Fine structures in sheared granular flows

PHYSICAL REVIEW E 66, 021303 ~2002!

Fine structures in sheared granular flows

William Polashenski, Jr.Lomic, Inc., 2171 Sandy Drive, Suite 130, State College, Pennsylvania 16803

Piroz Zamankhan, Simo Ma¨kiharju, and Parsa ZamankhanLaboratory of Computational Fluid Dynamics, Lappeenranta University of Technology, Fin-53851, Lappeenranta, Finland

~Received 28 November 2001; revised manuscript received 6 June 2002; published 23 August 2002!

Computer simulations were used to investigate shear flows of large numbers of viscoelastic, monosized,spherical particles in unbounded and bounded systems with solid fractions ranging from 0.16 to 0.59. Amodified hard-sphere model with inelastic, instantaneous particle interactions was found to replicate someresults predicted by kinetic theory in an unbounded shear flow at low and moderate solids fractions. This modelwas found to predict features such as particle lateral diffusive motion even for systems at solid fractions as highas 0.56. However, for higher solid fractions where phenomena such as jamming could occur, a particledynamics model accounting for particle contacts of finite duration has been developed, in which the viscoelas-tic behavior of the particles was represented using a nonlinear Hertzian model. The nonlinear viscoelasticmodel was found to give more reasonable predictions for cluster formation than previously reported linearmodels, especially when accounting for surface friction in the model. However, neither frictionless nor fric-tional particle models could predict particle ordering in unbounded flows. As such, simulations were performedfor bounded systems using both the modified hard-sphere model and the nonlinear particle dynamic model. Fora bounded shear flow, particle ordering could be predicted by the hard-sphere model even in the absence ofboth particle friction and gravity, with the local solid fraction and wall separation distance governing the flowstability. For these conditions chain formation was found to be quite likely in the disordered layers forfrictional particles. The interesting stick-slip dynamics could be clearly observed in the normal stress signal atthe bottom wall. Interpretations were proposed for the complex processes observed, which could lay thefoundation for further investigations in sheared dense granular systems.

DOI: 10.1103/PhysRevE.66.021303 PACS number~s!: 45.70.Mg, 02.70.Ns, 64.70.Dv

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I. INTRODUCTION

The progression of a nonequilibrium, dense, disordergranular assembly toward its steady state under shearingtion appears to be highly complicated. Recently, Cates, Wmer, Bouchaud, and Claudin@1# suggested that in a constanvolume shearing flow of a dense granular material inCouette geometry, jamming could occur indicating a trantion to the glass state which has an amorphous configuraThe dilatancy, which is the tendency of dense granular meto expand upon shearing@2#, could be viewed as theconstant-load counterpart of jamming. Jamming appearoccur due to the formation of anisotropic, nonstraight fochains@3,4#, which are linear strings of nearly rigid particlein contact that can support the shear stress along the cpressional direction indefinitely@1#.

In contrast to the metastable character of the jammprocess mentioned above, evidence has been reporte@5#which indicates that the initially disordered configurationshaken granular monolayers may exhibit a stable crystaorder. Moreover, experimental data@6#, such as that shown inFig. 1, suggest that an ordered phase may also be observrapid shear flows of smooth granular materials in a twdimensional planar Couette geometry. Thus, in additiontheir importance discussed in@7#, dense granular flows maserve as experimental models for investigating generaltures of the transition to an ordered state from a disordeconfiguration. This emphasizes the importance of being ato predict whether ordering or jamming would occur in

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sheared disordered granular assembly. It is worth mentionthat the surface properties had a strong effect on the expmental results of Elliot, Ahmadi, and Kvasnak@6#, since theydid not observe an ordered phase for shearing flow of rogranular materials. This observation highlights the imptance of the nature of the interactions between grainsgranular systems.

In a sheared, dry, granular system, gradients in the mflow induce interparticle collisions, which generate fluctutions in the local mean motion of the grains. The collisioinvolve a nearly elastic deformation of grains whose veloties change due to forces resulting from the deformation. Tcollision time, which is of the order of 1024 s or less, con-sists of a compression time during which deformation occand a restitution time during which the shape is restorSince kinetic energy is degraded into heat due to the inelanature of deformations, external energy should continuoube provided to granular systems to prevent the system fcollapsing into the solid state, which is characterized

FIG. 1. Snapshot of smooth steel balls from an experiment cducted in a two-dimensional Couette geometry. Note the preseof ordering.

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POLASHENSKI, ZAMANKHAN, MAKIHARJU, AND ZAMANKHAN PHYSICAL REVIEW E 66, 021303 ~2002!

strong fluctuations of the forces on individual grains.In the presence of contact friction between the grai

which induces fluctuations in the local mean rotational mtion of the grains, an additional dissipation of kinetic enerinto heat would be expected. Three types of friction forcmay be distinguished, which are due to rolling, sliding, aadhesion~sticking! of grains. Sliding friction forces and adhesive forces could accelerate or decelerate the rotatimotion of the grains, while rolling friction forces alwaydecelerates the rotational motion. However, the physicalevance of the frictional contact between the particles rema topic of controversy due to experimental evidence@8# thatsuggests that the dynamics of grains may be dominatedcollisions rather than sliding contacts even in slow deflows.

Computer simulations may play a valuable role in dscribing the macroscopic flow behavior resulting from taforementioned phenomena. Attempts have been madsimulate dense granular shear flows by modeling the gras inelastic rough spheres. One well-known methodsimulating granular particles is the particle dynamic meth@2,9–10#. This method assumes that the grains behaveviscoelastic particles that interpenetrate during a collisiresulting in the generation of restoring forces that canexpressed by Young’s moduli of the order of 107 kN/m2. Theshort-range interaction of particles, therefore, may resultlarge gradient of the interaction force, which implies that tinteraction force between the grains in contact should beculated about 1000 times during a collision to provide acrate results. Presently a three-dimensional simulation wilarge number of particles would require significant computional intensity. As such, a question arises as to whetheless computationally intensive model could predict the bafeatures of sheared granular flows.

Another method for the simulation of granular assembis the modified hard-sphere model of Alder and Wainwrig@11#, which has been frequently used@12,13#. In this ap-proach, collisions are assumed to be instantaneous anddissipation is introduced through a coefficient of restitutioeand a surface friction coefficientm and time advances fromone collision to the next, rather than with an imposed tistep as used in the particle dynamic methods. Althoughhard-sphere model appears to be highly idealized, it canproduce the major features observed in the experiments@13#.

In the present study, the hard-sphere model will be furttested against the particle dynamic model to find out hclosely the model can reproduce the interesting behasuch as jamming and ordering in sheared granular flowhigh densities. To this end two quantities are introducwhich may be suitable for monitoring changes in the initiadisordered state due to shearing motion. One indicator isquantityQ6 @14#, which characterizes the average bond oentations in a system of particles.Q6 is a rotationally invari-ant combination of the bond order parametersQlm , definedasQlm5Ylm@u(r ),f(r )#. In order to evaluate the bond ordeparameterQ6 , particles residing within a shell of 1.4s sur-rounding a given particle are considered as its near nebors. Heres is the particle diameter andYlm@u(r ),f(r )#represents spherical harmonics associated with a bond w

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midpoint is atr , Nb is the number of bonds,u(r ) andf(r )are the polar angles of the bond measured with respecfixed external Cartesian coordinates. An averaged valueQlm over a suitable set of bonds in the sample is given

Qlm5(1/Nb)(bondsQlm(r ). Therefore, the quantityQ6 is de-

fined asQ65@(4p/13)(m5266 uQlmu2#1/2.

A second indicator is the mean-square displacement ofparticles, which is represented as a function of time, defiby ^Dr 2(t)&5N21( i 51

N ^@r i(t)2r i(0)#2&, wherer i(t) is theparticle position,N represents the number of particles in tcomputational box, and the angular brackets, which denthe ensemble average, may be generated by using manyorigins.

Rintoul and Torquato@14# showed that the first indicatofor a disordered configuration approaches zero at the ratNb

21/2 with the expected width of fluctuations of 0.196Nb21/2.

For example, the value ofQ6 for a system with cuboctahedral symmetry is about 0.6@15#, which makes this indicatoruseful for distinguishing the glass state, for whichQ6

'1/ANb, from an ordered system, since for both of thestates the long-time value of^Dr 2(t)& @16# remains nearlyconstant@17#.

In his experimental study of mixing in granular flowBridgwater@18# suggested that the motion of spherical grain a simple shear cell is much like a random walk withoupreferred direction of movement. This observation suppothe use of the Einstein relationship@19# for the estimation ofthe diffusive motion, in an average sense, in these syste

Note that in the wall region large variations in velocimay occur over distances of the order of a few particleameters, which leads to particle migration in the directinormal to the plane of shear, where the particles moveregions of lower stress. As a result of this motion, whichmore pronounced in moderately dense granular sheaflows, nonuniform spatial distributions of particle concentrtion and granular temperature develop in the direction nmal to the mean flow, which results in ordinary diffusioalong the concentration gradients. In the presence of a stlateral granular temperature gradient, an extra diffusive flof particles may be observed. Under conditions whereflux becomes significant, the application of simplified moels could lead to different interpretations of measuremresults in roughly similar apparatus.

For example, in systems where gravity does not affectlateral particle diffusive motion, at nearly the same sofractions, a difference of four orders of magnitude exists@20#between the values of the dimensionless transverse diffucoefficient, defined asD* 5D/s2«, measured by Menon anDurian @8#, and the values calculated from the results of Ntarajan, Hunt, and Taylor@21#. Here,s represents the diameter of the particles,« is the shear rate, andD is the self-diffusion coefficient. It is unclear whether this differencedue to the inaccuracy of one~or both! of the experimentalprocedures~such as solid fraction measurements!, or to someother reason, such as misinterpretation of the results usimplified theories.

Analyzing the observations of Menon and Durian, Deniston and Li@22# pointed out that the value of^Dr 2(t)& at

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FINE STRUCTURES IN SHEARED GRANULAR FLOWS PHYSICAL REVIEW E66, 021303 ~2002!

long times could be considered as being almost constindicating that diffusion was very limited. This could be duto the particles being trapped in the almost permanent caformed by their neighbors. However, at long times the incator^Dr 2(t)& for a granular assembly in the gravity driveflow of Natarajan, Hunt, and Taylor@21# increases linearlywith time. This observation suggests a fluidlike behavior imonodisperse granular flow at high average solid fractioSurprisingly, they suggested that the average solid fractiothe system was in the range of 0.55,Fs,0.7, though a sys-tem with the solids’ fractionFs.0.6 can hardly be considered as a fluid. In this light, simulations of dense shear floof granular material may be of value to interpret their expemental observations and could even suggest a more reaable value for solids fraction in@21# based on the measurevalues ofD* .

Much of the present knowledge about particle diffusivin shearing flows of a granular assembly has been derfrom simulations. These include the calculation of the sediffusion coefficient in a bounded system by Savage and@23# and computer simulations of the anisotropic diffusitensor for an unbounded system by Campbell@12#. Using akinetic theory analysis, Savage and Dai@23# derived an ex-pression for the self-diffusion coefficient, which reducesthe classical Chapman-Enskog result@24# in the limit of per-fectly elastic particles. Apparently, the theoretical sediffusion coefficients were in excellent agreement with thoobtained from the simulations at low densities. However,applicability of the theoretical expression was questionedhigher densities, since the numerical result for self-diffusexceeded the theoretical value. This discrepancy mighattributed to an enhancement of the velocity correlationsto the excitation of slowly decaying collective motions in tflowing granular assembly. Note that Savage and Dai@23#did not consider particle diffusivity at high solid fractions,which either ordering or a glass transition might occur. Hoever, Campbell@12# reported the absence of particle diffusivmotion at a solid fraction of 0.56, although he did not discuwhether the granular assembly had undergone crystallizaor a transition to the glass state had occurred in his sysIn light of the above, a general study of the diffusion prcesses in granular materials appears to be lacking.

One of the aims of this paper is to set forth a preliminaframework for the analysis of shear-induced diffusion inbounded granular material at high shear rates. To this encomprehensive model was developed for the diffusive pcesses involved in a rapid shear flow using the recentlyveloped revised Enskog theory@25# to obtain formal expres-sions for the particle diffusion coefficient, particle thermdiffusion coefficient@26#, and coefficient of mobility.

To obtain an increased understanding of the aforemtioned mechanisms for the diffusive displacement of the pticles in bounded granular materials, it would be quite useto compare the theoretical predictions with the simulatresults. Another purpose of this study, therefore, is to invtigate further the issues such as ordering and jamming ingranular assembly. It has been reported that the parroughness, which induces fluctuations in rotational motihas a significant effect on the formation of force chains@1#.

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To address this issue, the simulations were divided intomain groups. In one set of simulations~rough particle simu-lations! the frictional contact force between the particles goerns the flow dynamics, and in the other group~smooth par-ticle simulations! the dynamics of grains is dominated binelastic collisions.

Starting from a low solid fraction, namely,Fs'0.16,where the behavior of the system appears to be less comthe solid fraction was increased carefully up toFs'0.58,with the goal of examining the conditions for which the fluwould become ordered and the conditions for which jaming would occur. For low solid fractions, there are a velarge number of disordered states in which spheres canrange themselves, which causes the fluid to be thermonamically stable. By contrast, for a dense system theremuch fewer allowable disordered configurations. Therefoat short times the instabilities could lead to the formationa nucleation point, which at longer times results in andered region surrounded by low density regions. As poinout by Johnson and Jackson@27#, not all the granular material is sheared uniformly in such a system. The orderedgion withstands shear, while in the low density regions,shearing motion is enhanced, reducing the tendency of jming of the system. Note that jamming can be viewed aspontaneous transition to a metastable disordered configtion that withstands shear.

Although the picture described above is quite detailed,actual behavior of sheared granular materials can be emore complex than has been indicated above. For examin the experiments of Miller, O’Hern, and Behringer@4#, theforce fluctuations were measured at the wall for a degranular flow in a Couette geometry. They observedstrongly non-Gaussian probability distribution in the time sries for the normal stress, which exhibits fat tails indicatiof an underlying complex dynamics with long correlatiotimes. Interestingly, a shell model of magnetohydrodynamturbulence produces a time series of energy dissipationdefined in @28#, with the intermittent spikes of dissipatioquite similar to the intermittent spikes in the data of MilleO’Hern, and Behringer@4#.

Another example is the observation of the stick-slip mtion only below a critical shear rate in the shearing of cohsionless glass spheres in an annular, parallel-plate sheaby Hanes and Inman@29#. In these systems a constanvolume shear flow may not be achieved due to the tendeof dense granular media to expand upon shearing. Analyzthe shearing flow of frictional disks confined between sowalls under a constant load, Thompson and Grest@2# pointedout that in the presence of gravity the system is unstablshear rates less than a critical shear rate, exhibiting a sslip motion consistent with the experimental observatioThis oscillatory motion could be due to periodic dilatantransition and gravitational compactification. It is remarkabthat a similar stick-slip dynamics has also been observethe shearing motion of a thin microscopic fluid film confinebetween solid walls where the role of gravity may notsignificant @30#. The similarity could indicate analogies between the statistical physics of these systems, suggestinga more comprehensive understanding of the flow dynam

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may be obtained by comparing the results of the sheaflow of a granular material in the absence and in the preseof the gravity. Such information is invaluable in the deveopment of advanced theories capable of describing flowhigh solid fractions, which are of real significance in facitating the design of practical granular flow systems.

The organization of the present paper is as follows.Sec. II, a description is given of the computer simulatimethods and the physical relevance of the simulations. Stion III presents the simulation results for unbounded gralar shear flows. Comparisons are made between the hsphere model and previous results from kinetic theory.higher solid fractions, simulations using the particle dynammodel are performed, including comparisons between linand nonlinear models of viscoelastic particle behavior.Sec. IV, simulations of bounded shear flows are performeinvestigate the formation of an ordered phase. This behahighlights the importance of the local solid fraction and wseparation on the stability of the shear flows of a granumaterial. In this section the results of the wavelet analysisalso presented, which may interpret the findings in the liof phenomena observed in physical systems such as theistence of stick-slip dynamics, characterized by harmofrequencies. Particle diffusion in low and moderately densystems is also analyzed using a comprehensive model bon the revised Enskog theory of granular fluids andGrad’s method of moments. A part of the formulation is psented in the Appendix, along with formal expressionsthe particle diffusion coefficient, particle thermal diffusiocoefficient, and coefficient of mobility@31#. The computersimulations were modified to include the effects of gravityinvestigate the bounded sheared systems under constanume as well as fixed normal load. Interesting features sucthe dilatancy of granular media and stick-slip dynamicsexplored and as a result fresh interpretations are proposethese complex processes. Finally, the concluding remarksgiven in Sec. V.

II. MATHEMATICAL MODELS

In the present study, simulations of granular assembwere carried out by means of the modified hard-sphmodel of Alder and Wainwright@11#. Simulations were thencarried out using the particle dynamic model@2,9,10#, andthe corresponding sets of results were compared. The mdevelopment for each approach will be described in thelowing sections.

A. Modified hard-sphere model

In the modified hard-sphere model@12,13#, the grains areassumed to interact repulsively@32,33#, and the inelasticityof the grains is taken into account through the normal coficient of restitution,e. The restitution coefficient relates thprecollision and postcollision relative velocities of the twpoints of impact on the particle surfaces:

~V i j8imp

• k!k52e~V i jimp

• k!k, ~1!

wherek is the unit vector directed from the center of partic

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indicates the postcollisional value of the relative velociThe normal restitution coefficient decreases from unity asnormal impact velocity,Vn5V i j

imp• k, increases from zero

@34#. Assuming that no plastic deformation occurs, Schwaand Po¨schel@35# used the generalized Hertz theory of elasimpact for the case of viscoelastic collisions and suggesan approximate functional form fore(Vn) as 12e;Vn

1/5,which appears to be in agreement with the experimental d@34#. Following this idea, the following heuristic functionaform was chosen for the coefficient of restitution:

e~Vn* !512~12e0!~Vn* !1/5, ~2!

where the dimensionless normal impact velocity is definedVn* 5Vn /V0 . Here, the reference coefficient of restitutioe0 , and the reference velocityV0 are adjustable parameteof the model.

Considering the particle rotational motion, the relative vlocity of the two points on the particle surfaces that comtogether at impact,V i j

imp , is given by Allen and Tildesley@36#,

V i jimp5~V i2V j !2

s

2k3~vi1vj !. ~3!

In the above equation,V i , V j , vi , andvj are the transla-tional velocity and the spin velocity ofi and j particles, re-spectively. For the description of the tangential forcestween grains at the contact zone, Lun and Savage@37#suggested that during a collision the tangential componeof V i j

imp are changed such that

k3~V i j8imp3 k!52b k3~V i j

imp3 k!, ~4!

whereb is the tangential coefficient of restitution. Then thconservation laws yield the following expression for the impulse:

J5mh1~ k•V i jimp!k1mh2k3~V i j

imp3 k!, ~5!

where h15 12 @11e(Vn)#, h25 1

2 (11b)k/(11k), k54I /(ms2), andm and I are the mass and the momentinertia of the particle, respectively. The first term on the rigside of Eq.~5! represents the normal impulse which is in tdirection of k, and the second term is the tangential impuwhich is in the direction perpendicular tok and lies in theplane of k and J. It is worth mentioning that the kineticenergy of the particles is not necessarily conserved in cosions due to the inelasticity and roughness of the particle

The Coulomb@38# friction law was chosen to model thfriction between two colliding grains with a surface frictiocoefficientm when the normal impact velocityVn is small,namely,h2uk3(V i j

imp3 k)u>mh1u( k•V i jimp) ku. Then, an ex-

pression for the tangential coefficient of restitution,b, maybe found as follows:

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Note that negative values ofb in the above expression indcate a reduction in the component of the postcollisional retive velocity perpendicular tok, without any change in itsdirection @37#.

When the normal impact velocityVn is large, namely,h2uk3(V i j

imp3 k)u,mh1u( k•V i jimp) ku, then either sticking or

rolling without slipping at impact may occur. During stickincontacts, both the magnitude and direction of the relatangential velocity may change. A phenomenological cstant, denoted byb0 , has been suggested by Lun and Be@12# to characterize the restitution of velocity in the tangetial direction for the sticking contact. The suggested valuethis parameter for steel pucks isb050.4 @12#. For rollingwith slipping, b50 @39#. In this case the average normforce results in a large tangential force which deceleratesprecollisional tangential velocity,k3(V i j

imp3 k), withoutslipping.

Applying the calculated values for the coefficients of retitution, the postcollisional velocities and spins, which arequired to create the trajectories of the particles in the ccan be determined using the following equations:

V i2V i85V j82V j5~h12h2!~ k•V i jimp!k1h2V i j

imp ,~7!

vi82vi5vj82vj522k

ks3@h2k3~V i j

imp3 k!#.

Following Allen and Tildesley@36#, a generalized relationwas developed for the collision time calculation of a pairparticles,i andw, of diameters, which are located at timetat r i and rw having velocitiesvi and vw , which experiencedifferent accelerationsai and aw . If these particles are tocollide at timet1t iw , namely,

Ur iw1V iwt iw11

2aiwt iw

2 U5s, ~8!

thent iw may be obtained by finding the smallest real positroot of the following quartic equation:

aiw2

4t iw4 1ciwt iw

3 1~n iw2 1diw!t iw

2 12biwt iw1r iw2 2s250.

~9!

Here,aiw andn iw are the values of the relative acceleratiand relative velocity vectors, respectively, between particliand w. The parameterciw is the dot product of these twvectors, namely,ciw5aiw•V iw , and the variabler iw denotesthe distance between the centers of the particles. The paeter diw5aiw•r iw represents the dot product of relative aceleration and position vectors. The dot product of the retive velocity and relative position vectors of particlesi,w, isrepresented bybiw , i.e.,biw5r iw•V iw . The solutions of Eq.~9! can be explicitly expressed by

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The expressions for the variablesk1 , k2 , andk3 are listed inTable I.

Neither, one, or both pairs of roots may be real. Tsmallest real positive root of the quartic equation~9! corre-sponds to particle impact. Obviously, the absence of anypositive root implies that the pair of particles will not immnently collide. In the case that the particles experiencesame acceleration~e.g., gravitational acceleration!, quarticequation~9! reduces to the quadratic equation given in@36#.

In this approach, in which the system evolves oncollision-by-collision basis @36#, the collision dynamicsmodel is implemented for the colliding pair and then a seais initiated for the next collision. Applying this methodthree-dimensional simulations of a large number of particcan be conducted in reasonably short run times.

B. Particle dynamics models

In the particle dynamics model, two stages occur duran impact. During the first stage, which is called quasielainitial compression, the particles undergo deformation utheir relative velocity vanishes. Recent experimentsGugan@40# demonstrated that in this period, the kinetic eergy of the relative motion of the particles may be storthroughout the strain field. Therefore, the elastic potenenergy function used by Hertz@41# may be valid for theinitial compression, suggesting that a normal repulsive ctact force on particlei due to particlej is produced, whosemagnitude is given by

uFn,i j u5Knd i j3/2. ~11!

In the above equation,Kn is a constant defined ass1/2/(3x)for a sphere on sphere and&s1/2/(3x) for a sphere onplane, x5(12n2)/E, n is the Poisson’s ratio, andE is

TABLE I. Definition of variables needed for the solution of thquartic equation.

Variable Definition

k1 4@(ciw /aiw)2223 (n iw

2 1diw)#/aiw2

k2 25/3@(22/3k4)/(k51k6)1/31(k51k6)1/3#/(3aiw2 )

k3 16@ciw(n iw2 1diw)/aiw

2 24(biw1ciw3 /aiw

4 )#/aiw2

k4 (n iw2 1diw)226ciwbiw13aiw

2 (r iw2 2s2)

k5 2(n iw2 1diw)3218biwciw(n iw

2 1diw)127aiw2 biw

2 19(r iw2

2s2)@3ciw2 22aiw

2 (n iw2 1diw)#

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FIG. 2. ~a! Side view of a cubic box containing about 10 000 identical nonoverlapping spherical particles, half light and half dark.show the direction of motion of the wall particles.~b! Dimensionless translational granular temperatureT* as a function of solid fraction.Smooth particles are represented by triangles and squares fore050.84 and 0.93, respectively. Squares with plus representT* for e50.93, where the coefficient of restitution is assumed to be constant. The upper and lower solid lines represent kinetic theory fie0

50.93 and 0.84. For rough particles, inverted triangles are fore050.84,m50.41, andb050 and diamonds are fore050.93,m50.123, andb050.4. The dashed lines are visual fits through the data. Inset: Distribution of dimensionless velocity as a function of dimendistance inz direction.~c! Dimensionless translational granular temperature for smooth particles as a function of dimensionless disz direction. Left and right triangles areTx* and Tz* at Fs'0.2 ande050.93, circles and diamonds areTx* and Tz* at Fs'0.45 ande0

50.93, triangles areTz* at Fs'0.2 ande050.84.~d! Graphs of the normalized probability density distributionspn for nx* , ny* , nz* , in theform of ln@2ln(pv)# versus ln(uV* u), wherepn5p(V* )/p(0), atFs'0.4 ande050.93. Triangles are fornx* , squares forny* , and invertedtriangles fornz* . The solid line indicates the fitted value for the angular coefficient, which is about 2.~e! Dimensionless translationagranular temperature for rough particles as a function of dimensionless distance inz direction. Right and left triangles areTx* and Tz* atFs'0.2 ande050.93,m50.123, andb050.4, respectively; circles and diamonds areTx* andTz* at Fs'0.4 ande050.93,m50.123, andb050.4, respectively; triangles areTz* at Fs'0.4 ande050.84, m50.41, andb050. ~f! Graphs of the normalized probability densidistributionspv for spins in the form of ln@2ln(pv)# versus ln(uvu), wherepv5p(v)/p(0), at Fs'0.4 ande050.93, m50.123 andb0

50.4. The triangles are forvx , squares are forvy , and inverted triangles are forvz . The solid line indicates the fitted value for the angucoefficient, which is about 2. The data ofvy , which has the mean value of 1.01 s21, was manipulated to obtain data with zero mean.

so

oeu

thumeth

than

abs

-izedur.ardmalldelp-

Young’s modulus;d i j is the amount of overlap defined as2ur i j u andr i j is the vector connecting the center of massparticle i to the center of mass of particlej.

A considerable part of the initial kinetic energy does ncontribute to the kinetic energy of recoil and is transforminto losses such as the thermal energy and energy of sooscillation. This transformation seems to occur duringsecond stage of impact, namely, after the time of maximcompression of the particles. During this period, the shapthe particles undergoes restitution due to the action ofelastic forces, and the potential energy of deformationagain transformed into the kinetic energy. At the end ofsecond period, the particles are no longer in contact,their recoil velocities are reduced after impact.

Gugan suggested that Hertz’s equations give reasonresults for inelastic collisions even when the energy los

02130

f

tdnde

ofe

ised

leis

large @40#. However, Haff@10# argued that for many applications the aforementioned contact scenario is not realand plastic flow, spallation, cracking, or fracture may occBy considering the fact that the constituent particles are hand the amount of deformation near the contact zone is scompared to particle size, Haff suggested a simplified mo@10#, which might capture the most important contact proerties. In this model for a binary collision betweeni and jparticles, the normal contact force is given by@2,10#

Fn,i j 5S knd i j 11

2mgndi j • kD k. ~12!

In the above equation,k is the unit vector directed from thecenter of particlei to that of particlej at impact,di j repre-

3-6

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nc

lesezeio

re

anr

ley

-binox-

ar-orhe

fileal

ro.t.

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effi-

ip-ra-te

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s

-nal

esowtionard-the

de-red-


sents the rate of change ofr i j , kn5(23105)mg/s is thestiffness of a spring that prevents the particles from interpetrating @2#, and gn is a constant which accounts for thenergy lost at contact due to inelastic processes. Thompand Grest@2# suggested that the coefficient of restitutionemay be defined as exp(21

2gntcol). Here tcol represents thecontact time, which is a constant defined asp(2kn /m2gn

2/4)21/2 @2#. However, the measurements@34,40,42# sug-gested thattcol appears to have a characteristic dependeon the impact velocity, namely,tcol;uV i j

impu21/5.During the collision of two slipping particles each partic

may be subjected to a tangential frictional force that oppoits rotational motion. The tangential force that characterithe frictional effects may be given by the Coulomb equat

Ft,i j 52muFn,i j uk3S V i jimp

Vi j3 kD . ~13!

For nonslipping contact, Thompson and Grest@2# suggestedthat the tangential force may be estimated using a diffeexpression,

Ft,i j 52 12 gsmk3~V i j

imp3 k!, ~14!

wheregs represents a tangential viscous damping constThe tangential coefficient of restitution,b, discussed earlieis defined as exp(2gstcol) @2#.

III. PARTICLE DIFFUSIVITY IN UNBOUNDEDSHEAR FLOWS

To simulate an unbounded system of spherical particsubjected to a uniform shear, which is characterized bvelocity gradient of the formV(z)5 «zex , the Lees-Edwards-type@43# periodic boundary conditions were applied to the bounding top and bottom faces of the cucomputational box. Simple periodic boundary conditio@36# were applied on other faces of the computational bFigure 2~a! illustrates the initial configurations for the simu

clc

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02130

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on

e

ss

n

nt

t.

sa

cs.

lations. About 10 000 identical nonoverlapping spherical pticles, consisting of two species differing only by a collabel, were placed randomly in the computational box. Tinitial velocity of each particle in thex direction was as-signed with a magnitude according to a linear velocity proV(z) plus a small random number within the [email protected](Lz)u,10.05uV(Lz)u#. The initial velocities in they and z directions as well as the spins were all set to zeHere Lz is the height of the box, andex represents the univector in thex direction, which is the direction of mean flowFor the geometry shown in Fig. 2~a!, Lz is set to unity.

Simple shear flows

To simulate an unbounded shear flow of a highly idealizsystem whose dynamics is dominated by collisions, the hasphere model discussed in the preceding section is usedwhich the particle surface frictionm is set to zero. In thiscase, the particle surface is characterized only by a cocient of restitution ofe. In the simulations, the solid fractionvaried from 0.16 to 0.58. In order to obtain the local descrtion of quantities such as solids fraction, granular tempeture, and velocity, first the box is divided into an approprianumber of layers in thez direction, using the criteria suggested by Loose and Ciccotti@44#. Then, the local values arcalculated as averages over the particles in the particularers. Finally, the local values are averaged for 200 configutions each separated byt* '1022. Here, the dimensionlestime t* is defined ast «.

Figure 2~b! illustrates that as the solid fraction in the computational box increases, the dimensionless translatiogranular temperatureT* 5T/( «s)2 decreases. Figure 2~b!also compares the simulation results for two different valuof e0 , namely, 0.93 and 0.84. As expected, the results shthat as thermal dissipation increases, the average fluctuaenergy of the particles decreases. The results from the hsphere model can be compared to those predicted bykinetic theory analysis for smooth spherical particles, aspicted by a solid line in Fig. 2~b!. Note that an expression foT* may be derived from the kinetic theory of unboundrapid granular flows@25# by assuming an isotropic distribution of collision angles between the colliding particles:

T* 51

24Fs~e0221!gc

H @11 45 Fsgc~11e0!#@ 5

2 1Fsgc~3e0212e021!#p

6Fsgc~11e0!~e023!2

8

5~11e0!FsgcJ , ~15!

asve-

s inra-theef-

act

wheregc5@12(Fs /Fm)4Fm/3#21 is the contact value of theequilibrium radial distribution function@45# and Fm is themaximum shearable solid fraction for the particles@46#. Ascan be seen from Fig. 2~b!, at higher solid fractions, thedimensionless translational granular temperatures produby numerical simulations exceed the values of those calated from the kinetic theory expression~15!.

To verify the accuracy of the algorithm, two extra ru

edu-

were performed, in which the coefficient of restitution wassumed to be independent of the magnitude of impactlocity and was set to a constant value ofe50.93. The resultsfrom these runs are marked by the squares with plus signFig. 2~b!. As can be seen, the translational granular tempetures were slightly lower in these cases, compared totemperatures calculated for similar cases in which the coficient of restitution depends on the magnitude of imp

3-7

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velocity, namely,e512(120.93)Vn*1/5, where settingVn*

5Vn /V0 to 2Vn gives a reasonable fit to the available daThis observation implies that the algorithm is correctly cculating the coefficient of restitution, which is higher tha0.93 for cases in which the impact velocity is lower than treference velocity in Eq.~2!.

Since the number of collisions for whichVn is lower thanV0 increases with the increasing solid fraction, the simution results for dimensionless granular temperature wouldexpected to be higher than the calculated values usingkinetic theory expression~15!.

Simulations were then performed using a system ofelastic rough spheres subjected to steady shear. The inconfiguration is that shown in Fig. 2~a!, and the solid fractionagain varied from 0.16 to 0.58. Two different sets of valufor the surface parameterse and m were used. The first sewas that suggested by Lun and Bent@12# for steel pucks,namely,e050.93 andm50.123. In this case of sticking binary collisions,b0 is set to 0.4. The second set was thsuggested by Drake@47# for glass beads, namely,e050.84and m50.41, with b0 taken to be zero. A comparison between the results of inelastic, rough particles and thosesmooth particles, as shown in Fig. 2~b!, indicates that at thesame solid fraction, shear rate, and restitution coefficient,frictional grains are cooler than the frictionless grains.

Figure 2~c! plots the variations of local dimensionlestranslational granular temperature in longitudinal and latedirections in the direction of the velocity gradient~namely,thez direction!. The nearly flat profiles in Fig. 2~c! are char-acteristic of unbounded simple shear flows with small amtude local fluctuations. Moreover, the distributions of dimesionless fluctuation velocities,nx* , ny* , nz* , are Gaussianwith standard deviations,Tx*

1/2, Ty*1/2, Tz*

1/2, respectively,about the mean value zero as shown in Fig. 2~d!. Here, thedimensionless fluctuation velocity is defined asv* 5v/s«. Acomparison between results of the numerical simulationsT* in the longitudinal direction, namely,Tx* , and those as-sociated with the lateral directions, namely,Ty* and Tz* , asdepicted in Fig. 2~c!, suggests that the translational granutemperature is not isotropic in simple shear flows of a gralar assembly. As shown in Fig. 2~e!, the translational granulatemperature profiles are also flat, implying that similar bhavior exists for unbounded simple shear flows of roughsmooth grains. Moreover, there is no effect of particle rouness on the form of velocity profile. It can be seen from Fi2~c! and 2~e! that the ratio of the dimensionless translationtemperature of smooth steel balls and rough steel puTzsmooth* /Tzrough* could be as large as 3 at a solid fractionFs'0.4. Therefore, the difference in translational tempetures suggests that shearing of the rough particles indrotational motion. It should be noted that for both of taforementioned systems the initial values of all spins wset to zero. The evidence for generation of rotational motin the shearing motion of an assembly of rough particlesillustrated in Fig. 2~f!, which presents the probability densidistribution of particle spins for a configuration of particlwith the surface parameters ofe050.93, m50.123, andb050.4 at the shear rate of 2 s21, taken after 23108 collisions.

02130

.-

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enis

Note that in this case the shearing motion in thex directioninduces a mean rotational motion in they direction ofvy'21.01ey s21. Here, ey represents the unit vector in theydirection.

Assuming that the particles are uniform solid spherthe rotational temperature is defined asQ5(ks2/12N)( i 51

N @(v ix2vx)21(v iy2vy)

2 1(v iz2vz)2#,

wherevx51/N( iv ix , vy51/N( iv iy andvz51/N( iv iz arethe average values of the spins in thex, y, andz direction,respectively. HereN represents the total number of particlin the computational box. The dimensionless rotational teperature in the direction of the velocity gradient may be dfined asQz* 5k( i(v iz2vz)

2/(4N«2). As seen in Fig. 2~f!, anormal distribution with zero mean and variance of 3.7which is shown by a solid line, provides good approximtions for spins in thez direction. Usingk5 2

5 for a uniformsolid sphere,Qz* for the aforementioned case was foundbe close to 0.1. In this case, the difference in total~rotationalplus translational! fluctuation energy of the systems of rougand smooth particles could be due to the fact that insystem of rough particles, the dissipation into thermal eneoccurs by means of both friction and inelastic collisions.

To investigate the signature of order in the sheared gralar system, the variation ofQ6ANb ~instead ofQ6! as a func-tion of solid fraction for both smooth and rough particlesgiven in Fig. 3. As mentioned earlier,Q6ANb'1 for a spa-tially uncorrelated system. Using this normalized valueeffect of having different number of bonds for different samplings can be removed. AsQ6ANb approaches 2.5, there istill significant disorder in the system. Note that the valueQ6ANb for a system with cuboctahedral symmetry couldeven higher than 100. Hence, the structural changes thacur during the process of increasing the solid fractionsmall and the system remains disordered even at high sfractions. This indicates that as the solid fraction is furthincreased the grains may become frozen and a noncrystasolid ~namely, a glass! may form. An increased understanding of the glass transition could provide insight into the dnamics of spatiotemporal fluctuations in a dense slowevolving granular system. Unfortunately, computer simution results such as those presented here may providelimited insight, since hard-sphere models may not beequate to capture cooperative motion, such as the rearra

FIG. 3. Variation ofQ6ANb with solid fraction for both smoothand rough particles. Note thatQ6ANb for a finite spatially uncorre-lated system is'1.

3-8

f

s

i-

-

s

l


FIG. 4. ~a! Left: Variations ofa normalized solids fraction olight particles as a function of di-mensionless distance in thez di-rection at different dimensionlestimes t* . Squares are att* 50,left triangles are att* 530, circlesare at t* 539. The total solidsfraction is Fs'0.2 ande050.93.The solid lines represent analytcal solutions~18! for which Tz* isgiven in Fig. 2~c! ~right triangles!ands* '0.034.~a! Right: Mixingof light and dark particles in thecomputational box att* 539. ~b!Left: Variations of a dimension-less solids fraction of light par-ticles as a function of dimensionless distance in thez direction atdifferent dimensionless timest* .Squares are att* 50; left trianglesare at t* 530; circles are att*539. The total solids fraction isFs50.45 ande050.93. Thesolidlines represent analytical solution~18! for which Tz* is given in Fig.2~c! ~diamonds! and s*50.043.~b! Right: Mixing of light anddark particles in the computationabox at t* 539.

u

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ain

ment of clusters of grains, which seems to dominate strtural relaxation at high solid fractions.

In sheared granular fluids where the interactions betwthe grains lead to correlations between the positions andlocities of different grains, the self-diffusion may be intepreted as a result of cooperative effects of macroscopic sfraction fluctuations. Thus an analysis of particle diffusimotion may contribute to the understanding of issues succooperative motion in a granular fluid. To this end, sheinduced dispersions of grains in thez direction at solid frac-tions of Fs'0.2 and 0.4 are illustrated in Fig. 4. The sidview of grains in the system before application of strainthe x direction is shown in Fig. 2~a!. As shown in Fig. 4, astime evolves, light grains migrate away from the uppergion, causing their solid fraction to decay from its initiprofile, with a higher rate of mixing for lower values of totsolid fraction.

To investigate the mixing behavior as shown in Fig. 4diffusion law may be obtained by assuming that the randowalk-type displacements of grains in thez direction are notcoupled to these in any other directions. Thus, a dimensless mass balance for the light particles may be given by@25#

]F l*

]t*2Dz* s* 2

]2F l*

]z* 2 50. ~16!

In the above equation,z* 5z/Lz , F l* 5F l /Fs , Dz*5Dz /s2«5p1/2Tz*

1/2/@8(11e0)Fsgc#, s* 5s/Lz , andFs

represents the total solid fractions of light and dark partic

02130

c-

ne-

lid

asr-

-

-

n-

s.

Here, the expression for the diffusion coefficient in thezdirection is given byDz5sp1/2Tz

1/2/@8(11e0)Fsgc#, whichmay be derived from the kinetic theory of unbounded ragranular flows@25#. The system shown in Fig. 2~a! is subjectto the three conditions

t* 50, H z* .0, F l* 51

z* ,0, F l* 50,

t* .0, z* 5 12 , F l* 5 1

2 , ~17!

t* .0, z* 5 14 ,

]F l*

]z*50.

The solution to this problem is given by@48#

F l~ t* ,z* !51

21 (

n51

`

e2p5/2n2s* 2Tz*1/2t* /2~11e0!Fsgc

3Fsin~np!

npcos~2npz* !

112cos~np!

npsin~2npz* !G . ~18!

A comparison between the analytically obtained sofraction distributions~18! and those obtained from numericsimulations, which are shown by symbols in Fig. 4, suggethat in the range of solid fractions given above, the gr

3-9

re--

e

r

r

r


FIG. 5. ~a! Left: Initial con-figuration of tracer grains in thecomputational box.~a! Right: Dis-persion of the grains. Squares ainitial positions and circles are positions of grains after a short period of time in a dimensionlessyzplane. ~b! Variations of ^Dz* 2&with the dimensionless delay timt* at Fs50.565 for the smoothparticles in an unbounded sheaflow. Triangles and circles aree0

50.84 ande050.93, respectively.The solid lines represent lineafits, with slopes of 2Dz* . ~c! Dz*as a function of solid fraction.Squares aree050.84, triangles aree050.93. The upper and lowesolid lines are the kinetic theoryresults, namely,Dz* 5p1/2Tz*

1/2/@8(11e0)Fsgc#, with the valuesof Tz* given in Fig. 2~c!, for e0

50.93, 0.84.

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ss

di-u-

de-icro-essntyly,

ee

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nalr-re,

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displacements in thez direction are of random walk typecharacterized by a self-diffusion coefficientDz .

To make the preceding remarks more specific, suppthatN0 is the number of tracer grains initially situated aboZ50 in the computational box, as shown in Fig. 5~a!. Toobtain a better visualization, the normal grains have not bshown in this figure. Due to shearing motion at timet.0, thetracers are spatially dispersed aboutZ50. The evidence ofdispersion is illustrated in Fig. 5~a!, which shows the posi-tions of tracer grains in ayz plane taken at two differentimes. At any timet.0 the second moment of the numbertracers per unit volume,N(z,t), represents the mean-squadisplacement of the traces in thez direction. That is, @z(t)2z(0)#2&5(1/N0)*z2N(z,t)dz. If the tracers are spatiallydistributed in a Gaussian such asN(z,t)5N0 /(2ApDt)exp@2z2/(4Dt)#, then the self-diffusion coefficienis given by@19#

D5^@Z~ t !2Z~0!#2&

2t5

^DZ2&2t

. ~19!

Equation~19! applies when the timet is large compared tothe average time between collisions of grains. Here thegular brackets denote the ensemble average and the samare taken at time intervalst apart.

Figure 5~b! illustrates the variations of the dimensionlemean-square displacement in thez direction with the dimen-sionless delay timet* 5t« at Fs50.56 for smooth particleswith surface parameterse050.84 andm50. The displace-

02130

set

n

n-ples

ments were evaluated excluding periodic boundary contions. At long times where the deviation from the free diffsion regime becomes apparent,^DZ2& behaves linearly int.This observation may suggest that the linear Fick’s lawscribes, on average, the dissipation of spontaneous mscopic density fluctuations. In this case the dimensionldiffusion coefficient in the direction of the velocity gradieis found to beDz* 50.05, which is close to that measured bBridgwater @18# in the simple shear apparatus, name0.057. Note that the solids fraction was not reported in@18#.As also seen in Fig. 5~b!, at a solid fraction of 0.56, using thvaluese050.93 andm50 for the surface parameters in thsimulation results inDz* 50.059. The dimensionless transltional temperature in the direction of the velocity gradieTz* , for this case is found to be 1.62. Considering the dimsionless diffusion coefficientsDz* given in Fig. 5~b!, alongwith a comparison between the corresponding translatiotemperaturesTz* , it appears that the diffusive motion of paticles is mainly controlled by the translational temperatuwhose value is a function of the coefficient of restitution,e.The results also indicate thate can be adjusted to obtain thvalue of the dimensionless transverse diffusion coefficimeasured by Bridgwater@18#, which was 0.057. Perhapsreasonable value for the coefficient of restitution describthe surface properties of phenolic resin balls would be inrange 0.84<e<0.93, but closer to the higher end of thrange.

Figure 5~c! compares the theoretical results obtaineding Dz* 5p1/2Tz*

1/2/@8(11e0)Fsgc# with those predicted by

3-10

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the simulations at two different coefficients of restitution.can be seen that the aforementioned expression forDz* rea-sonably fits the numerical results for systems with a sofraction lower than 0.45. However, the simulation resultsceed the theoretical value at higher solid fractions. Inpresent simulations, Eq.~2! was used to describe the inelasbehavior of the grains. Therefore, it is not clear whetherdeviations may be attributed to an enhancement of the veity correlations due to the excitation of slowly decaying coperative motions in a dense granular fluid, or to the usedifferent expressions for the coefficient of restitution in tsimulations and the kinetic theory expression.

The presence of long-duration frictional contact betweparticles was reported by Natarajan, Hunt, and Taylor@21#. Itmay be speculated that in their gravity driven flow, the gratational effects contribute insignificantly to the particle dfusive motion in the direction of the velocity gradient, whicis perpendicular to the direction of gravity. The obtainvalue of the dimensionless diffusion coefficientDz* , for thepresent simulations of a system of frictionless grains,somewhat larger than the values of the dimensionless ladiffusion coefficient measured in the moderately low shregion in the experiments@21#, which was 0.0397. Recalthat at the extremely high average solid fraction ofFs.0.6, a difference of four orders of magnitude exists btween the values of dimensionless diffusivity in the directiperpendicular to the main flow, measured by Menon aDurian @8#, and that calculated from the results of NatarajHunt, and Taylor@21#. Assuming that the solid fraction in thlatter case was much lower than that in the former casemay be concluded that the value ofDz* falls below its theo-retical value at solid fractions close to the glass transitio

Figure 6~a! plots the variations of the dimensionlemean-square displacement of rough particles in thez direc-tion, ^Dz* 2&5^Dz2&/s2, with a dimensionless delay timt* , a shear rate«54 s21, a solid fraction ofFs50.565, andsurface parameters of the second set mentioned abnamely, for glass particles. After 107 collisions, it was foundthat the value of the dimensionless diffusion coefficientnot depend on the choice of time origin, which indicates tthe system had attained a steady state. As shown in Fig.~a!,at long times^Dz* 2& behaves linearly int* , which may

FIG. 6. ~a! Variation of ^Dz* 2& as a function of dimensionlesdelay timet* at Fs50.565 and the surface parameterse050.84,m50.41, andb050. The solid line represents a linear fit.Tz* andDz* are 0.348, 0.038.~b! Dz* as a function of solid fraction. Thesquares aree050.84,m50.41, andb050; triangles aree050.93,m50.123, andb050.4.

02130

d-e

ec--of

n

-

sralr

-

d,

it

ve,

t

justify the use of the Einstein formula for calculating thmacroscopic dimensionless diffusion coefficient in the dirtion of velocity gradient. From Fig. 6~a! it is found thatDz*50.0394. This value does not vary with the shear rate. Inpresent cases, the average dimensionless translationalperatureTz* is found to be 0.352. Therefore, it can be cocluded that for these cases, the translational temperaturthe direction of the velocity gradient,Tz , was quadraticallydependent on the shear rate. The obtained value of themensionless diffusion coefficientDz* for the present simula-tions is very close to the average of the values measurethe moderately low shear region in the experiments@21#,which was 0.0397. This agreement, which supports thelier conjecture that the particle diffusive motion in their sytem is governed by shearing motion and that the effectgravity could be of minor relevance, is quite impressivMoreover, this finding may also suggest the value of sofractions in the experiments@21#.

It is worth mentioning that at the same solid fraction ashear rate, particles with smooth surfaces tended to smore diffusive motion in the direction of the velocity gradent than particles with rough surfaces. Comparing the resin Figs. 5~c! and 6~b!, the dimensionless diffusion coefficienDz* of the smooth particles is greater by a factor of aboutthan that of the rough particles. An explanation for the dference in the dimensionless diffusion coefficient is sugested by the observation that the dimensionless translatitemperature in the direction of the velocity gradient of trough particles is lower than that of the smooth particles

In the absence of any theoretical treatment for predictthe diffusive motion of rough grains in a sheared flow, tdimensionless lateral diffusive coefficient is extracted frothe data using the Einstein formula~21!. The results are il-lustrated in Fig. 6~b!, which plotsDz* as a function of thesolid fraction. The dimensionless lateral diffusion coefficiefor rough grains is smaller than that for smooth grains agiven value of the solid fraction. However, the present resdo not seem to indicate at what solid fraction the transitionthe glass state occurs.

Haff @10# suggested that hard-sphere models suffer frseveral limitations including the lack of a realistic model fthe contact forces between grains. Thus, the use of hsphere models may not be justified when the solid fractiovery high and the particle free times are shorter thanduration of the contacts. A related question can then be poregarding the extent to which a true model for the contforces between the grains affects the simulation results foassembly of uniform grains subjected to a shearing defortion. In the present simulations it is important to note thatprobability of finding free times shorter than 1024 s is quiterare even at a solid fraction of 0.58 for smooth particles,illustrated in Fig. 7~a!. According to the classical theory oimpact between frictionless elastic bodies@41#, the total du-ration of collision,Tc , of two identical spherical particleswith massm, diameters, and elastic coefficientx, is givenby Tc54.347@mx/s1/2#2/5Vn

21/5. ThusTc could be as smallas 831025 s for phenolic resin particles with diameters18.6 mm used in@18# having the normal impact velocity o

3-11

a

-

r

f


FIG. 7. ~a! Probability of par-ticle free time for two differentsolid fractions fore050.93. Thesquares areFs'0.2, triangles areFs'0.56. The straight line indi-cates the power lawt23. ~b! Di-mensionless mean free time asfunction of solid fraction. Thesquares aree050.84, triangles aree050.93. ~c! Probability of par-ticle free time for two differentsolid fractions for the surface parameterse050.84, m50.41, andb050. The squares areFs'0.2,triangles are Fs'0.57. Thestraight line indicates the powelaw t22.6. ~d! Dimensionlessmean free time as a function osolid fraction for e050.84, m50.41, andb050. The solid linesin ~b! and ~d! are visual fitsthrough the data.

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ultsticchive

e-

20 cm/s. At the same impact velocity, the value ofTc for1-mm glass particles is one order of magnitude lower ththat for phenolic resin particles. These estimated valuesTc suggest that the hard-sphere model could be of usilluminate the basic features of the shear flow of granumaterial even at high solid fractions. The corroborating edence is the agreement between the numerical results anmeasured values for the dimensionless self-diffusion coecient discussed above.

It is interesting that at the solid fraction of 0.58, the stistics of a timet during which a particle flies freely betweetwo successive collisions displays a power law distributiP(t);t2a, with exponenta'3. There is a short-time cutoff where the uniform distribution is observed ast ap-proaches very low values. However, as depicted in Fig. 7~a!,which compares the statistics oft at two different solid frac-tions, a nearly uniform distribution is observed for the whorange of particle free times at a low solid fraction of 0.2.

Figure 7~b! shows the variations of dimensionless mefree time as a function of solid fraction for smooth particleAt a specified solid fraction, the granular fluid with lowedissipation rate~and thus a higher granular temperature! isobserved to have a lower mean free time. Also, the meantime decreases sharply close to a solid fraction of 0.6, whthe glass transition may occur. The condition is even mpronounced in the case of rough particles at very high sfractions, namely, close to the glass transition solid fractFg . From Fig. 7~c! it may be concluded that a fairly higprobability exists of finding free times shorter than 1024 s at

02130

nortor

i-the-

-

,

.

eereeidn

solid fractions higher than 0.58. The results show thatstatistics of free timet displays a power law distributionP(t);t2a, with exponenta'2.6, and there is a very smasign of short-time cutoff where the uniform distributiocould be observed ast approaches very low values. Ashown in Fig. 7~c!, which compares the statistics of free timt at two different solid fractions, a nearly uniform distribution is again observed for the whole range of particle frtimes at the low solid fraction of 0.2. Figure 7~d! shows thevariations of dimensionless mean free time as a functionsolid fraction. At very high solid fractions the mean free timrapidly drops to extremely small values. The results psented in Fig. 7 suggest that particle dynamics models shbe used to obtain more reliable results at very high sofractions.

At this stage, it is worthwhile to investigate to what extethe Kelvin-type model of Haff@10# for linear viscoelasticgrains at contact mimics the real granular flows. It was sgested that a linear visoelastic behavior is often foundsmall deformations, and that a model based on this behapredicts a constant collision time. Recently, Gugan@40# mea-sured the duration of contact,Tc , as a function of normalvelocity at impact,Vn , for frictionless balls striking a flatsurface. Although the collisions were dissipative, the resfor Tc(Vn) are found to be consistent with Hertz’s elastheory of impact. In this light, a model is developed in whithe contact of frictionless grains produces a normal repulscontact force whose magnitude is proportional to the3

2 powerof the amount of overlap. In this model for a collision b

3-12

rc

eeng

uthteo

ee

ee

le

an-d

im.8thesing

there-

l istheisoc-de-rces

toels’ins

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rs

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eared.

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las

rs.ife-xi-ard

m


tween a ball and a flat surface, the normal contact foduring a collision is given byFn5(Knd3/21Gnd• k) k, whereGn represents the dynamic damping, which is a measurthe energy loss at contact due to inelastic processes. HerGnis a function of the impact velocity, which is calculated usithe e-Vn relationship given in@40#.

As given in Table II, using the nonlinear model the simlation results forTc are found to be in good agreement wimeasured values@40#. The predicted collision times are noconstant, but are a function of velocity at impact. Moreovthe coefficient of restitution decreases from unity as the nmal impact velocity increases from zero, in qualitative agrment with the results given in@34,42,49#. For most solidsubstances, the coefficient of restitution,e, is a function ofimpact velocityVn . Thus the primary issue regarding thmodels for the contact forces between grains is how closthe model can predict thee-Vn relationship. Limited infor-mation is available for thee-Vn relationship for solid sub-stances@34,42,49#.

The present results highlight the limitation of the particdynamics model for linear viscoelastic grains@2,9,10#, whichpredicts a constant collision time, which results in a constcoefficient of restitution. More limitations of the aforemetioned model are evident considering the results presenteFig. 8. Figure 8~a! shows the variations of thex, y, and zcomponents of the normal contact force as a function of tin a binary collision of two glass beads with diameter 4mm. The impact velocity is selected to be 0.56 m/s andcoefficient of restitution is set to be 0.84. The solid linindicate the results for linear, viscoelastic, frictionless graobtained using Eq.~12!, whereas those calculated usin

TABLE II. A comparison between results for contact time fromeasurements and simulations.

Vn

~m/s!Values measured@40#

Tc ~ms!Simulation results

Tc ~ms!

2.19 0.97 0.972.78 0.90 0.933.67 0.83 0.884.24 0.81 0.854.99 0.79 0.835.50 0.79 0.81

02130

e

of

-

r,r--

ly

nt

in

e

e

s

Fn,i j 5(Knd i j3/21Gndi j • k) k with Gn(Vn)51.36(Vn /V0)1/5

10.063(Vn /V0)2/5 kg/s are shown by a dashed line. HereV0

represents a reference velocity. A comparison betweenresults of linear and nonlinear models reveals that the pdicted maximum contact force using the nonlinear modeone order of magnitude higher than that found usinglinear model. Also a much shorter duration of collisionobserved for the nonlinear model. At the same impact velity, increasing the diameter of glass beads to 48 mmcreases the difference between the maximum contact fopredicted by the models as shown in Fig. 8~b!. Moreover, thenumerical values for the collision times become closereach other. Therefore, the differences between the modpredictions are more pronounced when the size of grabecomes smaller. For 48-mm particles a differentGn-Vn re-lation is used, which is Gn(Vn)5252.4(Vn /V0)1/5

1291.4(Vn /V0)2/5 kg/s.To show that even simple shear flows of frictionle

grains can produce complex structures even in simple sitions, simulations were performed for unbounded flow usabout 10 000 glass beads. The diameter of beads is54.8 mm with the material density ofr52500 kg/m3. Forthe geometry shown in Fig. 2~a!, the width, length, andheight of the computational box were each selected to becm. The beads were subjected to a uniform shear, whiccharacterized by a shear rate of«520 s21. The solid fractionis close to 0.58. For the linear model,kn and gn are set to63104 kg/s2 and 3162 s21, respectively. These parametegive a constant collision time of 1.131024 s ~for a binarycollision! and a constant coefficient of restitution of 0.8The second model is the nonlinear viscoelastic modelwhich a generalized form of the aforementioned nonlinmodel is used, by which multiple contacts can be predictIn this model the damping coefficient is given byGn(Vn)51.36(Vn /V0)1/510.063(Vn /V0)2/5 kg/s. The other modeparameter that controls the stiffness of the material, namKn , is set to 2.243109 Pa m1/2. The equations of motion areintegrated using fourth- and fifth-order embedded formufrom Dormand and Prince@50# with Dt'tcol/100.

Both models predict the formation of flattend clusteHowever, the nonlinear model predicts a shorter cluster ltime as compared to the linear model. However, the mamum intraparticle forces are much larger in the nonlinemodel. Recall that in the experiments of Miller, O’Hern, an

a

f

FIG. 8. ~a! Variations of forcesin the x, y, and z directions as afunction of time for 4.8-mm glassbeads using linear~solid line! andnonlinear ~dashed line! models.Here,X, Y, andZ denotex, y, andz directions. Inset: Schematic ofbinary collision of smooth par-ticles for the linear model.~b!Comparison similar to that in~a!for glass beads with a diameter o48 mm.

3-13

st

ete

fowie

icade

n

e

s

acmr

ino

nu

ikedeerer

ioaeb

rc

a

cern

ear

inof

ts.

d byrcelnu-

ibu-rac-s

del.all

e ofof21

tedn-oseof

thethe

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bu

na


Behringer@4#, in which 4-mm particles were used, the largepeaks in the normal stress signal were about ten timesaverage value of the normal stress. This observation appto be more consistent with the intraparticle forces predicby the nonlinear model.

In many systems of practical interest the presence olong-duration frictional contact between grains has beenserved@21#. Thus, the importance of surface friction in flodynamics of sheared granular flows should also be studIn order to model in a simple way the change of geometrcontacts during collisions of rough grains, the linear moof Thompson and Grest@2# was used. As suggested in@2# theparametersgn andgs , which ensure that collisions betweegrains are inelastic, are set asgn52gs , which means thatthe normal and tangential coefficients of restitution havvalue of 0.84. The static Coulomb friction coefficientm is setto 0.4. In this series of simulations the diameter of beads54.76 mm with the density ofr52500 kg/m3. Again thewidth, length, and height of the computational box were eselected to be 10 cm and the number of grains in the silations was 10 187. The grains were subjected to a unifoshear with a shear rate of«520 s21. The solid fraction isclose to 0.586. Contrary to the flattened clusters ussmooth particles, here the structures were found to be mlike elongated chains as shown in the inset of Fig. 9.

The nature of the interactions in a dense frictional gralar assembly under a shearing motion can be examinedanalyzing the formation and disintegration of the chainlcluster presented in the inset of Fig. 9. Using a linear moit was found that the effective lifetime of the chain is of ordof 1024 s. This result implies that the use of the hard-sphmodel, which predicts the mean free time of order 1025 s,may produce an incorrect picture of the complex behavinvolved in the rearrangements of the grains at the solid frtion of 0.58. Therefore, questions regarding the heterogenand cooperativity of granular dynamics in glasses mightanswered only by using a model that treats the contact fobetween the grains appropriately.

An important question to consider at this point is to whprecision could a linear model as suggested in@2# predict the

FIG. 9. Instantaneous cluster size probability density distrition in the computational box atFs50.586. Inset: A chainlike clus-ter consisting of eight frictional particles found in the computatiobox.

02130

thearsd

ab-

d.ll

a

is

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gre

-by

l,

e

rc-ityees

t

contact forces between the grains. To address this conthe differences between the linear model@2# and nonlinearmodel predictions are discussed below. In the nonlinmodel, during the contact of grainsi and j the former grainfeels a force, which is given by

Fi j 5~Knd i j3/21Gn~Vn!di j • k i j !k i j 2HminH mFi j • k i j ,Gt~Vn!

3F k i j 3S V i jimp

uV i jimpu

3 k i j D G•V i jimpJ JF k i j 3S V i j

imp

uV i jimpu

3 k i j D G .~20!

HereGt(Vn) represents the dynamic damping coefficientthe tangential direction, which characterizes the restitutionvelocity in the tangential direction for nonslipping contacHere it is assumed thatGt(Vn)5 1

2 Gn(Vn)50.68(Vn /V0)1/5

10.0315(Vn /V0)2/5 kg/s.Using a nonlinear model for beads withs54.76 mm and

densityr52500 kg/m3, the effective lifetime of the chain isfound to be as short as 331025 s, but the maximum intra-particle forces are quite large compared to those predictethe linear model. Again by considering the measured fofluctuations in @4# it appears that the nonlinear frictionamodel could predict the interactions of grains in dense gralar flows more precisely than a linear model.

Figure 9 illustrates the instantaneous cluster size distrtion in a sheared dense granular assembly with a solid ftion of about 0.586 for which particle-particle interactionwere modeled using the aforementioned nonlinear moSurprisingly, the changes in cluster size distribution are smwhen a linear model was used. However, note that the usperiodic boundary conditions may inhibit the occurrencelarger structures in a computational box whose length isparticle diameters.

As discussed above, the ordering of particles as illustrain Fig. 1 was observed neither in simulations of the ubounded system of particles with rough surfaces nor in thin which the particles were smooth. Therefore, simulationsbounded shear flows of a granular material for whichwall effects are significant are required to investigateexperimentally observed transition to an ordered state.

IV. BOUNDED SHEAR FLOWS

Methods such as photoelasticity@51# can be used to display the contacts between particles in dense bounded sflows. However, determining the contact duration betweparticles remains a difficult task. In this context, numericanalysis could provide some guidance. For instance,analysis of the impact between grains discussed in theceding section may be useful in calculating the durationimpact in terms of the size, relative velocity of impact, aphysical properties of the solid. The obtained results showthat the average duration of an impact in the experimeapparatus of@4# could be about 531025 s for glass beads odiameter 4 mm. It was also found that the duration of impis significantly longer for larger particles. The ratio of thimpact duration to the particle mean free time was identifi

-

l

3-14

lessthe

l


FIG. 10. ~a! Dimensionless normal stressP* exerted on the upper wall versus dimensionless timet* for smooth particles atFs'0.2 andthe apparent shear rate of«52 s21, for e050.93 ands* '0.032. Inset: Distribution of dimensionless velocity as a function of dimensiondistance inz direction. ~b! Variations of the normalized solid fraction of light particles as a function of dimensionless distance inzdirection at different dimensionless timest* . Squares are att* 50, triangles are att* 524, circles are att* 533. The solid lines are visuafits through the data. Inset: Variation of the total solid~light and dark! fraction as a function of dimensionless distance in thez direction. Thedashed line represents the average solid fraction.~c! Variations ofTz* as a function of dimensionless distance in thez direction.

ued

a

theoer

annuhiichee

minci

n

on62ret a

e

ai

Noass

true

i-sys-

telye of.2,res,

s

kenar-ngrmalthis

llm

t aheofuss

edin

.pre-ci-

as a criterion for selecting the relevant model for the simlation of sheared granular flows. In this light, a modifihard-sphere model, which is an event-driven algorithm, mnot be recommended for a system whose solid fractionhigher than 0.56. For more dense systems, for whichprecise information about the rearrangement of grains issential, a more computationally intensive force-driven algrithm based on the particle dynamics model is recommend

In this section, simulation results for the hard-sphemodel for bounded shear flows at moderately densedense solid fractions, for various values of the phenomelogical parameters, are compared qualitatively to the resof recent experiments. Moreover, the predictions of tmodel in the presence of gravity are also included, whcould help to distinguish the effects of different factors in tresults of the simulation. Then, the rough pictures providby the hard-sphere models are refined by performing silations using the particle dynamics model to obtain ancreased understanding of dense granular systems, espetheir transitions to a solid phase.

A. Fixed volume simulations

For the fixed volume bounded shear simulations, the omodification to the computational box shown in Fig. 2~a! isthe introduction of two rough walls with a fixed separatidistance, each comprised of an irregular cubic array ofmassive hemispherical particles with the same diametethe interior particles. Here, the interior particles are drivinto shearing flow by moving the top and bottom walls avelocity Vx0 in opposite directions along thex direction.Thus, there are no periodic boundary conditions in the dirtions normal to the walls, which are located atz56Lz/2.The simulations in this section are organized to investigunder what conditions the results presented in Fig. 1,which smooth steel balls were used, can be reproduced.that it is likely that the granular assembly in Fig. 1 whighly cooled, resulting in the formation of an ordered crytal. As a result, the granular assembly had undergone a stural arrest, where self-diffusivity of the grains becom

02130

-

yises--d.ed

o-ltssh

du--ally

ly

5asn

c-

tente

-c-

s

zero. In this light, the particle diffusive motion will be montored to distinguish a disordered state from an orderedtem.

The first case is the bounded shear flow of a moderadense system of about 10 000 smooth particles. The valuthe solid fraction including the wall particles is close to 0with the solid fraction of the wall particles slightly highethan that in the bulk. The particles with smooth surfachaving a coefficient of restitution ofe050.93 and surfacefriction of m50, are sheared at the apparent rate of 221

between walls separated by a distanceLz'30s. Here, thevalues of the particle surface parameterse andm, for colli-sions between interior particles and wall particles are tato be the same as those for collisions between interior pticles. To ascertain that the period of the run was loenough for the system to come to a steady state, the nostress at the upper wall was recorded. The time series forquantity in a dimensionless form,P* 5P/rps2«2, is illus-trated in Fig. 10~a!. Here, rp is the material density. Theeqiulibration period, which is the period during which amemory of the initial configuration is lost and the systecomes to a steady state, was found to bet* '20. Orderingwas not found to occur for this case, which implies thastrong particle diffusive motion would be expected in tdirection of the velocity gradient. Given the presencestrong mixing in the system of smooth particles, an obvioquestion is how would the diffusive motion of the graindiffer for the cases presented in Figs. 10~b! and 4~a!. Notethat the solid fraction of the bounded flow in Fig. 10~b! is thesame as that of the unbounded flow of Fig. 4~a!. To addressthis question, the diffusive motion of grains in a boundshear flow at moderately high solid fractions is analyzedthe following section.

1. Lateral diffusive motion in a moderately dense bounded flowof smooth grains

The local descriptions for the velocity fieldsV(z)*5@V(z)/Vx0#ex at t* '30, as illustrated in the inset of Fig10~a!, are obtained in the same way as described in theceding section. It is interesting to note that large slip velo

3-15

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furgn

hetunth

ec-le

a

thed

dal

o

enheux

nc

ion

in

r-eig.

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an-

mn,t ofren

ur-

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ionnu-igheg-

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ties exist at the walls. Since the rate of temperature gention is controlled by shear work, these slips result in a latranslational temperature near the wall, as shown in F10~c!. The energy is transferred by means of thermal difsion to the colder central region, in which the rate of enegeneration is lower due to the moderate velocity gradieThermal diffusion is caused by the relative motion of tparticles due to the presence of a translational temperagradient. The drift induced by the temperature gradient, idimensionless form, may be characterized in terms ofthermal diffusion coefficient@26# DT , which may be givenas

ruz* 52rDT*] ln T*

]z*. ~21!

In the above,r is the particle number density,uz* 5uz /s«represents the dimensionless diffusion velocity in the dirtion of velocity gradient,z* 5z/s is the dimensionless distance, « is the apparent shear rate, and the dimensionthermal diffusion is defined asDT* 5DT /s2«. In addition, ifnot all of the particles in the system are sheared uniformlymay occur in systems such as in@21# @also see Fig. 12~d!#,then particles could migrate from the high-shear region atwall to the low-shear region in the center. The drift inducby stress may be given as

ruz* 52B*]r U* ~z!

]z*, ~22!

where B* 5B« is the dimensionless particle mobility anU* (z)5U(z)/(s«)2 represents a dimensionless potentiApproximate expressions for the particle mobilityB and thepotential U for a moderately dense system comprisedsmooth particles are given in Appendix.

The above-mentioned drifts induce a particle number dsity gradient in the system, which results in a flux of tparticles from the central region towards the wall. This flcaused by Fickian diffusion may be given as follows:

ruz* 52Dz*]r

]z*. ~23!

The net flux, with respect to the center of mass refereframe, is given by the sum of the fluxes~21!–~23!:

ruz* 52S rDT*] ln T*

]z*1B*

]rU* ~z!

]z*1Dz*

]r

]z* D .

~24!

Substitution of this expression for the flux into the equatof continuity @52# yields the following result:

]r

]t*5V* ~z!

]r

]x*2

]

]z* FrDT*] ln T*

]z*

1B*]r U* ~z!

]z*1Dz*

]r

]z* G , ~25!

02130

a-e.

-yt.

reae

-

ss

s

e

.

f

-

e

where x* 52x/Lz represents the dimensionless distancethe direction of mean flow.

Equation~25! provides a convenient picture for the paticle diffusive motion in the lateral direction, for which thprofiles of temperature and solid fraction are shown in F10. Since no simple solution exists for Eq.~25!, it was sug-gested in@21# that a simplified form of Eq.~25!, namely,

]r

]t*52Dz*

]2r

]z* 2 , ~26!

might be of use in predicting the particle diffusive motion ffrom the walls in the fully developed part of their channeEquation~26! addresses the local aggregation and redispsion of particles in the above-mentioned regions, which cate local density fluctuations. Assuming that these flucttions dissipate according to Fick’s law, an expression mayobtained for the diffusion coefficient, namely, Eq.~19!,which should be applied to the time regions where the mesquare displacement varies linearly witht*

2. Particle diffusive motion and transition to order in densebounded flows

The validity of Eq.~19! is questionable for a dense systewhose local description for the velocity field, solid fractioand temperature are given in Fig. 11, where the coefficienrestitution ise050.84. However, in the absence of a moreliable theory for the treatment of particle diffusive motioin dense granular flows, Eq.~19! is used to estimateDz* forthe tagged particles initially located at20.1<z<0.1 in thecentral region, where the gradients may not be present. Ding the period of sampling oft* '3, no tagged particleswere found to enter the wall region, in which neither teffects of gradients can be neglected nor the use of linlaws such as those used to obtain Eq.~26! is justified. Figure11~f! illustrates the mean-square displacement in thez direc-tion for the tagged particles,^Dz* 2&, as a function oft* . Atlong times,^Dz* 2& appears to behave linearly int* withDz* '0.054, which is very close to the values of the dimesionless diffusivity measured in the experiments of Bridgwter @18#. It is worth mentioning that the calculation is baseon the local value of shear rate, namely, 1.5 s21. The solidfraction and dimensionless temperature in the central regare found to be 0.58 and 1.38, respectively. The high gralar temperature in the central region, which resulted in a hvalue for Dz* , namely, 0.054, could also play a role in thstability of the flow. This observation also highlights the sinificant role of T* in the particle diffusive motion in thelateral direction.

To focus on finding the dominant parameters in the sbility analysis for the bounded flow, in the above-mentionsimulation of smooth particles using the modified hasphere model, the solid fraction is increased to a value0.59 and the apparent shear rate is decreased to 2 s21 so thatthe local shear rate is nearly the same as that of the prevcase. Although this solid fraction is beyond the recomended range of applicability for the hard-sphere model,simulation results could nonetheless provide a rudimen

3-16

e


FIG. 11. ~a! Dimensionless normal stressP* exerted on the upper wall versus dimensionless timet* for smooth particles atFs

'0.565 and the apparent shear rate of«54 s21, for e050.84 ands* '0.048. ~b! Projection of interior particle positions onto thdimensionlessyz plane for the sample taken att* '60. ~c! Variations of the local solid fraction of the sample shown in~b! with thedimensionless distance in thez direction. The dotted line corresponds to the solid fraction obtained using 42 bins.~d! Variations of the localdimensionless axial velocityV(Z)* 5V(Z)/V0x with Z* . ~e! Variations ofTz* with Z* for the apparent shear«54 s21. The solid lines in~c!–~e! are fits.~f! Variations of^Dz* 2& with the dimensionless delay timet* for the tagged particles initially located at20.1<Z* <0.1.The solid lines represent linear fits.

nn

th

to

oacthigr-

th

, agsehr

oleedIn

wallcal

achg-ighhipes.eingto

s ofun-theove

arehisriti-ers

re

e-

insight into ordering in dense granular systems. As seeFig. 12~a!, the period of equilibration is long. The projectioof the position of the interior particles onto theyzplane of asample taken att* '100 is illustrated in Fig. 12~b!. Asshown in this figure, the system had become ordered withcrystallized regions visible at (20.3,0)3(20.1,0.2) and(0,0.3)3(0.21,0). Figure 12~c! plots the variations of solidfractions in the direction of the velocity gradient. Contrarywhat is observed in Fig. 11~c!, this profile exhibits spikes inthe central region, indicating the presence of layers ofdered particles in the system. Perhaps the most noticedifference between the present case and the previouscan be found by comparing the steady-state values ofdimensionless normal stress at the walls, as shown in F11~a! and 12~a!. It is surprising that the dimensionless nomal stresses at higher solid fractions tended to be lowerthose generated at lower solid fractions. The differencestresses could be due to the smaller wall slip velocitiesillustrated in Fig. 12~d!, which lead to the correspondinsmaller rates of energy production. At high solid fractionfor particles in a layer adjacent to the wall, an enhancemin the particle-wall collision rates may be observed. Tmovement of a tagged particle away from the wall aftecollision is limited due to the presence of a dense layermoving particles in the neighboring layer below the particUpon colliding with a neighboring moving particle, a taggparticle is likely to travel back towards the wall particles.

02130

in

e

r-bleasees.

anins

,nteaf.

this case, due to the enhancement in the rate of particle-collisions and also the limited displacement in the vertidirection, the particles are moved along horizontally withvelocity more correlated to that of the wall particles, whiresults in a smaller wall slip velocity. This observation sugests that the causes of smaller wall slip velocity at hsolid fractions may be understood in terms of the relationsbetween the particle-wall and particle-particle collision rat

As shown in Fig. 12~e!, the lower production rates at thwalls result in local temperatures that are below the melttemperature for this system, which leads to a transitionorder. The results obtained suggest that rapid shear flowsmooth particles, at a very high solid fraction, becomestable with respect to a density wave in the direction ofvelocity gradient. The phase transition as discussed aboccurs when variations of the density and velocity fieldscomparable to a particle diameter. In order to predict tbehavior a theory should be devised which captures the ccal wave number regime at intermediate wave numbwhere the static structure factorS(kz) has its maximum, asshown in Fig. 13. This figure illustrates the static structufactor of a sample illustrated in Fig. 12~b! taken aftert*'100. The static structure factorS(kz) was obtained fromthe initial value of the intermediate-scattering function, dfined as@53#

F~k,t !5^Q~k,0!Q~2k,t !&, ~27!

3-17

n-

r


FIG. 12. ~a! Dimensionless normal stressP* exerted on the upper wall versus dimensionless timet* for smooth particles atFs

'0.59 and the apparent shear rate of«52 s21, for e050.84 ands* '0.048.~b! Projection of interior particle positions onto the dimesionlessyz plane for the sample taken att* '100. ~c! Variations of the local solid fraction of the sample shown in~b! with Z* . The dottedline corresponds to the solid fraction obtained using 42 bins.~d! Variations of the local dimensionless axial velocityV(Z)* with Z* .~e! Variations of the local dimensionless translational temperature in the direction of velocity gradientTz* with Z* for the apparent shea«52 s21. The solid lines in~d!, ~e! are fits.

onrath-e

st isondesthe

om-thethe

d ishehe

r of00,

ig.ese

en-

dyalls

het theen

-

where Q(k,t)5N1/2( j 51N exp@ik•r j (t)#. Here, the attention

is confined tok vectors in thez direction. The smallest wavenumberkz that can be studied is 2p/Lz . The dimensionlessvariable kzs is used to express the structural correlatishown in Fig. 13, which provides a useful guide to structueffects on fluctuations. As seen in Fig. 13, the height offirst peak reaches a value of orderN. This indicates the presence of a near-perfect layering into planes with well-definlayer spacing as seen in Fig. 12~b!. The region of wave num-

FIG. 13. Static structure factorS(kz) of a sample whose projection of interior particle positions onto the dimensionlessyz plane isshown in Fig. 12~b!. Inset: Static structure factorS(ky) of the samesample.

02130

le

d

bers where structural effects are expected to be strongenear the first peak. It is interesting to observe that the secpeak inS(ky), as illustrated in the inset of Fig. 13, becomlonger than the first peak. This observation may suggestpresence of regions of cuboctahedral symmetry in the cputational box. Comparing the present results with Fig. 1,hard-sphere model appears to be capturing qualitativelyflow behavior for a wide range of flow conditions.

In order to show that the ordering transition persists annot merely an artifact of the cubic computational box, tgap between the walls is reduced by approximately half. Tinitial configuration of interior particles projected onto theyzplane is shown in Fig. 14~a!. To keep the solid fraction thesame as the previous simulation, namely, 0.59, the numbeinterior and wall particles was reduced to 4296 and 4respectively, withLz'10s. The results for this run indicatea transition to order, evidence of which is shown in F14~b!, which represents the configuration of interior particlprojected onto theyz plane. Using the smooth particles, thstick-slip dynamics is observed in the time series of dimsionless normal stress, as shown in Fig. 14~d!. This result isconsistent with the recent findings by Miller, O’Hern, anBehringer@4#, who have found that the stick-slip motion mavanish upon increasing the separation between the w@54#.

A similar behavior can be observed by comparing ttime series of dimensionless normal stresses obtained asame solid fraction, but with different separations betwe

3-18

ss


FIG. 14. ~a! Projection of the positions of the interior particles onto theyz plane at the onset of shear.~b! Projection of the positions ofthe particles onto theyz plane at the steady-state condition for rough particles atFs'0.59 and the apparent shear rate of«54 s21, fore050.84 ands* '0.1. ~c! Variation of the local solid fraction of the sample shown in~b! as a function ofZ* . The dashed line correspondto the solid fraction obtained using 17 bins. The solid line is a fit.~d! Dimensionless normal stressP* exerted on the upper wall versudimensionless timet* at Fs'0.59 and the apparent shear rate of«54 s21 for e050.84 ands* '0.1. ~e! Projection of the positions of theparticles onto theyz plane for smooth particles atFs'0.59 and the apparent shear rate of«54 s21, for e050.84, s* '0.1, ands«2/g'0.5.

icbetalelo

rdthith

oh

rlyrasitten-s-e.bdi

thIn

on

a

s atingtc-

lidithe

oflid

s is

ery

or-

ich

the walls, as shown in Figs. 12~a! and 14~d!. Stick-slip mo-tion has also been observed in studies of boundary lubrtion @55#. In the present system stick-slip motion mayassociated with the phase transition between ordered sand disordered kinetic states of the thin layer of particseparating the moving wall from the crystallized regioncated at (20.5,0.5)3(0.05,0.18).

The mean dimensionless normal stress is about one oof magnitude greater in the present simulation than inprevious simulation with the same solid fraction but wdifferent wall separation. The decreased wall separation mlikely resulted in a different instability behavior, whiccaused the system to undergo a transition to a mixturecrystalline and noncrystalline two-phase flow with a neaplanar interface normal to the direction of the velocity gdient. Clearly, an explanation of this phenomenon requiretheoretical stability analysis, where variations of the densand velocity fields on the length scale of a particle diameplay a key role in the instability scenario. It is worth metioning that the profile of the local solid fraction, as illutrated in Fig. 14~c!, appears to be very similar to that for thexperiments of@29#, in which partial shearing occurredHowever, in their system the formation of a nonshearasection adjacent to the bottom wall appears to be causegravity. In the presence of gravity, as can be seen in F14~e!, the nonshearable section, whose density is higherthat of the surrounding fluid, has migrated downward.order to take into account the effect of gravity, the collisitimes were calculated using the quartic equation~9! with ai ,which represents the acceleration vector of the interior p

02130

a-

tics-

ere

st

of

-ayr

lebyg.an

r-

ticles, set to22s«2ez m/s2 directed in the negativez direc-tion. The accelerations of the wall particles,aw , are assumedto be zero.

The absence of a disordered phase in the simulationthe high solid fraction of 0.59 suggests that the strong mixreported in@21# is unlikely to be predicted for a system athis solid fraction. For this reason, the mean bulk solid fration in @21# should be close to 0.565. Recall that at this sofraction the simulation of shear flow of smooth particles wa wall separation ofLz'21s resulted in a disordered phasin the central region withDz* '0.054.

At this stage it would be useful to investigate the effectwall separation distance on the flow stability. At the sofraction of 0.565, the apparent shear of 4 s21, the wall sepa-ration of Lz'10s, and the surface parameterse050.84, m50, the time series for the dimensionless normal stresshown in the inset of Fig. 15~a!. The mean value of thedimensionless normal stress in the interval of 150,t*,350 is found to be 15.5, which suggests thatTz* , whoseprofile is illustrated in Fig. 15~a!, is large enough to inducemixing. As seen from the inset of Fig. 15~b!, the local shearrate in the central region is nearly constant, with a value vclose to the apparent shear of 4 s21. Using Eq.~19! to esti-mate Dz* for the tagged particles initially located at20.1<Z* <0.1, where the effects of gradients may be unimptant, results in a value ofDz* '0.015. According to the pro-jection of the position of the interior particles onto theyzplane of a sample taken att* '150, as shown in Fig. 15~d!,some order in the central region can be observed, wh

3-19

t

tnt

e

ng

tot


FIG. 15. ~a! Variation Tz* with Z* for the ap-parent shear«54 s21. The solid line is a fit. In-set: Dimensionless normal stressP* exerted onthe upper wall versus dimensionless timet* forsmooth particles atFs'0.565 and the apparenshear rate of«54 s21 for e050.84 and s*'0.1. ~b! Variations of^Dz* 2& with the dimen-sionless delay timet* for the tagged particlesinitially located at20.1<Z<0.1. The solid linesrepresent linear fits. Inset: Variations ofV(z)* asa function ofZ* . The dashed line is a linear fiwhose slope is very close to that of the appareshear. The dotted line is a nonlinear fit.~c! Varia-tion of the local solid fraction of the samplshown in ~d! as a functionZ* . The dashed linecorresponds to the solid fraction obtained usi17 bins. The solid line is a nonlinear fit.~d! Theprojection of the positions of the particles onthe dimensionlessyz plane for a sample taken at* '150.

sy

inleanpu

s ocae.do

Thtoons

edth

on-hes,

ick-theics

theters

sshed athe

ings-irec-Inen-iffu-ter-ss

ing

explains the difference between the estimated values ofDz*in the present case and the previous case. Again, this obvation suggests that the separation between the walls plakey role in the stability of the flow.

3. Bounded shear flow of rough particles

In the preceding section, simulations with frictionless,elastic grains at a shear rate of 4 s21 revealed that an initiaconfiguration with random distributions of positions and vlocities of particles may finally evolve to a state whereordered solid phase is formed in a large part of the comtational box as illustrated in Fig. 14~b!. Thus, this configura-tion was selected as a new initial configuration for a seriesimulations, to investigate the effect of phenomenologiparameters on the vanishing of the ordered solid phasshould be noted that the creation of the solid phase is inpendent of the initial random state, but depends mainlythe wall separation distance and the mean solid fraction.view of the initial configuration of particles projected ontheyzplane as well as the variation of the local solid fractiin the direction of the velocity gradient are illustrated in Fig14~b! and 14~c!, respectively. Computations were performto explore the importance of the particle roughness on

02130

er-s a

-

-

-

flIt

e-ne

.

e

vanishing of the ordered structure present in the initial cfiguration. Two different sets of values were used for tsurface parameterse andm, for steel pucks and glass beadas introduced in the preceding section.

One signature of order in the system is the observed stslip motion in the normal stress signal, as discussed inpreceding section. Hence, the absence of stick-slip dynamin the normal stress signal could serve as a criterion ofmelting. Using the suggested values for surface paramee050.93,m50.123, as illustrated in Fig. 16~a!, the meltingof granular crystals occurred aftert* '170. The inset of Fig.16~a! plots the probability densities for the dimensionlenormal stress for both cases after the systems have reacsteady state. The observed shear-induced melting infriction-dominated system may be interpreted considerthe significance of rotational diffusion, which in a monodiperse system is a measure of the rate of change of the dtion of bonds joining a particle with its nearest neighbors.the aforementioned system the conversion of rotationalergy to translational energy enhances the translational dsion, which reduces the bond orientational order. This inesting combined translational-rotational diffusion procemay warrant further investigation. More evidence support

-

FIG. 16. ~a! Dimensionlessnormal stressP* as a function oft* . Inset: The probability densityfunction of the dimensionless normal stress att* .170. ~b! Varia-tions of the solids fraction of asample taken att* '170 withZ* .Inset: Variations of the corre-spondingVx* with Z* at shear rate«54 s21.

3-20

-


FIG. 17. ~a! Variations of P*with t* . Inset: Probability densityfunction of the dimensionless normal stress att* .70. ~b! Varia-tions of the solid fraction of asample taken att* '70 with Z* .Inset: Variations of the corre-spondingVx* with Z* at shear rateof «540 s21. The surface param-eters aree050.93,m50.123, andb050.4.

th

.erehe

nsththanreityin

ble

s

wpas

m

acactitho

na

th

usthisn

ion

ss

stemrenav-res,ingad

tedat-wnedless

thes-is aicle

ntialofif-s de-

entedre

or-

the aforementioned hypothesis is found consideringshorter melting time at an apparent shear rate of 40 s21, asshown in Fig. 17~a!, compared to that illustrated in Fig16~a!. For the same solid fraction and dissipation parametthe rate of melting increases at the higher shear rate, duhigher rotational energy production. The slip velocity at twalls also increases, as shown in the insets of Figs. 16~b! and17~b!.

It can be argued that the order-disorder transition canbe determined reliably by monitoring only the normal streat the wall, which represents a macroscopic quantity ofsystem that may not convey the precise information oflocal structure in the central region of the system. The vishing of the ordered phase may be more reliably monitoby considering the time variations of the invariant quantQ6 , which characterizes the average bond orientationssystem of particles. Initially, the value ofQ6 in the orderedphase@namely, 0.01<Z* <0.2 in Fig. 14~b!# was found to be0.5712, which is very close to the value ofQ6 for a systemwith cuboctahedral symmetry, namely, 0.574. It was oserved that the values ofQ6 decrease sharply for the particroughness parameters, such as those for glass beads orpucks, approaching steady-state values ofQ6 of 0.422 and0.414, respectively, at the shear rate of 4 s21. These valuesfor Q6 imply that the frictional impulse changes the flodynamics between the particles, resulting in a more rarate of vanishing of the ordered solid phase for the glbeads than for the steel pucks.

By comparing these results with those of real systesuch as the results shown in Fig. 1 in@4#, it appears thatneither the simulation performed using the particle surfproperties of glass beads nor that with the particle surfproperties of steel pucks fully reproduces the characterisof real granular systems. In fact, the Gaussian form forobtained probability distribution, as shown in the insetsFigs. 16~a! and 17~a!, will give an entirely incorrect pictureof the large-scale fluctuations observed in the experimeNote that the instantaneous local values of the normal wstresses have been found experimentally to be far largerthe mean, possibly as much as an order of magnitude@4#,which indicates that the occurrence of these events are mmore frequent than what would be predicted with the Gauian form. It turns out that the major features observed inexperiments@4#, as reported in their Fig. 1, such as the extence of instantaneous forces much larger the mean, ca

02130

e

s,to

otsee-d

a

-

teel

ids

s,

ee

csef

ts.llan

chs-e-be

reproduced using a system whose dynamics are collisdominated.

The plot of the probability density for the dimensionlenormal stress signals of Fig. 14~d!, as shown in Fig. 18,suggests that the occurrence of the large forces in the syin which the dynamics are collision dominated is much molikely with the exponential form than with the Gaussiaform. These considerations imply that in the absence of grity, bounded dense granular flows form ordered structuprovided the dynamics are collision dominated. Considerthat the probability distributions of force in stationary bepacks can be described by an exponential function@56#, thedifference between the signals of the collision-dominasystem and the friction-dominated system could well betributed entirely to the existence of a solid region as shoin Fig. 14~b!. More evidence supporting the aforementionconjecture can be found by considering the dimensionnormal stress signals in the quasifluid phase adjacent tobottom wall with the probability distributions of the Gausian form. Note that the aforementioned quasifluid phasephase having some correlations in the orientations of partclusters but is incapable of withstanding shear.

It appears that a hard-sphere model captures the essephysics for describing flow dynamics for a wide rangeflow even at high solid fractions. However, an important dference exists between the sheared granular assembly ascribed in Fig. 14 and the granular flows reported in Ref.@4#.The representative spectra for 2-mm glass beads as presin Fig. 2 of @4# do not display any sign of the periodic natu

FIG. 18. Probability density function of the dimensionless nmal stressP* of Fig. 14~d!.

3-21

o

n

obinangw-aleiea

afopr

th

f-exby

sa

ntelu

toign

intan

an

hatt

ea

icles

nsearwschthe

be-ear-ered

u-em

sed,

-ec-

bleall

ingwnsor-Fig.

esid-wise

.e

o-thehise

a-

am-nis

hlts


in the corresponding signal. Therefore, the nature of the nmal stress signal shown in Fig. 14~d! requires further inves-tigation. To address this concern the signal is further alyzed using a space-scale map of wavelet transforms@57#. Asimple understanding of the wavelet transform can betained by its analogy with the windowed Fourier transformthe similar time-frequency description of any given sign@57#. A wavelet function can be dilated or translated alothe time axist. These operations are characterized by tparameters, namely, scale,a, for dilation and center of wavelet, b, for translation. Any scale addresses a specific packof frequency contents in the wavelet. The stretched wavethat correspond to large scales contain low frequencwhile the compressed wavelets corresponding to small schave a high frequency content.

The wavelet transform of a functionf (t) using the wave-let c is simply defined as@57#

C~a,b!51

AaE f ~ t !cS t2b

a Ddt. ~28!

Here a and b are real numbers that vary continuously incontinuous wavelet transform. A suitable type of waveletthe present case is the Morlet wavelet, which has beenviously used in the analysis of sound patterns@58#. By usingsuch a complex-valued wavelet in the calculation ofwavelet coefficient modulusC(a,b), the spurious oscilla-tions @57# that could appear in visualizing the wavelet coeficient modulus may be eliminated. The mathematicalpression of the complex-valued Morlet wavelet is given@57#

cS t2b

a D51

p1/4expF2~ t2b!2

2a2 GexpF i5~ t2b!

a G . ~29!

As shown in Fig. 19~b!, the modulus of wavelet coefficientof the signal in Fig. 14~c! shows strong analogy to that ofsawtooth sweep over the frequencies in Fig. 19~a!. The saw-tooth signal is in fact a sine wave that is subjected tofrequency decrease beyond a certain time. This frequeslip has been manifested as an increase of scale paramethe corresponding map of the wavelet coefficient modunamely, larger scales represent lower frequencies. Thusexistence of higher frequencies can also be detected, shas the horizontal green thin line at a lower scale in F19~b!. Note that the scale axis of the wavelet coefficiemodulus map in Fig. 19~a! is identical for the rest of maps inFig. 19. The origin of the slight period changing shownthis figure might be associated with some dynamical insbility in the sheared granular flow. These long lived frequecies have not been observed in the continuous wavelet trform of the signal in Fig. 16~a! corresponding to thefrictional model, as shown in Fig. 19~c!. Here, the wavelettransform of Gaussian white noise is shown in Fig. 19~d! forcomparison. In light of the above, it may be concluded tthe nature of the stick-slip motion in the annular Coueapparatus of@4# could be different from that found in thsimulations. Apparently, the periodic nature of the signal

02130

r-

a-

-

l

o

getss,les

re-

e

-

acyr ins,hewn.t

--s-

te

s

illustrated in Fig. 14~d! could be associated with the periodshear-induced melting and reordering of the layer of particadjacent to the wall.

B. Effects of gravity

As speculated by Thompson and Grest@2#, the origin ofstick-slip dynamics could be periodic dilatancy transitioand gravitational compactification. They found that at shrates below a certain critical value, sheared granular floare unstable to gravitational compactification. For sucases, only a portion of the material was sheared, whilerest of the material appeared to be in rigid-body motionlow the shearing grains. This behavior gave rise to a shinduced phase boundary between ordered and disordstates.

In order to reproduce the aforementioned behavior, simlations were carried out using a collision-dominated systin the presence of the gravitational acceleration,g m/s2, withthe initial configuration as shown in Fig. 14~b!. In the simu-lations of this section, about 4300 glass beads were uwith a diameter of s55 mm and the density ofr52500 kg/m3. The width, length, and height of the computational box were selected to be 10, 10, and 5 cm, resptively. The value of the dimensionless parameters«2/g is setto 0.5 and the coefficient of restitution is set toe050.84. Sothat the simulations would as closely as possible resemthe experiments, the bottom wall was fixed and the top wwas set in motion with the velocity ofV0x522.5 cm/s pro-ducing an apparent shear rate of«54.5 s21. In the presenceof gravity, after a very short timet* '3, the ordered solidphase, whose density is higher than that of the surroundquasifluid phase, completely migrated downward, as shoin Fig. 14~e!. After the settling time, a steady motion waobserved where three layers of particles flowed over thedered solid phase. These layers are shown in the inset of20~a!, which plots the local solid fraction as a function ofZ.For these layers located in the regionZ.3.2 cm, it is foundthat the value ofQ6ANb is 1.85. For this value, the phasthat is formed over the ordered solid phase may be conered as a disordered phase. The profiles of the streamvelocity component and granular temperature in thez direc-tion for samples taken aftert* .3 are presented in Figs20~a! and 20~b!, respectively. At the low shear rates, thconstant-volume simulation with only the upper wall in mtion results in a decay in the height of the fluid phase topoint where no shearing motion is possible. To prevent tfrom occurring, simulations were performed in which thupper wall was maintained at a constant load ofW'3.0 g N and was connected by a Hookian spring withspring constantK to a driving motor and pushed with a constant velocityV0x in the horizontal direction.

The simulations were continued using the surface pareterse050.84,m50.41. As shown in the preceding sectioof simple shear flow of rough particles, a chainlike clusterlikely to be formed in this system. After 108 collisions thegeneralized version of the nonlinear model~20! in which theeffect of gravity is taken into account was utilized, whicshould be valid at any solid fraction. The preliminary resu

3-22


8 s.

.

FIG. 19. ~Color! Continuous wavelet transform.~a! Modulus of wavelet coefficients of a signal of period changing from 0.02 to 0.02~b! Modulus of wavelet coefficients of time series ofP* at shear rate of«54 s21 and the surface parameters ofe050.93 andm50. ~c!Modulus of wavelet coefficients of signal in Fig. 17~a! at 100,t* ,150. ~d! Modulus of wavelet coefficients of a Gaussian white noise

th

mioiro

p-d

heinunth

ionry

elythecan

tionor-asef the

fore,m-

of the dimensionless normal stress at the bottom wall insteady-state condition are shown in Fig. 20~c!. Apparentlythe signal is somewhat nonperiodic and the stick-slip dynaics can clearly be observed in the time series of dimensless normal stress. However, longer simulations are requto obtain a more refined picture of the shearing motiongranular materials.

It is worth mentioning that the vertical motion of the uper wall in the opposite direction of gravity was observeThis motion, which could be due to the dilatancy of tgranular material, might be linked to the formation of chain the granular assembly. Convincing evidence may be foby considering the presence of an ordered solid phase

02130

e

-n-edf

.

sdat

can withstand shear. The profiles of the local solid fractand the local velocity, as illustrated in Fig. 20, should be vesimilar to those for the experiments of@29# in which thedilatancy of granular assembly was reported. It is quite likthat in their system the nonshearable section adjacent tobottom was ordered. The signature of an ordered phasealso be seen in the background of the inset of Fig. 1 in@4#.Therefore, it may be speculated that during the compacperiod, while the upper wall moves downward, the disdered layers of particles flowing over the ordered solid phare compressed. At moderate shear rates, the thickness odisordered layers is about three particle diameters. Therea number of chainlike clusters could easily form in the co

3-23

ess.

a function

e as used


FIG. 20. ~a! Dimensionless velocityV* as a function of distance in thez direction for glass beads under gravitational compaction procThe top wall was set in motion horizontally with a velocity ofV0x522.5 cm/s producing a shear rate of«54.5 s21 with s«2/g'0.5. Thethickness of the nonsharable region is about 3.2 cm and the thickness of the disordered phase is 1.5 cm. Inset: Solid fraction asof distance in thez direction.~b! Dimensionless translational granular temperatureTz* as a function of distance in thez direction. The solidlines are nonlinear fits.~c! Typical time series for the dimensionless stressP* as a function of dimensionless timet* obtained using thegeneralized version of the nonlinear model in which the effect of gravity is taken into account. The simulation parameters are samin ~a!.

tmato

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mchele

ati

re

la6

teo

.58dy-

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ra-reis-emsilityalseddthedy-tion

slipionRe-in

sem-

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pressed disordered layers between the upper wall andordered phase, resulting in the cessation of the shearingtion. In order to have continuous shearing, the granularsembly would then have to dilate. It is of great interestfind out whether chainlike clusters would form during tdilation period. To this end the processes of formation adisintegration of clusters were carefully monitored. It apears that some arches could also form during the dilaperiod at the moderate shear rates mentioned above. Hever, at high shear rates where the thickness of the disordlayers is quite large, a more smooth continuous shearingtion would be expected. These observations enhance thederstanding of the complex processes involved in the expments of Refs.@4,29#. The stick-slip phenomenon in granulaflows is an interesting phenomenon that merits further invtigation.

V. CONCLUSIONS

The shear-induced motion of granular assemblies cprised of a large number of viscoelastic, monosized, sphcal particles in both unbounded and bounded systemsinvestigated using computer simulations. The solid fractwas varied from a low density system of 0.16 to a very hdensity system of 0.59. Two different models were usedpredicting the flow dynamics of granular assemblies, namthe modified hard-sphere model in which collisions betwethe grains are assumed to be instantaneous, and thecomputationally intensive particle dynamics model in whithe collisions have finite durations. A Hertzian-type modfor the treatment of the viscoelastic behavior of the particin contact was also introduced.

The modified hard-sphere model was shown to replicreasonably well some of the results predicted by kinetheory@25# at low and moderate solid fractions. Furthermothis model was found to predict the basic features suchparticle lateral diffusive motion in real sheared granuflows, even at a high value of the solid fraction of 0.5However, it was found that it would be difficult to simulacomplex phenomena, such as jamming in systems with c

02130

heo-s-

d-nw-redo-n-

ri-

s-

-ri-asnhry,nore

ls

tec,asr.

n-

stant volume, at values of the solid fraction larger than 0using hard-sphere models. For these systems, the particlenamic model was found to more closely predict the formtion of clusters when nonlinear viscoelastic grain behavwas assumed, as opposed to previously proposed linear mels @2,9,10#. An even more reasonable representation of clter formation was found by including the effects of surfafriction in the model. Comparing the simulation results withe results of experiments in which force fluctuations wemeasured@4#, it was found that a nonlinear model for thviscoelastic frictional grains could predict the behavior of tchainlike clusters more precisely than a linear model. Hoever, neither frictionless nor frictional particle models wecapable of predicting observed particle ordering@6# in un-bounded flows. It is likely that in the system in which thwall effects are not significant, the sheared granular assbly transforms to a glass state at solid fractions above 0

Particle ordering was predicted using the modified hasphere model for simulations of frictionless granular systein bounded shear flows even in the absence of gravity. It walso found that the local solid fraction and the wall sepation distance govern the stability of the flow. Values wedetermined for the solid fraction and the wall separation dtances at which a transition to order occurs. In these systthe presence of an exponential behavior of the probabdistribution function was found in the normal stress signexerted on the wall, consistent with experimentally observforce fluctuations@4#. Wavelet transform analysis was useto show the existence of a characteristic frequency innormal stress signal, indicating the presence of stick-slipnamics. For the range of shear rates studied, the simularesults for the rough model did not show the same stick-behavior due to the vanishing of the ordered solid regcaused by the presence of strong frictional interactions.call that in@6# no order was observed using rough particlesthe experiments.

The diffusive behavior of particles at moderately densolid fractions in bounded flows was analyzed using a coprehensive model based on the revised Enskog theor

3-24

aentr-ys

arialew

raroth

r tded

disinondeo

elstor

ro

ov

pleTontchb

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orfu

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enceheted.to

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a

nc-ecol-

blyn,mp-

oralar-edew

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granular fluids and on Grad’s method of moments. Formexpressions were derived for the particle self-diffusion coficient, particle thermal diffusion coefficient, and coefficieof mobility. Additional theoretical study is required to chaacterize further the particle diffusive motion in dense stems where cluster diffusion seems to be important.

Considering the ratio of the impact duration and the mefree time, the hard-sphere model was found to be appropfor solids fractions below 0.56. Above this value, a particdynamics model is recommended. However, it was shothat the hard-sphere model could provide a rough picturethe shearing flow of the granular assembly even at solid ftions above 0.56. It was shown that the rough picture pvided by the hard-sphere model could be refined usingnonlinear particle dynamics model. For example, in ordeconsider the effect of gravitational compactification, a stuwas performed in which gravity was considered and the vtical motion of the upper wall was allowed. The modifiehard-sphere model correctly predicted the formation of aordered layer of particles over the ordered solid phaseshearing motion of a granular assembly. The simulatiwere continued using a nonlinear particle dynamics mowhich predicted the cluster formation more precisely. Fthese conditions chain formation was found to be quite likin the disordered layers for frictional particles. The intereing stick-slip dynamics could be clearly observed in the nmal stress signal for the bottom wall.

Fresh interpretations were given for the complex pcesses observed in experiments of Refs.@4,29#. Additionalstudy is required to characterize further the flow dynamicsa granular assembly. Such a study should lead to the deopment of advanced theories capable of describing comprocesses in granular flows, which are of real significancfacilitating the design of practical granular flow systems.complement the simulation studies, additional experimeresults should be obtained, for which vital information suas solid fractions and particle surface properties shouldreported. The absence of such information limits thevances in the knowledge of the physics of granular flows tcould be obtained through comparisons of experimentalnumerical results.

ACKNOWLEDGMENTS

The authors would like to thank Dr. Goodarz Ahmadi fproviding the results shown in Fig. 1. They are also grateto Dr. Donald Gugan for helpful discussions. Computitime provided by the Finnish Center for High-PerformanComputing and Networking is gratefully acknowledged. Packnowledges the support of the Finnish Academy, GrNo. 74846.

APPENDIX

Consider a granular material consisting of a binary mture of differently colored smooth, spherical particles. Fthis system, light and dark colors are represented by sscripts l and d, respectively. The particles, which have thsame massesm and radiis but different number densitiesnl

02130

lf-

-

nte

nofc--e

oyr-

-asl,ry--

-

fel-exin

al

e-td

l

.nt

-rb-

andnd, are sheared at a high shear rate. Due to the presof gradients in the mean flow, random fluctuations in tlocal mean translational motion of the grains are generaSince the particles in a rapid shear flow behave similarmolecules in a dense gas, methods from the kinetic theordense gases@59# might be used to describe the motionparticles in this system. For a nonequilibrium granular fluin the limit of two-body forces between the particles, tprobability that at some instant of timet a particle of kindlwill be at locationr and have momentump within the re-spective limitsdr anddp may be governed by the followingequation@52#:

] f l ~1!

]t1cl

•

] f l ~1!

]r1Fl

•

] f l ~1!

]c52E E X l ,m

•

] f lm~2!

]cdrmdpm.

~A1!

Herecl is the instantaneous velocity of a particle of kindl,X lm is the force on a particle of kindl due to all other par-ticles, andFl is the force per unit mass acting on a particlekind l due to external fields.

Equation ~A1! involves the pair distribution function

f lm(2), which is the probability that there is simultaneously

particle of kindl in the space element (dr ldpl) about (r l ,pl)and another particle of kindm, which can be either light ordark, in the space element (drmdpm) about (rm,pm). Thus, itis needed to find an expression for the pair distribution fution in terms of single particle distribution functions. If thparticle contacts can be regarded as nearly instantaneouslisions, the distribution functions do not change appreciain a time interval comparable to the duration of a collisiothe gradients of inhomogeneities are small, and the assution of particle chaos remains valid, Eq.~A1! reduces to thegeneralized Boltzmann equation which governs the tempchange of the single particle distribution function of the pticle of kind l. Since the particle diffusion processes involvin a granular fluid are of interest in the present study, the nvariableCl5cl2V, which is the particle peculiar velocity, iintroduced in place ofcl into Eq. ~A1!. As a result, the fol-lowing kinetic equation is obtained@59#:

d fl ~1!~r ,C,t !

dt

5F2Cl•

]

]r1S ]u

]r D :S Cl]

]CD1S du

dt•

]

]C2Fl

]

]CD G3 f l ~1!

~r ,C,t !1 (m5 l

d E E s2@glm~r ,r1sku$ns%!

3 f l ~1!~r ,Cl ,t ! f m~1!

~r1sk,Cm,t !

2glm~r ,r2sku$ns%! f l ~1!~r ,Cl ,t ! f m~1!

~r2sk,Cm,t !#

3~clm•k!H~clm

•k!dk dcm, ~A2!

3-25

,

e-g

inths

onnh

tioteflothx

uich

vol-

e


Hered/dt5]/]t1V•]/]r is the substantial time derivative$ns%5$nl ,nd% are solid component densities,V5(m5 l

d mmcm/(m5 ld mm is the mean mass velocity of th

mixture, clm5cl2cm is the relative velocity of the two particles of kindl and kindm, k is the unit vector directed alonthe line from the center of the particle of kindm to the centerof the particle of kindl at contact, andH is the Heavisidestep function. Also, factorsglm are introduced to take intoaccount the difference in position of the colliding particlesbinary collision dynamics, and the resulting increase infrequency of collisions. Following van Beijeren and Ern@60#, glm are chosen to be the mixture radial distributifunctions at contact, which are evaluated as nonlocal futionals of the density fields of the two components in tgranular fluid mixture. Thus,

glm„r ,r6sku$ns~r !%…

5gclm„su$ns~r !%…

1(q5 l

d E drqHlmq„r ,r6sk,rqu$ns~r !%…~rq2r !•

]nq

]r

1O~¹2!. ~A3!

In the above equation,gclm is the equilibrium value of the

radial distribution function at contact for particles of kindlandm:

gclm~su$ns~r !%!

511 (q51

d

nq~r !E Vlmp~r ,r6skurq!drq1¯ .

~A4!

Here, Hlmq„r ,r6sk,rqu$ns(r )%…5Vlmq(r ,r6skurq)

1(q851m nq8(r )*Vlmqq8(r ,r6skurqrq8)drq81¯ , which is

symmetric under the interchange of superscriptl and m.

Also, V(r ,r6skurq) and V(r ,r6skurqrq8) represent Hu-simi functions. As discussed earlier, the above approximafor the radial distribution function at contact is not quiappropriate for the present study of a sheared granularwhere a spatial preference for collisions is imposed byvelocity field. However, there is presently no better appromation available.

To derive the hydrodynamic equations of a granular flmixture, which are valid outside the time regime for whiEq. ~A2! is exact. This equation is first multiplied bymc l ,wherec l is any property of the particles of kindl. The re-sulting equation is then integrated over the instantaneouslocity cl to give the equation of change for the assemblyparticles of kindl in terms of the mass-weighted mean vaues:

02130

et

c-e

n

we

i-

d

e-f

d

dt~rm

l F l c l !1rml F l c l

]

]r•V

5rml F l

dc l

dt2

]

]r•~rm

l F lClc l !2rml F l

dV

dt•

]c l

]C

2rml F l S ]V

]r D :Cl]c l

]C1rm

l F lFl•

]c l

]C

1rml F lCl

•

]c l

]r1 (

m51

d K x lm~c l !2]

]r•Q lm~c l !

21

2 E E E ms3F S k]~c l 82c l !

]Cl D :S ]V

]r D G3~clp

•k!H~clp•k! f m~1!

~r ,Cm,t ! f l ~1!~r ,Cl ,t !gc

lp

3H 11s

2k•

]

]r F lnf m~1!

~r ,Cm,t !

f t~1!~r ,Cl ,t !

G1¯J dk dcldcmL . ~A5!

Here rml and F l are the material density and solid volum

fraction of the particles of kindl. Expressions for theparticle-particle collisional fluxQ lm and the sourcelike termx lm are given as follows:

Q lm~c!521

2E E E m~c l 82c l !ks3~clm•k!H~clm

•k!

3gclmf m~1!

~r ,Cm,t ! f l ~1!~r ,Cl ,t !

3H 111

2sk•

]

]r F lnf m~1!

~r ,Cm,t !

f ~ l !~1!~r ,Cl ,t !

G1¯J3dk dcldcm,

x lm~c!51

2E E E m~c l 81cm82c l2cm!

3s2~clm•k!H~clm

•k!gclmf m~1!

~r ,Cm,t ! f l ~1!

3~r ,Cl ,t !

3H 111

2sk•

]

]r F lnf m~1!

~r ,Cm,t !

f l ~1!~r ,Cl ,t !

G1¯J3dk dcldcm for m5 l , ~A6!

x lm~c!5E E E m~c l 82c l !s2~clm•k!H~clm

•k!gclm

3 f m~1!~r ,Cm,t ! f l ~1!

~r ,Cl ,t !H 11 12 sk•

]

]r

3F lnf m~1!

~r ,Cm,t !

f l ~1!~r ,Cl ,t !

G1¯J dk dcldcm

for mÞ l .

3-26

an

ic

ho

ixelms

f

bytion

ofbyare


The equations of conservation of mass, momentum,energy for a granular assembly of kindl may be derived bytaking c l in Eq. ~A5! to be 1,Cl , andCl 2, respectively.

~i! Balance of mass

dr l

dt1r l

]

]r•V1

]

]r•~r lul !50, ~A7!

wherer l5rml F l is the apparent density andul is the diffu-

sion velocity of particles of kindl.~ii ! Balance of linear momentum

r ldV l

dt2V l

]

]r•~r lV l !52

]

]r•Pl1r l Fl2r l

dV

dt2r l

]V

]r•V l

1 (m51

d

x lm~Cl !, ~A8!

whereV l andPl are the velocity vector and the macroscoppressure tensor of particles of kindl, respectively.

~iii ! Balance of kinetic energy~pseudothermal energy!

3

2H nldT

dt2T

]

]r•~nlCl !J 52

]

]r•ql2g l2P l :

]V

]r1r lFl

•Cl

2r lCl•

dV

dt, ~A9!

whereT51/3Cl 2 is the mixture pseudothermal energy,ql isthe energy flux vector of particles of kindl, andg l is the rateof energy dissipation of particles of kindl per unit volume ofthe mixture due to the inelastic nature of the collisions. Teffect of the thermal energy on the particle elasticity is nconsidered in the present treatment.

Assuming that the rapid shearing motion of a binary mture of differently colored granular fluids can be adequatdescribed by the consideration of the 13-moment approxition @61#, expressions for the macroscopic pressure tenPl , energy flux vectorql , and sourcelike termsx lm(Cl) andg l are given as follows@25#:

Pl5~r lTI1r la†l !1 (m51

d

~11e0!r lnms3gclmH p

3TI

1p

15~a†l1a†m!2

4

15s~nT!1/2F ]V%

+

]r1

5

6S ]

]r•VD IG J ,

~A10!

ql5 52 r l~Tvl1 1

5 al !1 (m5 l

d

s3gclm~11e0!K 2s~nT!1/2

3H m

6T~12e0!S nm

]nl

]r2nl

]nm

]r D1 13 nmr l

]T

]r J1pnmr lT@ 1

6 ~um1ul !1 14 ~12e0!~um2ul !#

02130

d

et

-ya-or

1p

20nmr l@~am1al !1 1

2 ~12e0!~am2al !#L , ~A11!

x lm~C!5s2~11e0!gclmr lnmH p

3sF2T

nl

nm

]

]r S nm

nl D G2 4

3 ~nT!1/2~ul2um!1 115 S p

T D 1/2

~al2am!J ,

~A12!

g l5 (m5 l

d

2s2~12e02!gc

lmnmr lTF ~pT!1/22p

4s

]

]r•uG .

~A13!

In the above,a†m is the pressure deviator,am is the transportpseudothermal energy flux vector,e0 is the coefficient of

restitution, and]V%+

/]r is the nondivergent symmetric part othe mean mass mixture velocity-gradient tensor.

For steady shear flow, the energy supplied to the grainscontinuous shearing is balanced by the energy dissipadue to inelastic collisions, and Eq.~A9! reduces to the fol-lowing form:

05 (m5 l

d H rma^xy&†m 1(

p5 l

d

~11e0!rmnps3gcmpF p

15~a^xy&

†m

1a^xy&†p !2 2

15 s~pT!1/2«~z!G J «~z!

1 (m5 l

d

(p51

d

2rms2~12e02!gc

mpnpT~pT!1/2. ~A14!

In the above equation,«(z) is the local rate of shear.Expressions for the pressure deviatorsa^xy&

†l anda^xy&†d are

given as follows:

(p51

d

s2~11e0!gcpnnprn~pT!1/2^$ 2

3 1 15 @21~12e0!#%a^xy&

†n

1$ 15 @21~12e0!#2 2

3 %a^xy&†p &

52rnTK 11 (p51

d

ps3~11e0!gcnpnp

3$2 15 @11~12e0!#1 1

3 %L «~z!, n5 l ,d. ~A15!

Equation~A15! can be inserted into Eq.~A14! to obtain Eq.~15! for the mixture. It is worth mentioning that Eq.~A15!may be derived from the balance law for the deviant partthe mean of the second moment of velocity fluctuation,assuming that the spatial gradients of the mean fieldssmall, and that the dimensionless quantitiess/L, s/t0T0

1/2,ul /T0

1/2, ud/T01/2, a†l /T0 , a†d/T0 , al /T0

1/2, andad/T01/2 are all

3-27

la

hehem

n

hte-ux


of the same order of magnitude and small. Here,L, t0 , andT0 are the characteristic length, time, and mixture granutemperature, respectively.

Assuming thatuzl 'ux

l !(uxl 2ux

d), and that (uxl 2ux

d)T1/2

and (axl 2ax

d)T21/2 are of the same order of magnitude, in tpresence of nonuniform shear stress in the direction of sgradient, namely, thez direction, the balances of momentu~A7! in the direction of the shear plane for particles of kindl,for which nl@nd'n, may be reduced to

r luzl 5

]

]z H r la^xz&†l 1(

p5 l

d

~11e0!r lnps3gcF p

15~a^xz&

†l 1a^xz&†p !

2 215 s~nT!1/2«~z!G J . ~A16!

din

S

.

. Eid

da

Y.

a

d

Jr

v.

if-

02130

r

ar

Substituting fora^xx&†l and a^xx&

†d from Eq. ~A15!, assumingthat the parameterT/ «(z)2 does not change significantly ithe direction of the shear gradient@62#, gives a reduced formof this drift,

r luzl 52F sT1/2

15p1/2~32e0!«~z!2G ]

]z~r lU !, ~A17!

where the term in the brackets,B5sT1/2/15p1/2(32e0) «(z)2, is the particle mobility and U5$2(11e0)gcFs@1813e0(221p)2p#15p%«(z)2 is a potentialfield, which is a function of position. The term on the rigside of Eq.~A17! represents the contribution of inhomogneous shear to the drift as well as the extra diffusive flarising from viscosity gradients.

es,ionsles.by

-

A.

d

J.

l-

i-v.,

.

thevia

@1# M. E. Cates, J. P. Wittmer, J.-P. Bouchaud, and P. ClauPhys. Rev. Lett.81, 1841~1998!.

@2# P. A. Thompson and G. S. Grest, Phys. Rev. Lett.67, 1751~1991!.

@3# C.-H. Liu, S. R. Nagel, D. A. Schecter, S. N. Coppersmith,Majumdar, O. Narayan, and T. A. Witten, Science269, 513~1995!.

@4# B. Miller, C. O’Hern, and R. P. Behringer, Phys. Rev. Lett.77,3110 ~1996!.

@5# P. Pieranski, Am. J. Phys.52, 68 ~1984!; J. S. Olafsen and J. SUrbach, Phys. Rev. Lett.81, 4369~1998!; G. Strassburger andI. Rehberg, Phys. Rev. E62, 2517~2000!.

@6# G. Ahmadi~private communication!; more details including ageometrical description of the apparatus may be found in KElliott, G. Ahmadi, and W. Kvasnak, J. Non-Newtonian FluMech.74, 89 ~1998!.

@7# H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Phys. To49~4!, 32 ~1996!.

@8# N. Menon and D. J. Durian, Science275, 1920~1997!.@9# P. A. Cundall and O. D. L. Strack, Geotechnique29, 47 ~1979!;

K. Yamne, M. Nakagawa, S. A. Altobelli, T. Tanaka, andTsuji, Phys. Fluids10, 1419 ~1998!; C. M. Dury and G. H.Ristow, Europhys. Lett.48, 60 ~1999!.

@10# P. K. Haff, in Granular Matter, edited by Anita Mehta~Springer-Verlag, New York, 1994!.

@11# B. J. Alder and T. Wainwright, J. Chem. Phys.31, 459 ~1959!.@12# C. K. K. Lun and A. A. Bent, J. Fluid Mech.258, 335 ~1994!;

C. S. Campbell,ibid. 348, 85 ~1997!; P. Zamankhan, T. Tyn-jala, W. Polashenski, Jr., P. Zamankhan, and P. SarkomPhys. Rev. E60, 7149~1999!.

@13# C. Bizon, M. D. Shattuck, J. B. Swift, W. D. McCormick, anH. L. Swinney, Phys. Rev. Lett.80, 57 ~1998!.

@14# M. D. Rintoul and S. Torquato, J. Chem. Phys.105, 9258~1996!; P. Zamankhan, H. Vahedi Tafreshi, W. Polashenski,P. Sarkomaa, and C. L. Hyndman,ibid. 109, 4487~1998!.

@15# P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Re28, 784 ~1983!.

@16# At short times, namely, before collisions occur, a particle dfuses as a free particle andDr2(t)& grows ast2. At longtimes, after many collisions,Dr2(t)& grows ast, where the

,

.

.

y

a,

.,

B

Einstein relation, namely,Dr2&56tD, is obeyed. Here,Drepresents the coefficient of self-diffusion that characterizon average, the dissipation of spontaneous density fluctuatproduced by local aggregation and redispersion of particThe dynamics of density fluctuations may be monitoredcalculating of the self-space-time correlation functionGs(r ,t),which involves features of the radial distribution function combined with features of time correlation functions.

@17# P. Zamankhan, W. Polashenski, Jr., H. Vahedi Tafreshi,Shakib Manesh, and P. Sarkomaa, Phys. Rev. E58, R5237~1998!.

@18# J. Bridgwater, inGranular Matter ~Ref. @10#!.@19# A. Einstein, Ann. Phys.~Leipzig! 17, 549 ~1905!.@20# N. Menon~private communication!.@21# V. V. R. Natarajan, M. L. Hunt, and E. D. Taylor, J. Flui

Mech.304, 1 ~1995!.@22# C. Denniston and H. Li, Phys. Rev. E59, 3289~1999!.@23# S. B. Savage and R. Dai, Mech. Mater.16, 225 ~1993!.@24# J.-P. Hansen and I. R. McDonald,Theory of Simple Liquids

~Academic, New York, 1986!.@25# P. Zamankhan, Phys. Rev. E52, 4877~1995!.@26# J. M. Kincaid, E. D. G. Cohen, and M. Lopez de Haro,

Chem. Phys.86, 963 ~1987!; P. Zamankhan, Acta Mech.123,235 ~1997!; B. O. Arnarson and J. T. Willits, Phys. Fluids10,1324 ~1998!.

@27# P. C. Johnson and R. J. Jackson, J. Fluid Mech.176, 67 ~1987!.@28# G. Boffetta, V. Carbone, P. Dgiuliani, P. Veltri, and A. Vu

piani, Phys. Rev. Lett.83, 4662~1999!.@29# D. M. Hanes and D. L. Inman, J. Fluid Mech.150, 357~1985!.@30# R. G. Horn and J. N. Israelachvili, J. Chem. Phys.75, 1400

~1981!; D. Y. C. Chan and R. G. Horn,ibid. 83, 5311~1985!;J. N. Israelachvili, P. M. McGuiggan, and A. M. Homola, Scence240, 189~1988!; J. Van Alsten and S. Granick, Phys. ReLett. 61, 2570~1988!; R. G. Horn, S. J. Hirz, G. HadziioannouC. W. Frank, and J. M. Catala, J. Chem. Phys.90, 6767~1989!;M. L. Gee, P. M. McGuiggan, J. N. Israelachvili, and A. MNomola, ibid. 93, 1895~1990!; P. A. Thompson and M. Rob-bins, Science250, 792 ~1990!.

@31# See EPAPS Document No. E-PLEEE8-66-084208 foraforementioned appendix. This document may be retrieved

3-28

/e

of

rs

s-

h

s

S.

J.

fe,


the EPAPS homepage ~http://www.aip.org/pubservsepaps.html! or from ftp.aip.org in the directory lepapsl. See thEPAPS homepage for more information.

@32# W. A. M. Morgado and I. Oppenheim, Phys. Rev. E55, 1940~1997!.

@33# N. V. Brilliantov, F. Spahn, J. M. Hertzsch, and T. Po¨schel,Phys. Rev. E53, 5382~1996!.

@34# W. Goldsmith,Impact: The Theory and Physical BehaviourColliding Solids~Arnold, London, 1960!.

@35# T. Schwager and T. Po¨schel, Phys. Rev. E57, 650 ~1998!.@36# M. P. Allen and D. J. Tildesley,Computer Simulation of Liq-

uids ~Clarendon, Oxford, 1997!.@37# C. K. K. Lun and S. B. Savage, J. Appl. Mech.54, 47 ~1987!.@38# C. A. Coulomb, Acad. R. Sci. Mem. Math. Phys. Par Dive

Savants7, 343 ~1773!.@39# J. T. Jenkins and M. W. Richman, Phys. Fluids28, 3485

~1985!.@40# D. Gugan, Am. J. Phys.68, 920 ~2000!.@41# A. E. H. Love,A Treatise on the Mathematical Theory of Ela

ticity ~Dover, New York, 1944!.@42# K. J. Johnson,Contact Mechanics~Cambridge University

Press, Cambridge, 1987!.@43# A. W. Lees and S. F. Edwards, J. Phys. C5, 1921~1972!.@44# W. Loose and G. Ciccotti, Phys. Rev. A45, 3859~1992!.@45# A. Goldshtein, M. Shapiro, and C. J. Gutfinger, J. Fluid Mec

316, 29 ~1996!.@46# C. K. K. Lun and S. B. Savage, Acta Mech.63, 15 ~1986!.@47# T. G. Drake, J. Fluid Mech.225, 121 ~1991!.

02130

.

@48# H. S. Carslaw and J. C. Jaeger,Conduction of Heat in Solids~Clarendon, Oxford, 1976!.

@49# F. G. Bridges, A. Hatzes, and D. N. C. Lin, Nature~London!309, 333 ~1984!.

@50# J. R. Dormand and P. J. Prince, J. Comput. Appl. Math.6, 19~1980!.

@51# J. W. Dally and W. F. Riley,Experimental Stress Analysi~McGraw-Hill, Tokyo, 1978!.

@52# J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird,MolecularTheory of Gases and Liquids~Wiley, New York, 1954!.

@53# A. Rahman, Phys. Rev. Lett.32, 52 ~1974!.@54# B. Miller, Ph.D. thesis, Duke University, 1997, p. 54.@55# P. A. Thompson and M. O. Robbins, Science250, 792 ~1990!.@56# C. H. Liu, S. R. Nagel, D. A. Schecter, S. N. Coppersmith,

Majumdar, O. Narayan, and T. A. Witten, Science269, 513~1995!.

@57# M. Farge, Annu. Rev. Fluid Mech.24, 395 ~1992!; I.Daubechies,Ten Lectures on Wavelets~SIAM, Montpelier,Vermont, 1992!.

@58# R. Kronald-Martinet, J. Morelet, and A. Grossmann, Int.Pattern Recognit. Artif. Intell.1, 273 ~1987!.

@59# S. Chapman and T. G. Cowling,The Mathematical Theory oNon-Uniform Gases~Cambridge University Press, Cambridg1970!.

@60# H. van Beijeren and M. H. Ernst, Physica~Amsterdam! 68,437 ~1973!.

@61# H. Grad, Commun. Pure Appl. Math.2, 331 ~1949!.@62# S. S. Hsiau and M. L. Hunt, J. Fluid Mech.251, 299 ~1993!.

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Fine structures in sheared granular flows

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