Kinetic theories for granular flow predict the isotropy of the normal stresses, therefore,the solid curves in Figs. 3–5 and 7–9 are identical. However, anisotropy is observed inall direct numerical simulation results@Walton and Braun~1986!; Lun and Bent~1994!#,including the present simulation. We see from the figures that this anisotropy becomesmore pronounced in dense flows and for low particle restitution coefficients.

In a simple shear flow, the shear stress in Eq.~56! can be written assxz5 C2(R/a) exz. In a dilute case, this expression can be regarded as a definition for the

dimensionless effective viscosityC2 , although this expression is obtained for a densegranular system. Figure 11 shows the average value for coefficientC2 . In a simulation,because of fluctuations in positions and forces among the particles, the instantaneousvalues ofC2 also fluctuates. The values ofC2 shown in Fig. 11 and the stress in Figs.3–10 are averages over a sufficiently large number of time steps after a statisticallysteady state is reached. From bottom to top, the particle volume fractions are 0.232,0.347, 0.463, 0.521, 0.521, 0.579, and 0.695, respectively. At low particle concentrations(bD < 0.521),C2 is linear with shear rate, corresponding to a quadratic dependence ofshear stress on shear rate, in agreement with kinetic theories. At higher particle concen-trations (bD > 0.521), C2 is independent of the shear rate, corresponding to a lineardependence of stress on shear rate.

At low particle concentrations, the particle contact time is small compared to themean-free flight time. The stress is proportional to the product of the collision frequencyand the force during the collision. Both of them are proportional to the rate of strain,producing a stress that is quadratic in strain rate. For a dense granular system, particlesare consistently in contact and the collision frequency is saturated. For such prolonged

FIG. 11. Dimensionless effective viscosity vs shear rateg. From the bottom the particle volume fractionsbDare 0.232, 0.347, 0.463, 0.521, 0.521, 0.579, and 0.695. The two dashed lines forbD 5 0.521 correspondingto the two branches of theC2 value before and after the phase transition.



19:24:59

interaction time, the particle Deborah is small, and the force between particles is domi-nated by viscosity. If the averaged interaction force between a pair of particles is ex-panded in a Taylor series in shear rate, then the leading term in the force, and thus theshear stress, is proportional to the shear rate. Because of the symmetry of the simpleshear flow, the normal stress must be an even function of the shear rate. Therefore, thenormal stress must be quadratically dependent on the shear rate. This can also be seenfrom definition ~22! for the collisional stress. The leading term in the Taylor expansionfor the interaction force cancels through contributions from the upstream and downstreamsections of the particle surface. Indeed, the transition ofC2 occurs at random loosepacking densitybD ' 0.52, a minimum density at which every particle is in contactwith another@Shapiro and Probstein~1992!; Onoda and Liniger~1990!#. This is consis-tent with Figs. 3–10, namely, that the ratio of the shear stress to normal stress is quitelarge for the dense cases (bD . 0.521). At these volume fractions, the shear stress isproportional to shear rate while the normal stress is quadratic in shear rate. Therefore, thestress ratio is inversely proportional to shear rate and can be large for small shear rates.Again, in our force model, we assume that there is always a fluid film between particles,and solid–solid contact is neglected. Therefore, our current results will be most relevantin cases of a small particle Stokes number defined byrDa2g/m, wherem is the fluidviscosity. To consider solid–solid contact, we have performed similar simulations usinga force model of a parallel connection of a spring and a dashpot between particles. Thestress ratio is then between 0.1 and 1.0 and within the range commonly observed byexperiments of flows of large granular particles.

This transition ofC2 is similar to a phase transition in a molecular system. As shownin Fig. 11, at volume fractionbD 5 0.521, there are two branches forC2 . In oursimulations, we start from configurations in which particles are not in contact. The con-tact between particles is developed by the imposed shear motion. At the beginning of oursimulation, the shear stress, and thereforeC2 , stays at the lower branch for a certaintime, then moves to the higher branch. With increasing shear rate, the time during whichthe system remains on the lower branch decreases and eventually disappears at the shearrate where the line of the lower branch stops. A typical transition is shown in Fig. 12, forwhich the shear rate is 0.15. Figure 12 also shows fluctuations ofC2 . Each point in Fig.12 is an average ofC2 over 10 000 time steps corresponding to 0.5 time units. For thecase shown in Fig. 11, the lower branch value ofC2 is averaged over the duration ofdimensionless time between 40 and 150, and the upper branch value is obtained byaveraging over dimensionless time greater than 600. This transition only occurs when theimposed shear rate is greater than a certain value, below which~e.g., atg 5 0.1!, thetransition is not observed. At a small shear rate, the mean-free flight time is large, and aparticle pair completes their collision before a third particle approaches them. In thiscase, collisions are essentially binary, and kinetic theory applies. At a large shear rate,however, the mean-free flight time becomes short, and a pair of particles does not haveenough time to complete their collision before another one impinges them. The collisionis not binary and kinetic theory fails.

A steady-state solution can only be obtained for shear rates less than a certain criticalvalue that depends on the volume fraction and the spring and dashpot properties. Whenthe shear rate becomes greater than the critical value, the system becomes unstable, andformation of a shear band is observed. The physical reason for shear band formation isthe destabilizing effect of collisions. Assume that an initial flaw with a somewhat lowerparticle concentration is generated in a direction parallel to the flow. This is a weak bandwhere shear localizes. The relative motion of the particles in the band is greater than thatoutside, and the shear motion in the band tends to knock particles inside the band out-



19:24:59

ward. Thus, the particles with their centers at each side of the band tend to move furtherinto their respective sides, which widens the flaw. On the other hand, random fluctuationstend to diffuse particles into the band. The competition between these two mechanismsdetermines the formation of shear band. Indeed, in a system with a small particle volumefraction, particle fluctuations, corresponding to the streaming part of the granular stress,are larger than that in a dense granular system. As such, the tendency for band formationis reduced, and large shear rates can be sustained without system instability.

VI. CONCLUSIONS

In this paper we have developed a method to study dense granular flows. Finiteparticle interaction time and multiparticle interactions are considered. The evolutionequation of the collisional stress tensor is derived and leads to a viscoelastic model. TheJaumann time derivative of the collisional stress is obtained from the derivation directlywithout appealing to objectivity arguments. This derivative accounts for the rotation ofthe stress caused by the rotation of lines connecting the centers of the particles duringtheir contact. This causes the anisotropy of the normal stresses and is not included in akinetic theory.

Direct numerical simulations of granular systems undergoing simple shear have beenperformed to study the effects of finite particle collision time and the model. The forcebetween particles is modeled by a serial connection of a linear spring and a dashpotbetween contacting particles. In a dilute system, the shear stress is proportional to theproduct of the force during a particle collision and the frequency of collision. Thus, theshear stress is quadratic to the shear rate. In a dense system, particles are always incontact, and the shear stress depends only on the force between particles. The transitionbetween these regimes is similar to a phase transition in a molecular system and occurs atthe random loose packing particle volume fractionbD ' 0.52.

Depending on the competition between the destabilizing effect of collisional stress andthe stabilizing effect of diffusion caused by the fluctuations of particle velocities, the

FIG. 12. Transition of the dimensionless viscosityC2 at bD 5 0.521, and shear rate 0.15.



19:24:59

granular system can become unstable. Formation of shear bands is observed when theshear rate exceeds a certain critical value.

ACKNOWLEDGMENTS

One of the authors~D.Z.Z.! wishes to thank Professor Prosperetti at Johns HopkinsUniversity and Professor James Jenkins at Cornell University for many interesting andconstructive conversations about this work. One of the authors~D.Z.Z.! also wishes tothank Professor D. D. Joseph for inviting him to participate in an IMA workshop held inthe University of Minnesota, Spring, 1996, where he met Professor James Jenkins. Thetime integration scheme is derived from a code provided to the authors by Dr. Walton atLawrence Livermore National Laboratory. The authors wish to acknowledge his gener-osity. The authors are grateful to Dr. Francis H. Harlow for many interesting conversa-tions and encouragement to complete this work. This work was performed under theauspices of the United State Department of Energy.

APPENDIX

The product of the terms on the right-hand sides of Eqs.~17! and ~18!, can be repre-sented as a function of the position of particleb and the configuration relative to it,

~fP!~y~b!,y~a!,w~b!,w~a!,C N22;t ! 5 ~ fP!R~y~b!,RN22;t !, ~A1!

whereR(N21) is the configuration relative toy(b) defined as

R~N21! 5 $r ~a!,w~b!,w~a!,r ~g!,w~g!;H%, g Þ b, 1 < g < N, ~A2!

r ~g! 5 y~g!2y~b!; g Þ b. ~A3!

In writing y(b)1r (a)2r (a) for y(b) in the right-hand side of Eq.~A1!, we expandfunction (fP) by Rolle’s mean value theorem with respect to the first argumenty(b).Then, the product of the left-hand sides of Eqs.~17! and ~18! can be written

~fP!~y~a!,y~b!,w~a!,w~b!,C N22;t ! 5 2~ fP!R~y~b!1r ~a!,RN21;t !

1r ~a!•“y~b!~ fP!R~y~b!1r ~a!1h,RN21;t !,

~A4!

where h 5 h(y(a),r (a),RN22;t) and uhu , ur (a)u. Upon substitution of this into Eq.~15! and the use ofy(a) 5 y(b)1r (a), one finds

nfc 5 21

N! (@a,b#

E d3y~b!dRN21d~x2y~b!2r ~a!!~ fP!R~y~b!1r ~a!1h,RN21;t !

11

N! (@a,b#

E d3y~b!dRN21d~x2y~b!2r ~a!!

3r ~a!•“y~b!~ fP!R~y~b!1r ~a!,RN21;t !r ~a!. ~A5!

Since variabler (a) is independent ofy(b) in Eq. ~A5!, by differential properties of thedfunction, the integrand in the last integral can be written as

d~x2y~b!2r ~a!!r ~a!•“y~b!~ fP!R 5 2~ fP!Rr ~a!

•“y~b!d~x2y~b!2r ~a!!

5 ~ fP!Rr ~a!•“xd~x2y~b!2r ~a!! ~A6!

Upon substitution into Eq.~A6!, we find



19:24:59

nfc 5 21

N! (@a,b#

E d3y~b!dRN21d~x2z~a!!~ fP!~D

N;t !

11

N! (@a,b#

“x•E d3y~b!dRN21d~x2z~a!!~ fP!~D

N1h;t !r ~a!, ~A7!

where we have used Eq.~A1! again to substitute back (fP) by changing integrationvariables as follows:

DN 5 $za,va,zb,vb,zg,vg;H% 1 < g < N, g Þ a,b,

~A8!

DN1h 5 $za1h,va,zb1h,vb,zg1h,vg;H% 1 < g < N, g Þ a,b,

z~a! 5 y~b!1r ~a!, v~a! 5 w~b!

z~b! 5 y~b!12r ~b!, v~b! 5 w~a! ~A9!

z~g! 5 y~b!1r ~a!1r ~g!, v~g! 5 w~g!, g Þ a,b.

We note thatd3y(b)dRN21 5 dDN, and that the integral is independent of theintegration variables, thus Eq.~A5! can be written as integrals with integration variableC N,

nfc 5 21

N! (@a,b#

E dCNd~x2y~a!!~ fP!~y~a!,y~b!,w~a!,w~b!,C N22;t !

12“•~bDs!, ~A10!

wheres is the collisional stress tensor

bDs 51

2N! (@a,b#

N E dCNd~x2y~a!!f~C

N1h;t !~y~b!2y~a!!P~CN1h;t !,

~A11!

with C N1h defined similar toDN1h

CN1h 5 $y~1!1h,w~1!,y~2!1h,w~2!,...,y~N!1h,w~N!%. ~A12!

Comparison of this with Eq.~15!, leads to

nfc 5 2nfc12“•~bDs!, ~A13!

or Eq. ~21! in Sec. II.In this proof, the translation in Eq.~A4! is performed by keeping the relative positions

between particles unchanged, and whole configuration is translated rigidly by an amounty(a)2y(b), order of 2a, small in comparison with the macroscopic length scale. Only inthis sense is it legitimate to perform the Taylor expansion~20!. In this case, to a first-order approximation,h in Eq. ~A11! can be neglected to obtain Eq.~22!.

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a viscoelastic model for dense granular flows

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