Unsteady Couette granular flows

Unsteady Couette granular flowsMarijan Babic

Citation: Physics of Fluids (1994-present) 9, 2486 (1997); doi: 10.1063/1.869367 View online: http://dx.doi.org/10.1063/1.869367 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/9/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Unsteady MHD two-phase Couette flow of fluid-particle suspension in an annulus AIP Advances 1, 042121 (2011); 10.1063/1.3657509 Nonmodal energy growth and optimal perturbations in compressible plane Couette flow Phys. Fluids 18, 034103 (2006); 10.1063/1.2186671 Instability of the Plane Couette Flow by the Ghost Effect of Infinitesimal Curvature AIP Conf. Proc. 762, 258 (2005); 10.1063/1.1941547 Hydrodynamic stability of a suspension in cylindrical Couette flow Phys. Fluids 14, 1236 (2002); 10.1063/1.1449468 Some qualitative features of the Couette flow of monodisperse, smooth, inelastic spherical particles Appl. Phys. Lett. 71, 3790 (1997); 10.1063/1.120507

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Unsteady Couette granular flowsMarijan BabicDepartment of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame,Indiana 46556

~Received 6 November 1996; accepted 7 May 1997!

Unsteady Couette granular flows are investigated both theoretically and by the means of the discreteelement method~DEM!. Theoretical formulation of the problem is based on the kinetic theory forsmooth, nearly elastic particles. The resulting boundary value problem is solved numerically for twoparticular types of flows:~a! transient Couette flows, in which the wall velocity instantaneouslychanges from one constant value to another, and~b! cyclic Couette flows, in which the wall velocityis a harmonic function of time. The behavior of granular fluids in these flows is found to criticallydepend on the ratios of time scales that characterize the rate of change of the wall velocity andprocesses of momentum diffusion and energy relaxation. The limiting case in which the momentumdiffusion time scale is much smaller than the energy relaxation time scale is also analyzedanalytically by perturbation methods. In this limit granular materials behave as nearlyincompressible non-Newtonian fluids. Hence, under appropriate conditions, which correspond tosmall Mach numbers, mathematical analysis of rapid granular flows can be tremendously simplified.For both transient and cyclic Couette flows, predictions of the kinetic theory for disks are found tobe in a very good agreement with the corresponding DEM simulations at all time scales as long asthe coefficient of restitution is close to unity. However, as the coefficient of restitution decreases, theagreement deteriorates due to a gradual breakdown of assumptions used in the development of thenearly elastic kinetic theory. ©1997 American Institute of Physics.@S1070-6631~97!01109-4#

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I. INTRODUCTION

Granular materials are two-phase systems comprisediscrete solid particles dispersed in an interstitial fluid.many situations involving dry, coarse particulate solids,interstitial gas plays a minor role in the mechanics ofmaterial and can be neglected. Granular materials arepable of exhibiting both solid-like and fluid-like behavioThe fluid-like behavior of granular materials, which is commonly referred to as rapid granular flows, has received csiderable attention in recent years due to its numerous intrial and geophysical applications~Campbell,1 Savage2!.

The pioneering work on the rapid granular flows is cosidered to be Bagnold’s3 experimental study of rapidlysheared dense particulate suspensions. He observed thhigh bulk densities and shear rates both normal and sstresses were proportional to the square of the applied srate~grain-inertia regime!, while at lower bulk densities andshear rates the interstitial fluid effects became importantthe suspensions exhibited the Newtonian fluid behav~macroviscous regime!. He identified the principal stressgenerating mechanism in the grain-inertia regime to betransport of momentum in interparticle collisions and pposed a simple semiempirical theory that correctly predicthe quadratic dependence of the stresses on the shearThere have been several more recent experimental stuconcerned with the phenomenological behavior of dry pticulate materials subjected to steady shear flows in shcells ~e.g., Savage and Sayed,4 Hanes and Inman,5 Craig,Buckholtz, and Domoto6!. These experiments indicate ththe quadratic dependence of the stresses on the sheapersists throughout the range of concentrations over whthe particles interact primarily via brief, binary collisionFor extremely high concentrations and relatively low sh

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rates the particles are in continuous contacts with their nebors and the material exhibits a solid-like behavior~quasi-static regime!. The resulting shear stress versus shearrelationship depends on whether the shear cell experimeconducted under constant volume or constant normal stconditions. The constant-volume experiments yield a powlaw relationship in which the shear stress is proportionathe shear rate to a power less than 2, while the consnormal stress experiments yield the shear stress that is nindependent of the shear rate~Savage2!.

Rapidly flowing granular materials~granular fluids! arecharacterized by vigorous random motion of individual pticles that is analogous to thermal motion of molecules indense gas. One significant difference between granular fland dense gases is that granular particles are inelasticfrictional. These dissipation mechanisms are responsiblecontinuous conversion of the pseudothermal energy~i.e., ki-netic energy of the fluctuating motion! into the true thermalenergy~i.e., heat!. Kinetic theories of rapid granular flow~e.g., Jenkins and Savage,7 Lun et al.,8 Jenkins andRichman,9,10 Lun and Savage,11 Jenkins and Richman,12 etc.!extend the kinetic theory of dense gases to account fordissipation of the pseudothermal energy into heat due toelastic and/or frictional collisions. These theories are baon the evolution equation for the single-particle distributifunction ~s.d.f.!. The moments of this equation yield the baance laws for the mean field variables, such as the mdensity, mean velocity, and pseudothermal energy den~granular temperature!. This procedure also provides integrexpressions~constitutive integrals! for the fluxes and sourceof momenta and energy in terms of the s.d.f. and the pdistribution function ~p.d.f.!. The ‘‘molecular chaos’’ as-sumption is then employed in order to express the p.d.f

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terms of the s.d.f., and the approximate form of the s.d.fdetermined by perturbation methods. For smooth, slighinelastic particles, the s.d.f. is a slightly perturbed equilrium ~Maxwellian! distribution. Finally, the constitutive integrals are evaluated and the constitutive equations are dmined in a closed form. The resulting system of equatioresembles hydrodynamic equations for a compressible Ntonian fluid. The transport coefficients~viscosity, pseudot-hermal conductivity! are proportional to the square rootthe granular temperature, which is regarded as an indedent field variable. When applied to the steady shear flthe kinetic theory not only correctly predicts the quadradependence of the stresses upon the shear rate~which is, infact, a dimensional necessity if the collisions are binary ainstantaneous!, but also matches the observed stress matudes extremely well. Besides the steady simple shear flthe kinetic theory has also been applied to several stewall-bounded flows such as the Couette flow~e.g., Hanes,Jenkins, and Richman13!, vertical chute flow~e.g., Babic14!,and inclined chute flow~e.g., Richman and Marciniec15!.These analyses utilize the simplest version of the kintheory, which is applicable only to smooth, nearly elasparticles ~Jenkins and Richman9,10!, and the correspondingboundary conditions for bumpy walls developed by Jenkand Richman16 and Richman and Chou.17 A more advancedtheory that is not limited to nearly elastic particles butrestricted to dense and dilute flows has been developeJenkins and Richman.12

The current understanding of granular flows has bgreatly aided by the discrete element method~DEM!. DEMsimulations of steady unbounded shear flows in periodicmains have been carried out by Walton and Braun,18,19

Campbell,20 Babic, Shen, and Shen,21 Hopkins and Louge,22

etc. Simulations of steady, longitudinally uniform, wabounded granular flows in longitudinally periodic domaihave been performed by Campbell and Brennen,23,24 Camp-bell and Gong,25 Campbell,26,27Walton,28 Louge,29 etc. Mostof the attention has been focused on relatively simple, steparallel flows.

Louge, Jenkins, and Hopkins30 investigated unboundetransient granular shear flows of smooth, inelastic disksthese spatially homogeneous unsteady simple shear flowshear rateg5du/dy instantaneously changes att50 fromg0 to g`, i.e. g(t)5g0H(2t)1g`H(t), whereH(t) is theHeaviside unit step function. Their analysis is based onkinetic theory of Jenkins and Richman,12 in which the s.d.f.is assumed to be an anisotropic Maxwellian that dependthe second moments of the s.d.f. Considering the concention, shear rate, and second moments to be constant throout the flow, they obtained and numerically solved a systof ODEs that describe the time dependence of the secmoments~‘‘relaxation of the second moments’’!. They ob-tained an excellent comparison with the corresponding silations for nearly elastic particles, but observed significdeviations in the dense limit for inelastic particles. A simpfication of this problem occurs in the case of slightly inelatic particles for which it is sufficient to describe the timdependence of the granular temperature rather than offull second moments tensor. The steady-state flow is cha

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terized by a balance between the rates of productiondissipation of the fluctuation energy. During the flow trasient the energy balance equation states that the ratchange of the fluctuation energy is equal to the differenbetween the rates of its production and dissipation. Henthe time it takes to establish the new steady state mayestimated by equating orders of magnitude of the ratechange term and production/dissipation terms in the eneequation. This time scale, which will be referred to as tenergy relaxation time scale and denoted byt r , is the onlyrelevant time scale in this problem.

A generalization of the unbounded transient shear flproblem considered by Louge, Jenkins, and Hopkins30 leadsto the wall-bounded transient Couette flow problem. In tproblem the granular fluid is sheared between two parawalls at a distance 2L apart. The walls move with equal anopposite velocities of magnitude uw(t)5uw

0 H(2t)1uw

`H(t), i.e. the wall velocity instantaneously changfrom one constant value to another. The corresponding tsient Couette flow of an incompressible Newtonian fluidone of the classical problems in fluid mechanics. Analytisolutions of this problem are presented in most advanfluid mechanics textbooks~e.g., Panton31!. These solutionsindicate that the duration of the flow transient is of the ordof L2/n, wheren is the kinematic viscosity of the fluid. Thistime can also be estimated by equating orders of magnitof acceleration and viscous terms in the momentum equatThis time scale, which will be referred to as the momentudiffusion time scale and denoted bytd , is the only relevanttime scale in the incompressible Newtonian transient Couflow problem. In the transient Couette granular flow prolem, both momentum diffusion and energy relaxation timscales are expected to be important.

A further generalization leads to the general unsteaCouette flow problem in which the wall speed is an arbitrafunction of timeuw(t). In this case an additional~externallyimposed! time scale associated with the rate of change ofwall speed is expected to be important as well. In the pticular case of cyclic flows in which the wall speed is aoscillatory function of time, the externally imposed timscale is equal to the oscillation period. The solution torelated problem of a semi-infinite incompressible Newtonfluid bounded from below by a harmonically oscillating plais also presented in most advanced fluid mechanics textbo~e.g., Panton31!. This problem, known as the Stokes’ secoproblem, is one of the problems solved by Stokes32 in thecourse of his study of pendulum friction. This flow clearexhibits the intrinsic damping due to viscous diffusion. Tsolution of this problem takes a form of a damped wavewavelength (2p)(2n/V)1/2, wheren is the kinematic viscos-ity andV is the frequency of the wall speed oscillations. Tdamping is relatively strong so the motion is confined to‘‘penetrating’’ layer of thickness of the order of (n/V)1/2. Ifthe fluid is also bounded from above by a stationary platea distanceL from the oscillating plate, the problem becommore complex. IfL @ (n/V)1/2, the upper plate has noeffect on the flow; otherwise it does. Hence, the cyclic Coette flow of an incompressible Newtonian fluid dependsthe parameter (VL2/n), which can be interpreted as the rat

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of the momentum diffusion time scaletd 5 L2/n to the wallspeed oscillation time scaletw51/V. In the cyclic Couettegranular flows the relaxation time scalet r provides an addi-tional ~intrinsic! time scale so the flows are expected to dpend on the ratiostd /t r and td /tw .

In this study the unsteady Couette granular flow probldescribed above is investigated both theoretically and bymeans of the discrete element method~DEM!. Theoreticalanalysis is based on the kinetic theory for smooth, neaelastic particles~Jenkins and Richman9,10! and boundaryconditions for bumpy walls~Richman and Chou17!. Thestudy is focused on two types of flows:~a! transient flows, inwhich uw(t)5uw

0 H(2t)1uw`H(t), and ~b! cyclic flows, in

which uw(t)5&uwrms@H(2t)1cos(Vt)H(t)#.

The objectives of the study are~i! to develop an understanding of the time-dependent behavior of granular fluand, in particular, of the effects of the relevant time scaand their ratios;~ii ! to investigate possible simplifications othe problem that may occur for small ratios of the relevtime scales; and~iii ! to test the performance of the kinettheory through a comparison of theoretical predictions acorresponding DEM simulations. This is perhaps the mstringent test of the kinetic theory to date, since it has notbeen applied to and verified for time-dependent, wabounded flows.

This paper is organized as follows. A theoretical formlation of the problem, based on the kinetic theory for neaelastic particles, is presented in Sec. II. Finite-differenschemes developed for numerical analysis of the resulsystem of equations are described in Sec. III. The DEsimulations are described in Sec. IV. Perturbation analyleading to asymptotic solutions for the case in whichmomentum diffusion time scale is much smaller thanenergy relaxation time scale are presented in Sec. V. Reand conclusions are presented in Secs. VI and VII, resptively.

II. PROBLEM STATEMENT

Consider a class of unsteady granular shear flowsare driven by an equal and opposite time-dependent moof parallel bumpy walls~unsteady Couette granular flows!.The granular material is composed of uniform circular pticles ~disks or spheres! of diameters. The particles aresmooth and nearly elastic, i.e.h[12e→0, wheree is thecoefficient of restitution. The flow is bounded by bumpwalls located aty56L and moving with velocities6uw(t). The flow width 2L is considered to be fixed, so thCouette flows considered here are of the constant-volutype as opposed to flows of the constant-stress type in wthe normal stress would be fixed while the flow width woube allowed to vary. The flow is assumed to be uniformthe x direction. The problem described above is depictedFig. 1.

The governing equations of the kinetic theory~Jenkinsand Richman9,10! reduce to

]c

]t1v

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]y1c

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wherec is the concentration,u andv are thex andy com-ponents of the mean velocity,w5T1/2 is the mean fluctuationspeed, T is the granular temperature,n52 for disks,n53 for spheres, f p5apcx1c, f l5alc2x, f m

5amc2x$11am1@11am2 /(cx)#2%, f k5akc2x$11ak1@11ak2 /(cx)#2%, f g5agc2x, andx5(12axc)/(12c)n. Co-efficientsax , ap , al , am , am1 , am2 , ak , ak1 , ak2 andag ,which take different values for disks and spheres,given by ax

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Due to the flow symmetry about the centerline, the flowing boundary conditions are imposed aty50:

u~0,t !50, v~0,t !50,]w

]y~0,t !50. ~5!

A set of boundary conditions for bumpy walls has bedeveloped by Jenkins and Richman16 and Richman and

FIG. 1. Unsteady Couette flows.

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Chou.17 The bumpy walls are created by placing particlesdiameterd at a spacings along the walls. The balance olinear momentum in thex direction at the wall can be expressed as

g2~uw2u!1h1s ~]u/]y!

11h2~s/w!~]v/]y!5

~ f m / f p!s~]u/]y!

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~ f m / f p!s~]u/]y!~uw2u!2~ f k / f p!s~]w2/]y!

12~ f l / f p!~s/w!~]v/]y!

5g1hww2

11h2~s/w!~]v/]y!, ~7!

wherehw512ew , ew is the wall coefficient of restitutionh15g4(r 1 f b)2g2r , h25g3(r 1 f b)2g2r , r 5(s1d)/(2s), g1 , g2 , g3 , and g4 are functions off5arcsin@(d1s)/(d1s)#, and f b5ab1@11ab2 /(cx)#. Coefficientsab1 and ab2 , which take different values for disks anspheres, are given byab1

2D5p/(8&), ab1

3D5p/(12&),

ab22D55/8, ab2

3D55/8, while functionsg1 , g2 , g3 , and g4 ,which assume different forms for disks and spherare given by g1

2D5(2/p)1/2f cosecf, g22D5(2/p)1/2

3(f cosecf2cosf), g32D5(2/p)1/2(2(sin2 f)/322), g4

2D

5(2/p)1/2(2/3)sin2 f, g13D5(2/p)1/22 cosecf(12cosf),

g23D5(2/p)1/2(2/3)@(cosec2 f)(12cosf)2cosf#, g3

3D

5(2/p)1/2(sin2 f22), g43D5(2/p)1/2(1/2)sin2 f.

For simplicity, it is assumed thatsu]v/]yu/w!1. Underthis assumption, the boundary conditions~6! and~7! become

S u1 f Ms]u

]yD ~L,t !5uw~ t !, ~8!

s S ]w2

]y D ~L,t !5F f m

f kf MS s

]u

]yD 2

2 f Ehw2G~L,t !, ~9!

where f M5( f m / f p2h1)/g2 and f E5g1( f p / f k)(hw /h). Inaddition to these boundary conditions, it is also necessarimpose the no-flow boundary condition at the wall, i.e.

v~L,t !50. ~10!

The initial conditions may, in general, be arbitrary dtributions of c(y), u(y), v(y), andw(y), but in both spe-cific types of flows investigated here~transient and cyclicflows! they are taken to be distributionsc0(y), u0(y),v0(y), w0(y) corresponding to steady Couette flows wuw(t)5uw

0 . The initial conditions for the unsteady flows cathen be stated as

c~y,0!5c0~y!, u~y,0!5u0~y!, v~y,0!5v0~y!50,

w~y,0!5w0~y!. ~11!

Hence, the problem consists of four partial differentequations~1!–~4! for c(y,t), u(y,t), v(y,t), and w(y,t),three boundary conditions~5! at the centerline, three boundary conditions~8!–~10! at the wall, and initial conditions~11!. The wall motion functionuw(t) is prescribed as appropriate for the case under consideration.


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III. NUMERICAL ANALYSIS

Both explicit and implicit finite-difference methods fonumerical solution of~1!–~11! for arbitrary wall motionfunctions uw(t) have been developed. A staggered gshown in Fig. 2 is used. The spatial flow domain (0,L) isdivided intoN equal intervals of lengthdy5L/N. The set ofdiscrete variables includesN11 discrete values ofc at theedges of these intervals, i.e. at pointsyj 11/25 j dy ( j50,...,N), and 3N discrete values ofu, v, and w at thecenters of these intervals, i.e. at pointsyj5( j 21/2)dy ( j51,...,N). The staggering of the grid allows central spatdifferencing of both field equations and boundary conditioand is necessary in order to achieve second-order accuraspace. If a nonstaggered grid were used, forward or baward differencing of the boundary conditions would havebe used. This would result in the need to increaseN in orderto achieve the same accuracy as with the staggered grid.finite-difference approximation of the mass balance equa~1! are written at N11 points yj 11/25 j dy ( j 50,...,N)while the finite-difference approximations of the balanequations~2!–~4! are written at atN pointsyj5( j 21/2)dy( j 51,...,N). Six additional equations for the values ofu, v,and w at the fictitious nodes 0 andN11 are obtained bycentral-difference approximations to the symmetry contions ~5! at y50 and the boundary conditions at the wa~8!–~10!.

A variety of numerical schemes for temporal differening have been developed and tested. The explicit schemfirst-order accurate in time and is subject to stringent stabrequirements. The exact stability requirements are difficulfind—even the simplest local von Neumann analysis of learized difference equations leads to a matrix eigenvaproblem, which has to be solved numerically for a givenof physical and numerical parameters. Based on experimtation with the model, the stability criterion appears tosimilar to the heuristic stability criterion of the FTCS schemfor one-dimensional 1-D nonlinear diffusion problems~e.g.,Press et al.33!, i.e. dt,minj@dy2/(2Dj)#, where D j

5(swj /cj )max@fm(cj),fl(cj),(2/n) f k(cj )# is the maximumdiffusion coefficient~of x momentum,y momentum, or en-ergy!. In the present study the explicit method has been ufor computation of the transient flow solutions. The fulimplicit scheme is first-order accurate in time and appearbe unconditionally stable for the present set of equatioThis method is more efficient than the explicit schemeproblems in which the solution varies on a time scale thasignificantly larger than the maximum allowed time stepthe explicit scheme. In the present study this method w

FIG. 2. Staggered grid used in the finite-difference method.


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used for computation of the steady flow solutions~initialconditions! and cyclic flow solutions. The Crank–Nicholsoscheme is second-order accurate in time and is uncondially stable for the 1-D linear diffusion problem. However, fthe present set of equations it is not unconditionally staand the stability requirements appear to be similar tostability requirements of the explicit scheme. Sinceamount of work per time step is typically much greater thin the explicit scheme, this method is generally more ticonsuming and has not been used for computation ofsolutions reported herein.

Both explicit and implicit schemes become computatioally intensive if the spatial domain is finely discretized~largeN!. In the explicit scheme the time step becomes very sm(dt;N22), while in the implicit scheme both the numberiterations required to solve the nonlinear system of diffence equations each time step and the size of the lineartem that has to be solved each iteration rapidly increaseN. The convergence of the models with respect toN hasbeen carefully tested for a number of different flows andhas been found that the results obtained withN510 are veryclose to those obtained withN520. With N510 both mod-els are quite manageable and most computations take ofew minutes on a Sparc10. All results presented herein wobtained withN510. The ability to perform the computations on a relatively coarse spatial grid is a consequencthe fact that the analyzed flows do not have small lenscale features as well as of the second-order accuracspace that has been achieved by the use of the staggered

IV. DEM SIMULATIONS

A. Methodology

Numerical simulations of unsteady Couette flows weperformed with an extension of the soft-particle DEM moddescribed in Babic, Shen, and Shen.21 A hard-particle modelwould be applicable and probably more efficient in this caA soft-particle model was used because the computer twas not a real concern and only a soft-particle model wavailable. The results of hard-particle and soft-particle mels are equivalent provided that soft particles are sufficierigid and this was carefully verified, as discussed in SIV B.

The equations of motion for particlep are: xp5vp andvp5r p/mp, wherexp is the position vector of the center oparticlep, vp is the velocity of particlep, mp is the mass ofparticlep, andr p is the resultant interaction force acting oparticlep. Bumpy walls are created by placing semidisksdiameterd at a distances apart along the solid walls. Thcenters of boundary particles are located aty56@L1(s1d)/2#. The velocities of boundary particles are set6uw(t). Periodic boundary conditions are implementedx56Lx . The equations of motion for each particle are ingrated step by step by an explicit second-order accurate lfrog algorithm. The resultant interaction forcer p is obtainedas the sum of all forces acting on particlep, i.e., r p5(fpq,wherefpq is the force exerted by particleq on particlep. Theinteraction force between two particles is a function of threlative positionxpq5xp2xq and their relative velocityvpq


129.115.103.99 On: Wed,

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r

5vp2vq. The contact force model used in this study is tlinear viscoelastic~spring-dashpot! model of Cundall andStrack.34 This model can be expressed as

fpq5@Kpq~r p1r q2uxpqu!1Cpq~vpq–npq!#

3npqH~r p1r q2uxpqu!, ~12!

where Kpq is the contact stiffness,Cpq is the dampingcoefficient,npq52xpq/uxpqu is the contact normal,r p andr q

are the radii of particlesp andq, andH(j) is the Heavisidestep function. The damping coefficientCpq can be expressedas Cpq5zpq(Kpqmpq)1/2, where zpq52 ln(epq)/@p2

1ln2(epq)#1/2 is the dimensionless damping coefficient,epq isthe restitution coefficient, andmpq5mpmq/(mp1mq) is thereduced mass. In this study all particles are identical wmassm and restitution coefficiente. If both p andq are flowparticles,epq5e and mp5mq5m, so mpq5m/2. If q is aboundary particle,epq5ew and mp5m, mq5`, so mpq

5m. The duration of a binary collision istcol5p/vd , wherevd5$Kpq@12(zpq)2#/mpq%1/2 is the contact mechanism frequency. In the present modelvd is set to be the same foflow contacts and boundary contacts. The value ofvd is aninput parameter that has to be selected carefully, as discubelow. Oncem, vd , e, and ew are specified, the normastiffnesses and damping coefficients can be determined frelations presented above.

B. Parameters

The parameters that define the physical problemscribed in Sec. II arec0 ,L,m,s,d,s,e,ew ,uw

0 ,us ~which isequal touw

` in transient flows anduwrms in cyclic flows!, and

V in cyclic flows. Additional parameters required for thsoft-particle model are the contact mechanism frequencyvd ,time stepdt, and the periodic cell half-lengthLx . The di-mensionless parametervdL/us affects the rigidity of par-ticles and the size of overlaps between particles that arelowed in the soft-particle model. The overlap/diameter rais of the order ofR5ws /(svd), wherews is the character-istic fluctuation velocity scale. Furthermore, the probabilof multiple contacts is proportional totcol /tmfp , wheretmfp isthe mean travel time between successive collisions. Stmfp;s/ws and tcol;1/vd , the ratiotcol /tmfp is also propor-tional toR. From scaling arguments presented in Sect. V,characteristic fluctuation velocity scale can be estimatedws5sus /@L(12e)1/2#. Hence, vdL/us5R21(12e)21/2.Provided thatR is sufficiently small, the soft-particle modeoperates with small overlaps and a small probability of mtiple contacts, and hence approaches the rigid-particle moIn the present study the value ofR51023 was selected. Thevalue ofdt, which affects the accuracy of numerical integrtion of the equations of motion, was selected asdt50.05tcol . The value ofLx , which affects the total numbeof particles in the system and hence the quality of calculastatistical averages, was selected asLx510L.

C. Averaging

During the course of the DEM simulation the complemicroscopic description of the system~all particle positions,velocities, and interparticle forces! is available. Macroscopic


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~continuum mechanical! variables such as the mean densivelocity, granular temperature, stresses, etc., can be dmined by an appropriate averaging procedure. In ragranular flows the averaging procedure consists of bothtial and temporal averaging. A detailed development ofaveraging theory for granular materials is presentedBabic.35 The relationships for calculation of the average cocentrationc, velocity v, granular temperatureT, and stresstensorS are

c~x,t !5 E`

(p

wpVp dt8, ~13!

v~x,t !5*`

`(pwpvp dt8

*``(pwp dt8

, ~14!

T~x,t !5*`

`(pwp~vp2v!–~vp2v!dt8

n*``(pwp dt8

, ~15!

S~x,t !52 E`

(p

wpm@~vp2v! ^~vp2v!#dt8

2 E`

(p

(q.p

wpq~xpq^fpq!dt8, ~16!

wherex and t are, respectively, the position vector and timto which the average value is assigned,Vp is the particlevolume ~area for disks!, n52 for disks,n53 for spheres,and

wp5wf~xp2x,t82t !, ~17!

wpq5E0

1

wf~xq1sxpq2x,t82t !ds. ~18!

In the above equations,w f(r ,t) is the weighting function.The weighting function used in this study corresponds tosimple volume–time averaging. The formal expressionwf( r ,t) is

wf~r ,t !51

2Lx Dy Dt FHS t2Dt

2 D2HS t1Dt

2 D G3FHS r y2

Dy

2 D2HS r y1Dy

2 D G , ~19!

whereH(j) is the Heaviside step function,Dy is the aver-aging strip thickness, andDt is the averaging time intervalWith this choice,wp5(2Lx Dy Dt)21 if ~i! the center ofparticle p is located within the averaging strip of thickneDy centered aty @i.e., if y2Dy/2,yp(t8),y1Dy/2#, and~ii ! t8 is located within the time-averaging interval oduration Dt centered att ~i.e., if t2Dt/2,t8,t1Dt/2!.Otherwise, wp50. Also with this choice, wpq

5(2Lx Dy Dt)21spq, wherespq is the fraction of the lengthof the vectorxpq that is located within the averaging strip othicknessDy centered aty, if ~i! particlesp and q are incontact at timet8, ~ii ! a portion of vectorxpq is locatedwithin the averaging strip of thicknessDy centered at pointyat time t8, and ~iii ! t2Dt/2,t8,t1Dt/2. Otherwise,wpq

50.


129.115.103.99 On: Wed,

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There is a number of philosophical and practical issuassociated with the averaging of simulation data as welwith comparing such averages with predictions of a cotinuum theory~e.g., kinetic theory!, especially for unsteadyflows. First, if it were possible to find ranges ofDy andDtfor which the calculated averages would not depend onues of Dy and Dt, the average variables would be scaindependent and the continuum hypothesis would be vaUnder the ergodicity hypothesis~the equality of ensembleand space/time averaging! there would be no philosophicaproblem in comparing the scale-independent average vables obtained from the simulations with the predictions ocontinuum theory. However, if the scale independence cnot be established, the averaged variables have to begarded as scale dependent.

In the rapid granular flows the situation is the followinIf the ensemble-averaged mean flow is spatially inhomoneous and unsteady, there are characteristic length andscales associated with the spatial and temporal flow featuIn order to resolve these features in the analysis of simuladata, the averaging length and time intervals must be smathan the characteristic length and time scales, respectivelthe selected averaging space–time element there will bfinite number of collisions. The average variables are bacally determined by averaging of the appropriate micscopic quantities over all collisions that actually occur in tselected averaging space–time element. The larger the nber of collisions in this space–time element, the more likit is that these average variables are good approximationthe corresponding ensemble averages. However, sincenumber of collisions is finite, the average variables willdifferent from the ensemble averages. They can be thouof as superpositions of a signal~true ensemble averages! andnoise. If the noise to signal ratio is small, comparison ofnoisy data to the predictions of a continuum theory is meingful for all practical purposes. However, if the noisesignal ratio is large such a comparison is not meaningThe amount of noise depends on the number of collisionthe selected space–time interval, but the strength of thenal depends on the ensemble-averaged mean flow. If thenal itself is weak, the noise to signal ratio will be large, evif the noise itself is not particularly large. For this reascertain relatively small features of unsteady flows~e.g.,transversal velocity, deviations of concentration and granutemperature from the width-averaged values! cannot be reli-ably detected in the simulation data. On the other hand,steady flowsDt can be made arbitrarily large whileDy canbe made arbitrarily small, as long as the number of collisioin the space–time element is sufficiently large. Hence, stial distributions of all relevant variables can be accuratdetermined in steady flows.

V. PERTURBATION ANALYSIS

In this section asymptotic solutions to the unsteady Cette flow problem are obtained by perturbation analysesthe limit of a small ratio of the momentum diffusion anenergy relaxation time scales. It turns out that this limit cresponds to small values of the flow Mach number and tgranular fluids in this limit behave as nearly incompressi


03 Sep 2014 07:20:57

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non-Newtonian fluids at all time scales. Hence, under apppriate conditions, mathematical analysis of rapid granuflows can be tremendously simplified. In the present casesimplification actually leads to analytical solutions, but thoare not so importantper se. More important is the identifi-cation of the regime in which the incompressibility assumtion is appropriate and the development of simplified goveing equations that can be solved, if not analytically thnumerically, much easier than the full system of equation

A. Regular perturbation analysis

The analysis begins with a scaling of the governiequations. The governing equations are scaled as follofunctionsf a (a5p,l,m,k,g) are scaled by a common scaf s , the y coordinate is scaled by the flow half-widthL, thestreamwise velocityu is scaled by a characteristic wall velocity us , which is taken to beuuw

`u for transient flows anduw

rms for cyclic flows, the fluctuation speedw is scaled byws5sus /(Lh1/2), chosen such that the rate of shear woand the rate of energy dissipation terms in the energy eqtion appear to be of the same order of magnitude, the timtis scaled by the relaxation time scalet r5s/(hwsf s), chosensuch that the unsteady term and the dissipation term inenergy equation appear to be of the same order of magnitand the transversal flow velocityv is scaled by a scalevs

5(hLwsf s)/s, chosen such that all terms in the conservtion of mass equation appear to be of the same ordemagnitude. The present analysis is applicable to modeconcentrations, for whichf s can be chosen as unity, and higconcentrations~dense limit!, for which f s is large @ f s

;x(c0);(12c0)2n#.Hence, let f a5 f sf a , t5t r t, y5Ly, c5 c, u5usu, v

5vsv, andw5wsw. The resulting scaled equations are

] c

] t1 v

] c

] y1 c

] v] y

50, ~20!

e cS ]u

] t1 v

]u

] yD5]

] y S f mw]u

] yD , ~21!

e cS ] v] t

1 v] v] y D52

1

eb2

]

] y~ f pw 2!1

]

] y S f lw] v] y D ,

~22!

encw S ]w

] t1 v

]w

] y D5]

] y S f kw]w 2

] y D2e f pw 2] v] y

1e2b2 f lw S ] v] y D 2

1eF f mwS ]u

] yD 2

2 f gw 3G , ~23!

v~0,t !50, v~1,t !50, ~24!

u~0,t !50, S u1b f M

]u

] yD ~1,t !5uw~ t !, ~25!


129.115.103.99 On: Wed,

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]w

] y~0,t !50,

S ]w

] y D ~1,t !5eb

2 F f M

f m

f k

1

w S ]u

] yD 2

2 f EwG ~1,t !, ~26!

whereb5s/L ande5h/b2.The momentum diffusion time scaletd is defined such

that if the time in the momentum equation were scaled bytd

instead oft r , the unsteady term and momentum diffusioterm would appear to be of the same order of magnituHence, td5L2/(swsf s). Since t r5s/(hwsf s), the param-etere can be interpreted ase5td /t r . Perturbative solutionsare sought in the limite→0, which corresponds to a smaratio of the momentum diffusion to the energy relaxatitime scale. This limit occurs forh→0 and a small but finiteb5s/L. The smalle limit is well within the practical range.For example, in the shear cell laboratory experiments wglass beads of Savage and Sayed,4 e'0.97 andL/s'3, so arepresentative value ofe'0.1.

Another interpretation of the parametere is the follow-ing. The flow Mach number, Ma, can be defined as the raof the characteristic velocity,us , to the speed of propagatioof compressibility waves, which is proportional to the chaacteristic fluctuation speed,ws . Hence, Ma2;(us /ws)

2

5hL2/s25e. This implies that flows with smalle are likelyto be nearly incompressible. In such flows the transvevelocity, which is associated with compression waves, wobe expected to be very small. In the scaling employed abvs /us5e1/2s/L, which is indeed small ife is small. In anadhocmanner one could therefore assume that for smalle con-centration is constant and equal toc0 , that the transversavelocity is zero, and that the granular temperature is indepdent ofy and dependent ont alone. The following perturba-tion analysis rigorously shows that these assumptionsindeed correct to the zeroth order ine.

A perturbative solution is constructed as a series of poers ofe, i.e.

~ c,u,v,w!5 (k50

ek~ ck ,uk ,vk ,wk!. ~27!

Substitution of the perturbation series~27! into ~20!–~26!and setting coefficients of the successive powers ofe to zeroyield a sequence of boundary value problems for the expsion functionsck , uk , vk , wk . The zeroth-order problem~coefficients ofe0! is

] c0

] t1 v0

] c0

] y1 c0

] v0

] y50, ~28!

]

] y S f m0w0

]u0

] y D50, ~29!

]

] y~ f p0w 0

2!50, ~30!

]

] y S f k0w0

]w 02

] y D 50, ~31!

v0~0,t !5 v0~1,t !50, ~32!


03 Sep 2014 07:20:57

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u0~0,t !50, S u01b f M0

]u0

] y D ~1,t !5uw~ t !, ~33!

]w0

] y~0,t !5

]w0

] y~1,t !50, ~34!

where f a0 5 f a(c0).Equations~31! and ~34! imply that w05w0( t ). Equa-

tion ~30! then implies thatc05 c0( t ). Integration of~30!with respect toy from 0 to 1 and use of~32! then indicatesthat c0 must be a constant and thusc05c0 . Equation~28!then implies thatv050. Finally, Eqs.~29! and ~33! implythat

u05uw~ t !y

11b f M0. ~35!

Hence, the zeroth-order solution corresponds to an unstesimple shear flow in which the concentration, granular teperature, and shear rate are all constant across the flow.shear rate is instantaneously adjusted to the wall mofunction. This is the ‘‘instantaneous diffusion’’ limit, inwhich changes in the wall motion are immediately fthroughout the flow.

The first-order problem~coefficients ofe1! is

] c1

] t1c0

] v1

] y50, ~36!

05 f m0w0F]2u1

] y 2 1uw

11b f M0S 1

w0

]w1

] y1

f m08

f m0

] c1

] y D G ,~37!

05 f p0w 02 S 2

w0

]w1

] y1

f p08

f p0

] c1

] y D , ~38!

nc0w0

dw0

dt52 f k0w 0

2 ]2w1

] y 2 1 f m0w0S uw

11b f M0D 2

2 f g0w 03, ~39!

where, for example,f a08 [ (d fa /dc)(c0). The boundaryconditions are

v1~0,t !5 v1~1,t !50, ~40!

u1~0,t !50,

S u11b f M0

]u1

] y1

b f M08

11b f M0uwc1D ~1,t !50, ~41!

S ]w1

] y D ~0,t !50,

S ]w1

] y D ~1,t !5b

2 S f M0

~11b f M0!2

f m0

f k0

u w2

w02 f E0w0D . ~42!

The governing equation forw0( t ) is found by integrat-ing the first-order energy equation~39! over the width andutilizing the associated first-order boundary conditiowhich yields

w0

dw0

dt5a2u w

2 w02b2w 03, ~43!


129.115.103.99 On: Wed,

dy-hen

,

where

a25f m0

nc0~11b f M0!, ~44!

b25f g01 f k0b f E0

nc0. ~45!

The first term on the right-hand side of Eq.~43! representsthe total rate of work per unit mass done by the mean shflow and the slip velocity, while the second term represethe total rate of energy dissipation per unit mass in the inrior and boundary collisions.

The first-order problem also yields the following firsorder solutions forc1 , u1 , v1 , andw1 :

w15w0F h1 kS y 2

22

1

6D G , ~46!

c1522f p0

f p08kS y 2

22

1

6D , ~47!

v151

3

1

c0

f p0

f p08k8~ y 32 y !, ~48!

u15u0F2k

6 S 122f m08

f m0

f p0

f p08 D S y 22113b f M0

11b f M0D

12

3

kb f M08

11b f M0

f p0

f p081

1

6

u w8

uw

y 2

f m0w0G , ~49!

whereu w8 5duw /dt, h is the average value ofw1 /w0 acrossthe flow, k85dk/dt, andk is the slope ofw1 /w0 at the wallgiven by

k5b

2 F f M0

~11b f M0!2

f m0

f k0S uw

w0D 2

2 f E0G . ~50!

The governing equation forh( t) is found by integratingthe second-order energy equation over the width and uting the associated second-order boundary conditions, wyields

dh

dt1~a2u w

2 1b2w 02!

h

w0

52 f k0w0k

3nc0F k1~2k1b f E0!S f M08

f M0

2f m08

f m0D f p0

f p08G

1uwu w8

3nc0w 02~11b f M0!2

. ~51!

Examination of the regular perturbation solution prsented above reveals two deficiencies:~i! the zeroth-ordervelocity distribution is discontinuous if the wall motion function is discontinuous, and~ii ! the termu w8 in ~49! must be oforder unity or less. The first deficiency invalidates the sotion neart50 in transient flows, while the second deficieninvalidates the solution if the wall motion time scale bcomes much smaller than the energy relaxation time s~e.g., if Vt r@1 in cyclic flows!. Since the discontinuity inthe wall motion function can be viewed as an infinite val


03 Sep 2014 07:20:57

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of u w8 it can be concluded that the regular perturbationlution is uniformly valid only for unsteady flows in whichO(uu w8 u) < 1.

In the following sections specific results for steaflows, transient flows, and cyclic flows are presented. Fortransient flows the deficiency~i! is resolved by performing asingular perturbation analysis, while for the cyclic flows tdeficiency~ii ! for Vt r@1 is resolved by performing a multiple scale analysis.

B. Steady flows

For steady flowsuw(t)5uw`5const. With us5uuw

ù,uw( t )51. The first-order solution is given byv50 and

c5c0H 12eF k`

c0

f p0

f p08 S y 221

3D G J , ~52!

w5w0`H 11eF h`1 k`S y 2

22

1

6D G J , ~53!

u5y

11b f M0H 11eF2

1

6k`S 122

f m08

f m0

f p0

f p08 D3S y 22

113b f M0

11b f M0D1

2

3

b k` f M08

11b f M0

f p0

f p08 G J , ~54!

where

w0`5

a

b, ~55!

k`5b

2 F f M0

~11b f M0!2

f m0

f k0S 1

w 0`D 2

2 f E0G , ~56!

h`5f k0k`

6nc0b2 F k`1~2k`1b f E0!S f M08

f M02

f m08

f m0D f p0

f p08 G .~57!

It can be seen that the first-order corrections are oforder of eb5h/b rather than of the order ofe. Hence, theperturbation solutions for steady Couette flows have a wrange of applicability than those for unsteady Couette floThe solution presented above is quite simple: it predparabolic concentration and fluctuation speed profiles ancubic mean velocity profile. Qualitative shapes of these pfiles depend only on the value ofk`, which in turn dependsonly on the mean concentrationc0 and boundary parameterd/s, s/d, andhw /h. For k`.0 the flux of fluctuation en-ergy is directed from the wall to the centerline and viversa.

The steady solution obtained above can be compawith the exact solution for steady Couette flows in the delimit developed by Haneset al.13 It can be shown that thepresent solution is consistent with the expansion ofHaneset al.13 solution into a power series in terms of parameter h/b. The present solution can also be compared wthe approximate solution for steady Couette flows develoby Richman and Chou.17 In their solution, which is approxi-mately valid over the full range of concentrations, the cocentration was replaced by the mean~width-averaged! con-


129.115.103.99 On: Wed,

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centration wherever it appeared in the energy equation. Twere then able to analytically solve the resulting linear ODfor w and subsequently determine the mean velocity aconcentration profiles from conditions that both shear anormal stresses are constant throughout the flow. It mayverified that the present solution agrees with the solutionRichman and Chou17 in the limit h/b→0. It can also beshown that errors in the solution of Richman and Chou17 areof the order of (h/b)2, just like in the present solutionTherefore, the present theory yields an approximate solufor steady Couette flows that is much simpler than aequally accurate as the previous approximate solutionRichman and Chou.17

C. Transient flows

For transient flowsuw( t)51 for t.0. Since steady solutions for w are proportional tous , the initial condition forw0 can be expressed asw0(0)5w 0

05(uuw0 u/uuw

ù)w 0` .

Equation~43! can be easily solved in a closed form by sepration of variables. The result is

w0

w 0`

512C exp~22abt !

11C exp~22abt !, ~58!

where

C5uuw

0 u2uuwù

uuw0 u1uuw

ù. ~59!

The zeroth-order outer problem solved above is vsimilar to the unbounded transient granular shear flow prlem studied by Louge, Jenkins, and Hopkins.30 The only dif-ferences are due to the contributions of the slip work aenergy dissipation in boundary collisions to the total prodtion and dissipation terms.

The zeroth-order outer solution for the mean velocfield u0( y, t )5 y/(11b f MO) cannot match the zeroth-ordeinitial condition u0( y,0)5u w

0 y/(11b f MO) because of thediscontinuity in the wall velocity. There will be a short initiatransient~initial layer!, during whichu0 changes very rapidlyand the ‘‘relaxation’’ scaling is no longer valid. The resoltion of the initial layer, which is expected to be of durationthe order oftd , requires a singular perturbation analysis. T‘‘outer’’ part of the analysis has already been completed athe results are given by~35! and ~58!. In order to completethe analysis, it is necessary to rescale the governing etions, develop an ‘‘inner’’ solution valid for small times (t!1), and asymptotically match this inner solution with th‘‘outer’’ solution, which was developed above and is valonly for t@e. To this end, the governing equations are recaled using a time scaletd instead of t r . This results inreplacement ofe]/] t by ]/] t in Eqs. ~20!–~23!, where t5t/td5 t/e. The solutions are again assumed in the forma power series ine:

~ c,u,v,w!5 (k50

ek~ ck ,uk ,uk ,wk!, ~60!

where (ck ,uk ,uk ,wk) are functions ofy and t5t/td . Intro-ducing perturbation series~60! into ~20!–~26! and setting


03 Sep 2014 07:20:57

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coefficients of the successive powers ofe to zero yields asequence of boundary value problems for the expanfunctions ck ,uk ,vk ,wk . The zeroth-order solution isc0

5c0 , w05w 00 and

u0~ y, t !5 (m51

`am

11b f M0@11~ u w

0 21!e2lm2 Km t #sin~lmy!,

~61!

whereKm5( f m0w 00)/c0 , lm’s are roots of the transcenden

tal equation tan(lm)1bfMOlm50, andam’s are given by

am52~sin lm2lm coslm!

lm2 2lm sin lm coslm

. ~62!

The zeroth-order inner problem solved above is vsimilar to the classical transient Couette flow problem forincompressible Newtonian fluid. The only differences adue to the effect of the slip at the wall. If the slip at the wwere neglected,f M would be zero, solm5mp and am

52(21)m11/(mp). In this case one would recover the clasical solution for an incompressible Newtonian fluid.

The matching of the zeroth-order inner and outer sotions is easy. The uniformly valid solution foru is obtainedby the standard asymptotic matching procedure, i.e. by aing the outer and inner solutions and subtracting the inlimit of the outer solution. Since the inner limit of the out

e

f

-


129.115.103.99 On: Wed,

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d-r

solution is equal to the outer solution itself, the uniformvalid solution is equal to the inner solution and is therefogiven by Eq.~61!.

The first-order outer solution can be easily determinfrom ~46!–~51!. However, the first-order inner solution turnout to be rather complicated and cumbersome to evaluatespite of this, this solution has been determined in orderverify the consistency and compatibility of the outer ainner expansions and to validate the scaling of the governequations used in this study.

D. Cyclic flows

1. Slow oscillations, O (Vt r )<1

As explained in Sec. V A, the regular perturbation anasis is applicable only ifO(uu w8 u)<1. For cyclic flows withuw(t)5&uw

rms@H(2t)1cos(Vt)H(t)#, this condition is satis-fied if O(Vt r)<1. This case will be referred to as the slooscillations. The analytical solution of~43! for this case isdescribed below.

For t,0 ~steady Couette flow withuw5&!, the solu-tion of ~43! is w05(a/b)&. For t.0 @cyclic Couette flowwith uw5& cos(Vt)# Eq. ~43! can be transformed into aquasilinear second-order Mathieu equation. Followingmethod of solution outlined in Abramowitz and Stegun36

one obtains

w0

w0`

512C exp~22maabt !@Pa~2Vt !/Pa~Vt !#@w0

`~2Vt !/w0`~Vt !#

11C exp~22maabt !@Pa~2Vt !/Pa~Vt !#, ~63!

eofre

le

d

where

w 0`~s!5

a

b S ma11

a

d ln@Pa~s!#

ds D , ~64!

C5w 0

`~0!2w0~0!

w0`~0!1w0~0!

, ~65!

ma is a function ofa5ab/V, andPa is a periodic functionof period p that depends ona only. The value ofma iscalculated from the relationma5@1/(pa)#arcosh@y(p)#,wherey(p) is found by numerically solving the initial valuproblem d2y/ds25a2(11cos 2s)y, y(0)51, (dy/ds)(0)50 on the intervalsP(0,p). The functionPa can be ob-tained in the Fourier series form

Pa~s!5ReS (k52`

`

C2k exp~2iks!D . ~66!

The coefficientsC2k are obtained as follows. Substitution o~66! into ~63! yields a recurrence relationD2kC2k5C2k22

1C2k12 , where D2k522@11((2k2 ima)/a)2#. Let G2k

5C2k /C2k22 andH22k5C22k22 /C22k . The recurrence relations for G2k and H22k are G2k51/(D2k2G2k12) and

H22k51/(D22k222H22k22). Therefore,G2k andH22k canbe expressed in terms of continued fractions as

G2k51

D2k2

1

D2k122•••,

H2k51

D22k222

1

D22k242•••. ~67!

As t→`, w0 approaches the limit cycle solutionw 0` .

The period of the limit cycle solution is equal to a half of thperiod of the wall speed oscillations. Qualitative featuresthe limit cycle solutionw 0

` depend exclusively on parametea, which is proportional to the ratio of the oscillation timscale to the relaxation time scale. The approach ofw0 to thelimit cycle solution w 0

` is characterized by a time scat r /(2maab).

The limiting form of w 0` for an infinite value ofa ~very

slow oscillations! can be obtained by setting the left-hanside of ~43! to zero. This yields the quasi-steady solution

w 0`~s!5

a

b@&ucos~s!u1O~a21!#. ~68!


03 Sep 2014 07:20:57

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2. Fast oscillations, O (Vt d)51

In order to explore the flow behavior in the casewhich the oscillation time scale is of the same order of mnitude as the momentum diffusion time scale~which is muchsmaller than the relaxation time scale!, the perturbationanalysis must be modified. It is assumed here thatV[Vtd

;1. The analysis in this limit is performed using the methof multiple scales~Bender and Orszag37!. In this method theperturbative solutions are assumed in the following form

n

he

b

ly


129.115.103.99 On: Wed,

-

~ c,u,v,w !5 (k50

ek~ ck ,uk ,vk ,wk!, ~69!

where (ck ,uk ,vk ,wk) are functions of y, t5t/t r and t5t/td . Even thought and t are related byt5e t, in themethod of multiple scalest and t are treated as independevariables. Time derivatives are evaluated according to

f

]

] t~ c,u,v,w!5 (

k50ekS ]

] t~ ck ,uk ,vk ,wk!1

]

] t~ ck ,uk ,vk ,wk!

dt

dtD5 (

k50ek

]

] t~ ck ,uk ,vk ,wk!1

1

e (k50

ek]

] t~ ck ,uk ,vk ,wk!. ~70!

Introducing ~69! and ~70! into ~20!–~26! and setting coefficients of successive powers ofe to zero yields a sequence oboundary-value problems for (ck ,uk ,vk ,wk). The zeroth-order solution is given byc05c0 , w05w0( t ) and u05U11U2 ,whereU1 andU2 are the limit cycle and transient parts of the solution, respectively. These functions are found to be

U15ReS & sinh~My!exp~ i Vt !

sinh M1b f M0M coshM D , ~71!

U25 (m50

`

bm exp~2lm2 Km t !sin~lmy !, ~72!

whereM5(11 i )@(Vtd)/(2Km)#1/2, Km5 f m0w0 /c0 , lm’s are roots of transcendental equations tanlm 1 bfM0lm 5 0, andbm’s are given by

bm52&~sinlm2lm coslm!/~11bfM0!2Re@lm

2 ~sinlmM coshM2lm coslm sinhM!/~M21lm2 !~sinhM1bfM0M coshM!#

lm2 2lm sinlm coslm

.

~73!

f

Equation~72! suggests that the duration of the transieof the velocity field to its limit cycle is of the order oftd .Since in this casetd;1/V, the limit cycle velocity field isexpected to be reached during the first cycle.

The governing equation forw0( t ) can be found fromthe first-order energy equation. The result is

dw0

dt5a2 ReS 11b f M0

~ tanhM !/M1b f M0D 2b2w0

2. ~74!

The cyclic flow problem has now been solved for tcases of slow oscillations (V;e) and fast oscillations (V;1). The case of intermediate oscillations for whichV@eand V!1 is considered next. This intermediate case canobtained, for example, withV;Ae. Fortunately, it is notnecessary to perform a new perturbation analysis~in whichthe solutions would be assumed as series in powers ofAe!.Rather, it is sufficient to examine the limitsV@e and V!1 of the slow and fast oscillation solutions, respectiveFor V @ 1 (a ! 1) it can be deduced from Eqs.~20.3.17!and ~20.3.18! of Abramowitz and Stegun36 that ma'1, Pa

'1. Hence,~63! reduces to

t

e

.

w0

w 0`

512C exp~22abt !

11C exp~22abt !, ~75!

wherew 0`5a/b andC5@w 0

`2w0(0)#/@w 0`1w0(0)#. On

the other hand, forV!1, uM u!1 and (tanhM)/M→1.Hence,~74! reduces to

dw0

dt5a22b2w 0

2. ~76!

It is easily verified that the solution of~76! is given by~75!,which also happens to be the solution of~43! for the transientflow problem. Furthermore, forV!1, Eq.~73! indicates thatbm→0, so the transient part ofu0 becomes negligible. Thelimit cycle part, given by~71!, reduces to

u05& cosVt

11b f M0y, ~77!

which agrees with~35!. Therefore, to the zeroth order oapproximation, theV@1 limit of the slow oscillation solu-tion and theV!1 limit of the fast oscillation solutions are incomplete agreement.


03 Sep 2014 07:20:57

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VI. RESULTS

A. Transient flows

Numerical results have been obtained for the casedisks. The following parameters are held constant:c050.5,d/s51, s/d50, ew /e51, anduw

0 /uw`50.5. Four cases are

discussed below: Case I, withe50.99 and L/s55(e50.25); case II, withe50.96 andL/s55(e51); case III,with e50.84 and L/s55(e54); and case IV, withe5 0.96,L/s 5 10(e 5 4). For all cases predictions of thekinetic theory are obtained by the finite difference methoddescribed in Sec. III and compared with results of the DEsimulations described in Sec. IV. These results are shownFigs. 3–10.

Figures 3, 5, 7, and 9 describe the transient Couette flosolutions for cases I–IV. These figures consist of six panethat depict the time dependence ofc, du/dy, v, w, and

FIG. 3. Transient Couette flow, case I~e50.99, L/s55, e50.25!. Solidlines: width-averaged values; dashed lines: values near the wall; dot–daslines: values near the centerline; plus symbols: width-averaged DEM da

FIG. 4. Steady Couette flow, case I~e50.99, L/s55, e50.25!. Lines:numerical solution; symbols: DEM data. In panels~d!, ~e!, and ~f!, solidlines and1 symbols: total stresses; dashed lines and x symbols: collisionstresses; dot–dashed lines and o symbols: transport stresses.


129.115.103.99 On: Wed,

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dimensionless stressesSyx5Syx /(rswsusb) and Syy

5Syy /(rsws2), wherers is the particle density. In each pane

there are three lines: solid, which shows the time dependeof the width averages of these functions; dashed, whshows the time dependence of these functions at the w@except v for which the dashed line showsv(12d y, t ),sincev is zero at the wall#; and dot–dashed, which shows thtime dependence of these functions at the centerline@exceptv, for which the dashed line showsv(d y, t ), sincev is zeroat the centerline#. The DEM data, which are shown by1symbols, include only the width averages ofw, Syx , andSyy . These data should be compared with the correspondsolid lines. As discussed in Sec. IV C, spatial and tempoaveraging of the results from a single simulation yields aerage variables that can be thought of as superpositionsignals~true ensemble averages! plus noise. The noise is aconsequence of a finite number of collisions in the averagi

eda.

al

FIG. 5. Transient Couette flow, case II~e50.96, L/s55, e51!. Solidlines: width-averaged values; dashed lines: values near the wall; dot–dalines: values near the centerline; plus symbols: width-averaged DEM da

FIG. 6. Steady Couette flow, case II~e50.96, L/s55, e51!. Lines: nu-merical solution; symbols: DEM data. In panels~d!, ~e!, and~f!, solid linesand 1 symbols: total stresses; dashed lines and x symbols: collisiostresses; dot–dashed lines and o symbols: transport stresses.


03 Sep 2014 07:20:57

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space–time element. The magnitude of the noise can bepreciated from scatter in the width-averaged data duringportion of the flow that is supposed to be approximatesteady~large t !. If the noise to signal ratio for a particulavariable is high, this variable cannot be reliably determinfrom the simulation data. In the present problem this is tcase with the transversal deviations ofc, w, Syx and Syy

from their width-averaged values as well as withdu/dy andv. For this reason only the width-averages ofw, Syx , andSyy determined from the DEM simulations are shown. Evethough not all aspects of theoretical solutions could be copared with the simulations, those that could~i.e., width av-erages! are sufficient to assess the performance of the timdependent kinetic theory. Figures 4, 6, 8, and 10 describesteady Couette flow solutions for cases I–IV. These figuconsist of six panels that depict the transverse profiles ofc,w, u, Syx , Sxx , andSyy . In thec, w, andu panels theoret-

FIG. 7. Transient Couette flow, case III~e50.84, L/s55, e54!. Solidlines: width-averaged values; dashed lines: values near the wall; dot–dalines: values near the centerline; plus symbols: width-averaged DEM da

FIG. 8. Steady Couette flow, case III~e50.84,L/s55, e54!. Lines: nu-merical solution, symbols: DEM data. In panels~d!, ~e!, and~f!, solid linesand 1 symbols: total stresses; dashed lines and x symbols: collisiostresses; dot–dashed lines and o symbols: transport stresses.


129.115.103.99 On: Wed,

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ical predictions are shown by solid lines while the DEM daare shown by1 symbols. In the stress panels the tostresses are shown by solid lines and1 symbols, transportstresses by dot–dashed lines ands symbols and collisionalstresses by dashed lines and x symbols.

The results for case I are shown in Figs. 3 and 4. In tcasee5td /t r50.25, so the ratio of the momentum diffusioscale to the energy relaxation scale is relatively small. Thefore, the perturbation theory developed in Sec. V is expecto apply reasonably well to this case. It can be seen from3 thatw andSyy are approximately constant across the widthroughout the flow transient~solid, dashed, and dot–dashelines are indistinguishable!, which is consistent with the perturbation theory. Figure 3 also demonstrates the existenctwo distinct time scales, namely the momentum diffusiand energy relaxation time scales. It can be seen that theit takes forw to approach its steady state is of the order oft r

eda.

al

FIG. 9. Transient Couette flow, case IV~e50.96, L/s510, e54!. Solidlines: width-averaged values; dashed lines: values near the wall; dot–dalines: values near the centerline; plus symbols: width-averaged DEM d

FIG. 10. Steady Couette flow, case IV~e 5 0.96, L/s 5 10, e 5 4!.Lines: numerical solution; symbols: DEM data. In panels~d!, ~e!, and ~f!,solid lines and1 symbols: total stresses; dashed lines and x symbols:lisional stresses; dot–dashed lines and o symbols: transport stresses.


03 Sep 2014 07:20:57

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( t;1), while the time it takes for the velocity profile tapproach its~approximately linear! steady state is significantly shorter~of the order oftd5et r!. Computations witheof 0.1, 0.01, and 0.001, which are not presented here, hfurther verified that the ratio of times it takes foru andw toreach their respective steady states~within 1%! is indeedproportional toe. Compressibility effects can be observedthe behavior ofc2c0 andv. These effects are small and amore important during the momentum diffusion stage. Imediately after the change in the wall velocity the conctration near the wall decreases, and the transverse mofrom the wall towards the centerline begins (v,0). Theconcentration near the centerline increases and reachmaximum. At this time the transversal motion changes dirtion (v.0) and the concentration begins a gradual approtoward its steady-state distribution~which is nearly uniformfor a smalle!. The DEM data shown in Fig. 3 were obtaineby the averaging procedure described in Sec. IV C followby subsequent averaging over the width of the flow. A sstantial scatter in the DEM data can be observed duringportion of the flow that is supposed to be steady. This iconsequence of a finite number of collisions within the aeraging volume during the averaging time interval, whiwas selected asDt50.1t r in order to resolve the flow transient. The overall agreement between the theory and simtions is very good. Figure 4 shows the steady flow profifor case I. The DEM data were obtained by the averagprocedure described in Sec. IV C. The averaging strip lenwas selected asDy50.25s, while the time-averaging interval was between 5t r and 15t r (Dt510t r). It can be seen thathe DEM data forc exhibit rapid variations~with a charac-teristic length scale of the order of particle diameter! near thewalls. These variations are caused by the presence ofwalls, which influences geometrical arrangements of pticles near the walls. Similar concentration distributions habeen previously observed and reported by Campbell,26,27

Louge,29 and others. The kinetic theory used in this studoes not take this phenomenon into account, and thuspredicted concentration distribution does not exhibit suvariations. At first sight the data forw does not seem to agrevery well with the theoretical predictions. However, note ththe range ofw in this figure is very small~0.38–0.40!, so thediscrepancies are only about 2%. This plot also illustrateseffect of statistical noise. Even with the relatively long timaveraging intervalDt, the transversal variations ofw ~signal!are apparently too small to be reliably detected in the DEdata. Hence, the predicted shape of thew profile could not beverified from the DEM data. Theu profile is approximatelylinear, with the dimensionless shear rate less than 1 duthe slip at the wall. Both transport and collisional stresexhibit rapid variations near the wall similar to those inc.However, the total stressesSyx and Syy are remarkably con-stant throughout the width of the flow~as they should be duto the momentum conservation!. This result validates the averaging procedure and proves beyond doubt that rapid vations near the wall are genuine flow features that have ning to do with statistical noise. It is also interestingobserve thatSxx is not equal toSyy and that the totalSxx doesexhibit variations near the wall. This is not inconsistent b


129.115.103.99 On: Wed,

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cause there is no momentum conservation constraint forSxx .The results for case II are shown in Figs. 5 and 6. In t

casee5td /t r51, i.e. the momentum diffusion and energrelaxation time scales are equal. The transient flow featuare similar to those in case I, except that the approach ovariables toward their steady-state distributions takesproximately the same amount of time. Deviations ofw fromits width average can now be observed, though it couldargued that they are still relatively small. The normal streis still approximately uniform across the flow at all timeTheoretical predictions are in a very good agreement wthe DEM simulation. Steady flow profiles shown in Fig.are similar to those for case I shown in Fig. 4. Howevercan be seen that transversal variations ofw are now largerthan in case I~the noise to signal ratio is smaller! and thepredicted shape of this profile can be verified from the DEdata.

The results for case III are shown in Figs. 7 and 8. In tcasee5td /t r54. Duration of the transient is now of thorder oftd . Concentration and transversal velocity behavea qualitatively similar way as in cases I and II. However,this casew exhibits significant variations across the widtFurthermore, even the normal stress exhibits variatiacross the width, which are a consequence of the transvemotion. Normal stress is higher at the wall than at the cterline during the time whendv/dt,0, and vice versa. As itcan be seen from both Figs. 7 and 8, the agreement betwthe theoretical predictions and the DEM data is poor. Dcrepancies in magnitudes of the width-averaged values ow,Syx , andSyy are of the order of 50%. Qualitative shapesthe steady flow profiles are similar.

The results for case IV are shown in Figs. 9 and 10.this casee5td /t r54 as in case III, whilee50.96 as in caseII. This flow is qualitatively similar to the flow in case III. Ican be seen that the duration of the flow transient is oforder of td as in case III. Quantitative differences are duethe different values ofb. The agreement between the theretical predictions and the DEM data is good. It can be ccluded that the kinetic theory is capable of correctly preding steady and transient Couette flows of nearly elaparticles~e.g.,e50.99,e50.96!, regardless of the value oe. However, ase decreases the agreement deteriorates, asbe seen in case III (e50.84). This disagreement is due to thfact that the nearly elastic version of the kinetic theory hbeen used. Ase decreases, terms of the order of (12e)2,which have been neglected in this theory, become moremore significant, and the theory gradually breaks down.

B. Cyclic flows

Numerical results have been obtained for the casedisks. The following parameters are held constant:c050.5,d/s51, s/d50, and ew /e51. The problem is uniquelyspecified by assigning values ofb, e, and VL/uw

rms, or,equivalently, by assigning values ofb, e, V ~or V!. A para-metric study focused on the effects of the ratios of relevtime scales was performed. The results for 12 cases arecussed below. Each case is labeled by a letter~A–D! and anumber~1–3!. The letter part of a label is used to identifye


03 Sep 2014 07:20:57

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and L/s as follows: in group A,e50.99 andL/s55 (e50.25); in group B,e50.96 andL/s55 (e51); in groupC, e50.84 andL/s55 (e54); and in group D,e50.96 andL/s510 (e54). The number part of a label is used to idetify V as follows: in group 1,V50.5; in group 2,V52; andin group 3,V 5 8. The values ofV can be determined fromV5V/e. For each case predictions of the kinetic theory wobtained by numerical solution of Eqs.~1!–~4! using the im-plicit finite-difference method. Numerical simulations weperformed using the soft-particle DEM method. Asymptosolutions based on the perturbation theory presented inIV were obtained for group A (e50.25).

Some general features of all cyclic flow solutions adiscussed first. Ast→` each flow tends to a limit cycle~periodic flow!. Mathematically, the limit cycle solutioncould be obtained by solving the governing equations~1!–~4! subject to the periodicity conditions (c,u,v,w)(y,t)5(c,u,v,w)(y,t12p/V) instead of the steady flow initiaconditions~11!. In the limit cyclesc, v, w, andSyy oscillatewith a period equal to one-half of the period of the woscillations~i.e., p/V!, while u and Syx oscillate with a pe-riod equal to the period of wall oscillations~i.e., 2p/V!. Anexplanation for this can be traced back to the governequations~1!–~4!. Equations~2! and ~8! indicate thatu de-pends on both the magnitude and direction ofuw , but sincein the rest of the governing equationsu appears only in(du/dy)2, c, v, andw depend only on the magnitude but non direction ofu ~anduw!.

In group A ~e50.99, L/s55! the ratio of the momen-tum diffusion to energy relaxation time scales is relativesmall (e5td /t r50.25), so the asymptotic solutions deveoped in Sec. IV are expected to apply reasonably well. Hethe solutions in group A are discussed in relation to thasymptotic solutions. These asymptotic solutions are difent for V;e ~‘‘slow’’ oscillations! andV;1 ~‘‘fast’’ oscil-lations!. For e!1 it has been shown that these two solutiotend to the same intermediate limit in whiche!V!1 ~e.g.,V;e1/2!. However, whene is small but finite the boundarybetween these two solutions is not clear cut. The slooscillation asymptotic solution, obtained by the regular pturbation theory, corresponds~to the zeroth order of approximation! to an unsteady simple shear flow in which tconcentration, granular temperature, shear rate, and streare all constant across the flow. AsV increases the amplitudof oscillations ofw decreases until, in the intermediate limthese oscillations vanish and the limit-cycle value ofw be-comes independent ofV. The fast-oscillation asymptotic solution, obtained by the multiple-scale analysis, correspo~also to the zeroth order of approximation! to a flow in whichthe concentration and granular temperature are consacross the flow, but the velocity field no longer corresponto the unsteady simple shear flow. According to tasymptotic solution~71! the velocity field in the limit cycleis a harmonic function of time. This function can be epressed in a formA( y)cos@Vt1F( y)#, where A( y) andF( y) are the amplitude and the phase lag, respectively, bof which are complicated functions ofy. The shear ratedu/dy can be expressed in a similar way. In this case


129.115.103.99 On: Wed,

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amplitude of the oscillations ofw is zero, but the limit-cyclevalue of w is not independent ofV. For both slow- andfast-oscillation solutions the duration of the transient ofw tothe limit cycle is of the order oft r .

Results for cases A1, A2, and A3 are shown in Figs. 112, and 13, respectively. These figures consist of six panthat depict the time dependence ofc, du/dy, v, w, anddimensionless stressesSyx5Syx /(rswsusb) and Syy

5Syy /(rsws2). In each panel there are three lines: solid

dashed, and dot–dashed, which show the width-averagvalues, wall values, and centerline values of these functiorespectively. The exception isv, for which the dashed anddot–dashed lines show the values at the nodes nearest towall and centerline, respectively~sincev50 at both the walland the centerline!. The DEM data, which are shown by1

FIG. 11. Cyclic Couette flow, case A1~e50.99,L/s55, e50.25,V50.5!.Solid lines: width-averaged values; dashed lines: values near the wall; ddashed lines: values near the centerline; plus symbols: width-averaged Ddata.

FIG. 12. Cyclic Couette flow, case A2~e50.99,L/s55, e50.25, V52!.Solid lines: width-averaged values; dashed lines: values near the wall; ddashed lines: values near the centerline; plus symbols: width-averaged Ddata.


03 Sep 2014 07:20:57

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symbols, include only the width averages ofw, Syx , andSyy . These data were obtained by time averaging ovefraction of the period of oscillations and volume averaginover the entire flow domain. This volume–time-averaginelement is generally sufficiently large so that statistical nois a relatively small fraction of magnitudes of variables ointerest.

Figure 11 shows the results for case A1 (V50.5). Thissolution has general features of the slow-oscillatioasymptotic solution. The width-averaged values, wall valuand centerline values ofw andSyy are nearly identical~solid,dashed, and dot–dashed lines coincide!. Deviations of cfrom c0 and magnitudes ofv are small. However, the wallvalues ofdu/dy and Syx exhibit a slight phase shift com-pared to the width-averaged and wall values~which are ap-proximately equal to each other!, indicating a slight depar-ture from the slow-oscillation asymptotic solution. It can alsbe seen from Fig. 11 that the duration of the transient ofw tothe limit cycle is of the order oft r ( t;1), as predicted bythe asymptotic theory. In this case this time is equal tofraction of the period of wall oscillations. Finally, it can beseen that the predictions of the kinetic theory are in a exclent agreement with the results of the DEM simulations.

Figure 12 shows the results for case A2 (V52). In thiscase the solution has general features of the fast-oscillasolution: concentration and granular temperature are neaconstant across the flow, but shear rates at the wall and atcenterline have significantly different amplitudes and phalags. The shear rate at the centerline has a smaller amplitthan and lags behind the shear rate at the wall. The shstress behaves in a similar way. The amplitude of oscillatioof w is significantly smaller that in case A1 shown in Fig. 11It can also be seen from Fig. 12 that the duration of ttransient ofw to the limit cycle is also of the order oft r ( t;1), as predicted by the asymptotic theory. However, in thcase this time is equal to several periods of the wall speoscillations. Finally, it can be seen that the predictions of t

FIG. 13. Cyclic Couette flow, case A3~e50.99,L/s55, e50.25, V58!.Solid lines: width-averaged values; dashed lines: values near the wall; ddashed lines: values near the centerline; plus symbols: width-averaged Ddata.


129.115.103.99 On: Wed,

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kinetic theory are in a good agreement with the DEM daHowever, the data exhibit more statistical noise due tofact that the ratio of the time-averaging interval to the metime between collisions becomes smaller asV increases.

Figure 13 shows the results for case A3 (V58). Thephase lag and amplitude difference betweendu/dy and Syx

at the wall and at the centerline are much greater than in cA2. A new feature that can be observed is thatw andSyy areno longer approximately constant across the flow. Herew atthe wall is significantly higher thatw at the centerline, andthere also seems to be a slight phase lag between theThe differences betweenSyy at the wall and at the centerlinare smaller but noticeable. The agreement between thedictions of the kinetic theory and DEM data does not appto be as good as in the previous cases. However, note tharange of the vertical axes forw andSyy is much smaller thanin Figs. 11 and 12. The relative ‘‘discrepancy’’ is still quitsmall, in spite of an increased amount of statistical no~especially inSyy data!. This case indicates that the ‘‘fast’oscillation asymptotic theory gradually breaks down asV

becomes large. WhenV becomes extremely large the kinettheory itself is not applicable since the number of collisioper particle per oscillation cycle becomes extremely sma

Results for group B are similar to those for group Athe same values ofV. The velocity fields are particularlysimilar, indicating a small effect ofe on the velocity field.However, there are also important differences. First,transversal variations ofw are much more significant ingroup B than in group A. Furthermore, this effect becommore pronounced asV increases. The reason for this is thwhen e;1, the diffusion, production, and dissipation termin the energy equation are all of the same order of magtude. WhenV!1 the flow is basically quasi-steady and thgranular temperature profile takes the same shape assteady flow. The instantaneous values ofw are proportionalto the current wall velocity, so there should be no phasebetweenw at the wall and at the centerline. However, whV;1 the unsteady term in the energy equation is of the saorder of magnitude as the diffusion, production, and dissition terms. Hence, the shape of instantaneousw profiles de-pends on time. This can explain that the phase lag andference in amplitudes betweenw at the wall and at thecenterline increases asV increases. The second differencbetween groups A and B is the amplitude of oscillations inwand Syy . These amplitudes are similar in cases A1 and Bbut those in cases B2 and B3 are much greater than in cA2 and A3, respectively. Hence, the amplitude ofw ap-proaches zero asV increases only for smalle and not ingeneral. The predictions of the kinetic theory for group B ain all three cases in very good agreement with the DEM da

Results for cases C1, C2, and C3 are shown in Figs.15, and 16, respectively. The velocity and shear stress fiare remarkably similar for different values ofe at the samevalues ofV. The w and Syy fields in group C are qualitatively similar to those in group A at the same ofV. Theduration of the transient ofw to the limit cycle is now of theorder of td ( t;e). The predictions of the kinetic theory ar

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in all three cases in a poor quantitative agreement withDEM data. The magnitudes of the width-averagedw, Syx ,and Syy predicted by the theory are about twice as highthose determined from the DEM simulations. For cases DD2, and D3,e54, as in group C, whilee50.96, as in groupB. These cases are qualitatively similar to those in groupbut in this case the predictions of the kinetic theory are agin a very good agreement with the DEM data, unlike fgroup C. Hence, it can be concluded that cyclic flows at tsame values ofe and V are qualitatively similar to eachother. It can also be concluded that the kinetic theorycapable of correctly predicting cyclic Couette flows of nearelastic particles~e.g., e50.99, e50.96!, regardless of thevalue of e. However, ase decreases the agreement deterrates, as it can be seen for group C (e50.84). This disagree-

FIG. 14. Cyclic Couette flow, case C1~e50.84, L/s55, e54, V50.5!.Solid lines: width-averaged values; dashed lines: values near the wall; ddashed lines: values near the centerline; plus symbols: width-averaged Ddata.

FIG. 15. Cyclic Couette flow, case C2~e50.84, L/s55, e54, V52!.Solid lines: width-averaged values; dashed lines: values near the wall; ddashed lines: values near the centerline; plus symbols: width-averaged Ddata.


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ment is due to the fact that the nearly elastic version of tkinetic theory has been used.

VII. CONCLUSIONS

Unsteady Couette granular flows were analyzed botheoretically and by the means of the discrete elememethod. Specific results were obtained for the transieflows, in which the wall speed instantaneously changes froone constant value to another, and the cyclic flows, for whithe wall speed is a harmonic function of time. Theoreticformulation of the problem is based on the kinetic theory fosmooth, nearly elastic particles. The resulting boundavalue problems were solved numerically by finite-differencmethods.

Transient Couette flows are characterized by two timscales, namely momentum diffusion (td) and energy relax-ation (t r) time scales. The flow behavior critically dependon the ratio of these two time scales, i.e. parametere5td /t r . The duration of the approach of the mean velocifield to its steady state is of the order oftd , while the ap-proach of the fluctuation energy field to its steady state isthe order oft r . The duration of the transient flow is of theorder of the greater of these two time scales. Cyclic Coueflows are characterized by three time scales, namely momtum diffusion (td), energy relaxation (t r), and wall oscilla-tion (tw) time scales. The flow behavior is strongly influenced by the relative magnitudes of these time scales,expressed by parameterse5td /t r andV5td /tw . In general,cyclic Couette flows tend to limit cycles~periodic flows! ast→`. The duration of the transient to the limit cycle is othe order of the larger of the time scalestd and t r . In thelimit cycle the concentration, granular temperature, antransversal velocity fields oscillate with a period equal tohalf of the period of wall oscillations while the longitudinavelocity field oscillates with a period equal to the period owall oscillations.

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FIG. 16. Cyclic Couette flow, case C3~e50.84, L/s55, e54, V58!.Solid lines: width-averaged values; dashed lines: values near the wall; ddashed lines: values near the centerline; plus symbols: width-averaged Ddata.


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Asymptotic solutions were obtained by perturbatianalysis for the case in which the momentum diffusion tiscale is much smaller than the energy relaxation time scThis limit occurs when the parametere5h/b2 is small,which corresponds to nearly elastic particles and a finite pticle diameter to flow width ratio. This limit also corresponto small values of the flow Mach number, and the flowsthis limit are nearly incompressible. In this limit the transieflows consist of a relatively brief momentum diffusion stagduring which the mean velocity profile rapidly adjusts to tnew wall motion, and a relatively long relaxation stage, ding which the granular temperature gradually changes frthe initial to the final steady granular temperature. Tozeroth order of approximation, in the momentum diffusistage the granular fluid behaves as an incompressible Ntonian fluid with a constant viscosity proportional to thsquare root of the initial granular temperature. The only dference with respect to the classical solution for an incopressible Newtonian fluid is due to the effect of the slipthe wall. At the end of the momentum diffusion stage, tvelocity field is compatible with the new wall motion whilthe granular temperature is still compatible with the old wmotion. In the relaxation stage the time dependence ofgranular temperature is governed by an ODE that statesthe rate of change of the fluctuation energy equals the difence between the rates of its production and dissipationto both internal and boundary mechanisms. To the zeorder in e, the relaxation stage is similar to the unboundtransient granular shear flow studied by Louge, Jenkins,Hopkins.30 The mean velocity profile is linear, concentratioand granular temperature are constant throughout the fland the transversal motion is negligible. The only differenwith respect to the unbounded flow problem is due tocontributions of the slip work and energy dissipationboundary collisions to the total production and dissipatterms, respectively. The agreement between the full numcal solutions and zeroth-order perturbation solutions forstresses and granular temperature is excellent fore<0.25.Transient flows withe51 resemble in many ways the smae solutions, except that momentum diffusion and energylaxation stages are no longer distinct but simultaneous.normal stress is still approximately uniform across the flfield. As e increases the flows become more complicatedparticular, the normal stress is no longer constant acrossfield, indicating importance of transversal accelerationsthe flow dynamics.

Asymptotic solutions were also obtained for the cycflows for the case in whiche!1. The cyclic flows in thislimit are also nearly incompressible, except at extremhigh frequencies. Within this limit two frequency rangwere considered separately: ‘‘slow’’ oscillations~V;e, i.e.tw;t r! and ‘‘fast’’ oscillations~V;1, i.e. tw;td!. The solu-tion to the intermediate case~e!V!1, i.e. td!tw!t r! wasfound by taking theV@e andV!1 limits of the slow- andfast-oscillation solutions, respectively. TheV!e limit of theslow-oscillation solution corresponds to the trivial quasteady case. The case of very fast oscillations~V@1, i.e.tw!td!, which is essentially a vibrating-boundary proble


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was not considered since in this case the number of csions per particle per oscillation cycle becomes small andboundary conditions of the kinetic theory would have tomodified before approaching this problem. To the zerothder of approximation, the slow-oscillation flows~e!1 andV;e! are basically unsteady simple shear flows. The ccentration and granular temperature and are approximaconstant across the flow, transversal motion is negligible,longitudinal velocity profile is linear, and at every instacompatible with the current wall motion~i.e., the same as ina steady Couette flow in which the wall speed is the samthe current wall speed!. This is the instantaneous diffusiolimit, in which the velocity field is instantaneously adjusteto the current wall motion. However, the granular tempeture is not instantaneously adjusted to the current velofield, except in the quasi-steady limitV!e. The time depen-dence of the granular temperature is governed by a ODEstates that the rate of change of the fluctuating energy eqthe difference between the rates of its production and dipation due to both internal and boundary mechanisms.amplitude of the oscillations of the granular temperaturecreases asV increases until it vanishes in the intermedialimit e!V!1. To the same order of approximation, in thfast-oscillation flows~e!1 and V;1!, the concentrationand granular temperature are also constant across thebut the longitudinal velocity profile is nonlinear. The veloity solution consists of a transient part and a periodic limcycle part. The duration of the transient to the limit cycleof the order oftd . The limit cycle velocity field is a har-monic function of time, but the amplitude and phase lagdifferent at different locations in the flow. Both the amptude difference and phase lag between the shear rates awall and at the centerline increase asV increases. The timedependence of the granular temperature is also governean ODE that states that the rate of change of the fluctuaenergy equals the difference between the rates of its protion and dissipation due to both internal and boundmechanisms. The difference with respect to the slooscillation case occurs in the production term, as the velofields responsible for the production of the fluctuation eneare different in the two cases. In this solution the granutemperature varies smoothly at a time scale proportional tt r

~oscillations of the granular temperature are a first-orderfect!. Hence, in the limit cycle the granular temperatureconstant both in space and time. Its value increases aVincreases. The duration of the transient to this limiting vais of the order oft r . This solution is in many ways similar tothe corresponding solution for the cyclic flow of an incompressible Newtonian fluid. One difference is due to the effof slip at the wall and the other is due to the fact that granuviscosity is time dependent during the transient stage. Hever, in the limit cycle the granular viscosity is constantspace and time and the only remaining ‘‘non-Newtoniaeffect is due to the slip at the wall. It should be emphasizthat this similarity between granular flows and incompreible Newtonian fluid flows exists only ift r is much greaterthan bothtd andtw . Comparison of the asymptotic solutionand numerical solutions of the full system of equations


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excellent in both slow- and fast-oscillation limits fore ashigh as 0.25. However, for such a~finite! value of e theboundary between the two limiting solutions is not clear cAt the intermediate frequencies both solutions have advtages and disadvantages one over another. The soscillation solution underpredicts the magnitude whilefast-oscillation solution underpredicts~ignores! the oscilla-tions of the granular temperature. Of the two solutions,fast-oscillation solution is to be preferred in the intermedifrequency range because it correctly predicts the velofield while the slow-oscillation solution does not.

For finite and large values ofe the asymptotic solutionsare no longer applicable so the flow behavior must be alyzed on the basis of numerical solutions of the full setequations. For a finitee the velocity fields are similar tothose fore!1 at the same values ofV. The effect ofe onthe velocity field is relatively small while the effect ofV isof primary importance. For smallV, the flow is quasi-steadyand the velocity profile is mildly nonlinear and compatibwith the current wall motion. For finiteV the limit cyclevelocity field is a harmonic function of time, with differenamplitudes and phase lags at different locations in the flBoth the amplitude difference and phase lag betweenshear rates at the wall and at the centerline increase aVincreases, just like fore!1. The granular temperature fieldmuch more significantly affected by the value ofe. First, thevariations of the granular temperature across the flowmore pronounced. Second, the amplitude of the oscillatiof the granular temperature is much greater than in the!1 case, at the same values ofV. Both effects become morpronounced ase increases.

Discovery of the regime in which granular flows cantreated as incompressible is important. Mathematical simfications resulting from the incompressibility assumptionsubstantial and should be exploited when appropriate.requirements for incompressibility, though more restrictthan for regular fluids, are not overly restrictive and canachieved in practice.

In comparison with the DEM simulations, the kinettheory has performed very well for both transient and cycflows for nearly elastic disks~e.g., e50.99 ande50.96!,regardless of the value ofe. This is a remarkable achievement of the wall-bounded, time-dependent kinetic theoThe value ofc0 used~0.5 for disks! corresponds to the casin which both transport and collisional contributions to tstresses and fluctuation energy flux are of equal importaso both had to be modeled correctly in order to achieve sa good quantitative comparison. However, ase decreasesthe agreement gradually deteriorates. This is to be expebecause the nearly elastic (e→1) version of the kinetictheory has been used. The quantitative discrepancies fe50.84 are quite significant~of the order of 50%–100%!. Theauthors of the nearly elastic kinetic theory have carefully aconsistently emphasized the limitations of this version oftheory. The present study clearly illustrates these limitatiand reinforces the point that the nearly elastic kinetic theworks for nearly elastic particles but fails to produce acrate results when the particles are not nearly elastic. In


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present study we show that particles withe50.96 can beconsidered to be ‘‘nearly elastic’’ while particles withe50.84 cannot, at least for the particular class of flows unconsideration. As far as the values ofe between these tworeference points are concerned, several calculations andresponding simulations withe50.9 have been performedThe discrepancies were found to be of the order of a halthe discrepancies fore50.84, indicating that the loss of accuracy is gradual.

The present study has lead to a better understandinunsteady rapid granular flows. This study represents a nastep between the development of governing equationstheir application to complex problems of practical impotance. The unsteady Couette flows considered here aretively simple, yet they are significantly more complex thsteady uniform flows that have been primarily consideredthe past. It is encouraging that the kinetic theory is ableaccurately predict the behavior of these wall-bounded, timdependent flows.

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Unsteady Couette granular flows

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