Kinetic theory of fluidized granular matter

PHYSICAL REVIEW E APRIL 1997VOLUME 55, NUMBER 4

Kinetic theory of fluidized granular matter

Byung Chan Eu* and Hikmat FarhatDepartment of Chemistry, McGill University, Montreal, Quebec, Canada H3A 2K6

~Received 8 July 1996; revised manuscript received 23 September 1996!

In this paper, we present a statistical treatment of fluidized, elastic granular matter and a kinetic equation thatdescribes the evolution of macroscopic properties of such matter. The present kinetic theory recognizes that theeffects of excluded volume become dominant in the dynamic evolution of an assembly of granules andaccordingly takes them into account in the formulation. On the basis of the equilibrium solution of the kineticequation, a thermodynamics-like mathematical structure is constructed for the Boltzmann entropy of granularmatter. The meaning of temperature in this mathematical structure is fixed by the shear rate. The equilibriumsolution is shown to yield a density distribution comparable with the experimental data of Clementet al.@Europhys. Lett.16, 133~1991!#. The shear viscosity of granular matter is shown to increase with the packingfraction. This behavior is in qualitative agreement with experimental result by Haneset al. @J. Fluid Mech.150,357 ~1985!#. The viscosity also increases with the shear rate since the ‘‘temperature’’ increases with the shearrate in the case of granular matter. Consequently, the granular matter is shown to be dilatant, as is experimen-tally known. @S1063-651X~97!01504-3#

PACS number~s!: 05.20.Dd, 05.60.1w, 46.10.1z, 83.70.Fn

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I. INTRODUCTION

Granular matter has unusual properties and behavior@1,2#that defy easy comprehension from the conventional waythinking and theories of continuum matter known tothrough our studies of condensed matter up until now. Tsubject has been studied in engineering@3–6# and has beenlately drawing attention in physics@7–9#. There have beensome kinetic theories@10–14# proposed for such mattealong the lines of the classical Chapman-Enskog theorygases@15#, but they appear to be only a first step towatheories of a deeper, perhaps more appropriate, nature.recent experimental article and computer simulationsgranular matter, Clement and Rajchenbach@16# and Gallas,Herrmann, and Sokolowski@17#, respectively, report that adensity distribution of granules subjected to an acoustic pturbation at the bottom of its pile exhibits a Fermi-like ditribution. Although this feature appears to be what is initively expected and intriguing, it requires some thought wregard to the underlying dynamical and kinetic theoreprinciples. It is the aim of the present paper to presenviewpoint on the kinetic theory of granular matter whichsufficiently fluidized to treat it as if it is a fluid. To make thtreatment as simple as possible without sacrificing the estial aspects of granular matter, we assume that the granare elastic, since the assumption can still describe dissipain the system. The inelasticity of collisions between granuis important, but not of primary importance in a statisticdescription of granular matter, since a similar effect canachieved by means of momentum correlations, as willshown later. A truly particulate, statistical treatment of tsubject should take into account the internal states of gules and their evolution at the kinetic theory level, but suctreatment would be rather involved in formalism. Here,

*Also at the Centre for the Physics of Materials, DepartmenPhysics, McGill University, Montreal, Quebec, Canada H3A 2K

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set aside the question of internal states of granules by tring them as an elastic substance and pay attention totranslational motions only. Nevertheless, we believe thattheory presented in this work still has sufficient validity asstarting point for further investigations on granular mattThe present theory is quite different from the statistical trement given recently by Bernu, Delyon, and Mazighi@18# ongranular matter and those in Refs.@9–14#.

In Sec. II, we examine granular matter for its featuredistinct from the ordinary molecular fluids, with a specifiaim of guiding us in developing a kinetic theory of fluidizeelastic granular matter. Based on what we have foundgranular matter in Sec. II, a kinetic theory is then developfor the fluidized granular matter in Sec. III. The balanequations can be derived for mass, momentum, and enerthe conservation laws—from the kinetic equation if the mamomentum, and internal energy densities are statisticallyfined appropriately. Such definitions are given in Sec. IV, bthe explicit forms for the balance equations are referred tothe literature, since they are in the same forms as thosethe ordinary continuum matter. However, in applying thein flow problems the existence of a basic, finite length scof the granules must be carefully taken into account. TheHtheorem is also briefly discussed together with the equirium solution of the kinetic equation in the same section. Tequilibrium solution is used to construct a thermodynamanalogy in Sec. V, since thermodynamics of granular mais unavailable at present. In Sec. VI, the parameters apping in the equilibrium solution are corresponded to obseables such as shear rate and so on. We thereby construanalogy of thermodynamics by means of the equilibriumlution of the kinetic equation. The calculated density distbution is compared with experiment in Sec. VII. The viscoity of granular matter is calculated in Sec. VIII. It shows thdilatancy of the granular matter and a qualitatively corrdensity dependence reminiscent of experiment. The concing remarks are given in Sec. IX.f

4187 © 1997 The American Physical Society

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4188 55BYUNG CHAN EU AND HIKMAT FARHAT

II. GRANULAR MATTER IN CONTRASTTO ORDINARY MOLECULAR FLUIDS

To acquire a clear picture of how distinct the behaviorgranular matter is from the ordinary molecular fluids, wcompare characteristics of the two kinds of matter in qution. First of all, for a given volume there is a differencemany orders of magnitude in number between the ordinmolecular fluids and granular matter, since there are ab1019 molecules in 1 cm3 of a molecular fluid whereas thercan be, for example, only 104 ~of course, depending on thsize of the grain! in the case of granular matter. This impliethat considerable caution must be exercised when a statistheory is applied to granular matter, since the charactercally small number density of granules can give rise to sable fluctuations in the mean values calculated. Closelylated to this feature is the fact that granular particlesmacroscopic in size and mass. This also gives rise toquestion under what condition can granular matter be treas a continuum and calls into question the validity of cotinuum equations such as the mass, momentum, and enbalance equations. In general, since grains are massivegravitational force cannot be left out in general when gralar matter is considered for its dynamic properties, wherthe gravitational force plays an insignificant role in the caof molecular fluids except, perhaps, for interfacial phenoena. Unlike molecular fluids, attractive forces betwegranular particles are negligible and play an insignificant rin granular dynamics. Dynamics in granular matter is domnated by the hard, repulsive forces and the excluded-volueffects come into play significantly in the case of granumatter, whereas in the case of molecular fluids it becomimportant only when the density of the latter is close toclose-packed fluid value. In the case of molecular fluithermodynamics provides a well-defined continuum desction of their behavior and such a continuum description cbe furnished by a molecular theory with the help of statistimechanics and, in particular, kinetic theory. In fact, in tcase of molecular fluids, these two lines of theory are mually complementary in the sense that, whereas the thermnamic theory is given molecular foundations by the statical theory of the system, the latter is elevated to a phystheory from a purely mathematical probability theory withe help of thermodynamics. If we wish to develop a statical theory of granular matter, it is therefore importanthave a phenomenological thermodynamic theory of the stem firmly founded on the principles of thermodynamics aif such a phenomenological theory is absent, on an anathat is well thought through at least from the viewpointthermodynamic principles. Since it is not clear at preswhat the thermodynamics of processes in granular mashould be like, this question should be resolved beforeattempt to formulate statistical and kinetic theories for gralar matter. We will return to this question later.

The consideration made earlier on granular matter sgests that the notion of finite size and excluded volumesociated with granules play a crucial role. This of coursewell recognized in the phenomenological studies@1–6,8#,statistical treatments@10–12,15,18#, and computer simulations @9,17#, but the subtle aspects of excluded volumenot seem to have been fully and clearly elucidated at

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dynamical level when the granular assembly is in motioWe believe that the excluded-volume effect and, especiathe momentum correlations arising therefrom should be bter accounted for than has been so far in the kinetic-thetreatments mentioned. For definiteness of our discussionconsider an assembly ofN spherical, elastic granular particles of diameters and massm in this work. The totalvolume ofN such granules then is

V05Nv0 , ~1!

wherev0 is the volume of the granule

v054p

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2 D 3. ~2!

Note thatV0, however, is not the actual volume of the sytem. If the actual volume of the assembly in a configuratis denoted byV, then the number densityn at the givenconfiguration of the granular assembly is given by

n5N/V. ~3!

The packing fraction of the assembly may be defined by

f5 16nps3. ~4!

Unlike molecular fluids, the number density of granular mter sensitively depends on its configuration, since it cansume an unusual metastable form~e.g., arches and voids!with a volume larger than a close-packed volume, whichthe minimum volume the system can assume. The clopacked volume of the granular matter will be denoted byVc ,which yields the volume per particlevc5Vc/N that may bewritten as

vc516pRc

3. ~5!

HereRc , defined by Eq.~5!, is the diameter of imaginaryspheres making up the volumeVc . The ratioRc/s is largerthan unity. The volume of configuration space ofN granularparticles in the close-packed configuration is therefore emated to be

Gconf5~ 16pRc

3!N. ~6!

The diameters of the granule orRc sets the basic lengthscale in the description of the granular assembly of intehere. All the distances in the theory of granular matter csidered here can be reckoned in this scale. The differencthe configuration space volumesVc andV0 is

DGconf5@ 16p~Rc

32s3!#N, ~7!

which ranges in the interval

@ 12ps2dRc#

N<DGconf<@ 12pRc

2dRc#N. ~8!

Here dRc5Rc2s. Therefore, the minimum volume uncetainty of the excluded volume of the system, i.e.,the granu-lar matter, is given by

GR[DRcN 5@ 1

2ps2dRc#N. ~9!

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55 4189KINETIC THEORY OF FLUIDIZED GRANULAR MATTER

Left unperturbed, granular matter in a stable or metastaconfiguration~state! does not flow and the velocities of thparticles are equal to zero. Only an external force suchshearing or other means of forcing produces a motion ofassembly in part or as a whole. If the frequency of the pturbation isv and the characteristic distance associated wit is amplitudeA, then the momentum of a granule of massmassociated with the motion is

pA5emAv, ~10!

where e is a dimensionless proportionality constant. If wdenote the velocity of the particle byc and the motion is inthe direction of the gravitational field, then at equilibrium

mgA5 12mc2,

whereg is the gravitational acceleration. That is, the charteristic distance in this instance is given by

A5c2

2g. ~11!

Bagnold@2# found that the mean velocity, or more precisethe speed of a granular assembly subjected to shearinproportional to shear rateg. Therefore, since the characteistic frequencyv is proportional tog in the case of shearingwe may take

c5a8g5av. ~12!

Here a is a constant parameter with the dimension of dtance which may be seta5A2eA. By combining Eqs.~10!–~12!, we obtain

pA5UgmAv, ~13!

where

Ug5Av2

g. ~14!

This parameterUg characterizes the motion of granular paticles subjected to the gravitational field and an external pturbation that displaces the particles from their positions istatic configuration. Therefore, the volume of the momentspace ofN granular particles going through such a memotion is given by

DpN5~UgmAv!3N ~15!

and the corresponding phase volume is then given by

Gc[DcN5~DpDRc

!N5@ 12psA

3Ug3~mAv!3#N, ~16!

where

sA5Vc1/3s,

Vc5dRc

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s21. ~17!

Therefore, it is possible to interpretsA as the effective diam-eter of the granule swelled by the factorV c

1/3 associated with

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the perturbation applied to the granular system. ThisGc is theminimum phase volume associated with the granular assbly, which under the gravitational field is perturbed byexternal force of frequencyv and amplitudeA in a close-packed configuration. This phase volume must be proptaken into account in the development of a kinetic theoryfluidized granular matter, since it must serve as distancemomentum scales for the phase volume of the system instatistical description of the granular matter of interest. TDc is essentially the phase volume which accommodategranular particle of the mean momentumpA . This basicphase volume per granuleDc is not negligible compared withthe actual phase volume in the case of granular particThis point is crucial in determining the behavior of an asembly of granular particles.

Let us denote the phase volume element of the granassembly around the phase point, or simply the pha~r (N),p(N)! by dG5dr (N)dp(N), where r (N) and p(N) collec-tively denote the center-of-mass position and momentvectors of granular particles in a fixed coordinate systeThen, the statistical description of such a system can be vonly if this phase volume of the system is much larger ththe basic phase volumeGc presented earlier. This will begenerally the case if the granular matter is sufficiently fluized. Thus, we will assume that the phase volume ofsystem contains a large number of the basic phase volumGcmentioned, and the former is reckoned in the scale oflatter,Gc . The number of basic phase volumes in unit phavolume is then equal to 1/Gc . And such a number is anelement that must be carefully taken into consideration wthe collision term in the kinetic equation is calculated on tbasis of granular collision dynamics.

III. KINETIC EQUATION FOR FLUIDIZEDGRANULAR MATTER

We have now come to realize that a granular assemshould behave qualitatively differently from the behavioran isolated pair of granular particles, which should stricobey the laws of motion within the framework of Newtoniamechanics. It is because of the very fact that the partichave a sizable excluded volume, which causes strong colations of momenta, that there are severe limitations oncollective motion of the particles in a congested assemblygranular particles. In the case of granular matter, becausthe relatively large volume fraction of the excluded volumthe aforementioned limitations become a dominant factorcollective motions of granular particles. We believe that tmomentum correlations take precedence over the inelastof collisions in the hierarchy of dominant effects in granumatter. We now take this feature into account in formulatia kinetic theory of such matter. A cogent picture for the kiof difference in the behaviors of an isolated pair of granuand an assembly of them can be seen in the example useJaeger, Nagel, and Behringer in their recent article@19#.These authors compare, say, the motions of a bead wisack of many beads that falls on a glass plate. A single bfalling on the glass plate repeatedly bounces off the pbefore it comes to rest, whereas the sack falls dead onplate and remains there. The basic difference is attributemany inelastic collisions that the beads in the sack

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through, but it is basically the excluded-volume effectbeads in the sack that frustrates the bouncings of thebeads hitting the plate which are predicted to occur byNewtonian laws of motion. In the case of the beads insack—a congested assembly of beads—the individualtions of beads are highly correlated and constrained, unisolated beads. In this connection, it must be pointed outthe friction forces on contact are indispensable inmolecular-dynamics simulations of granules by Gallas, Hrmann, and Sokolowski@17#, and that they are a kind omomentum correlation that springs into action when the pticles come into contact. We thus see that a congested asbly of granules puts certain statistical constraints on thenamical evolution that isolated individual particles wouhave followed according to the Newtonian mechanical laand that the normal Newtonian mechanical evolutionbeen thereby frustrated. That is, the statistical evolutionsuch an assembly, therefore, cannot be inferred to be clcal in the sense of the Boltzmann kinetic theory. This iscrucial point we would like to stress in this work.

We are interested in the probability of finding a granuparticle with momentump at position r in phase volumeelementdr dp at time t and the spatiotemporal evolution othe probability distribution function. That is, the objectinterest is the single distribution functionf ~v,r ;t! of velocityv5p/m at time t. If this distribution function is integratedover the entire phase volume, it is normalized to the tonumber of particleN:

E dr dv f ~v,r ;t !5N. ~18!

Therefore, when integrated over the momentum spacegives

E dv f ~v,r ;t !5n, ~19!

wheren5N/V is the number density. If we count the number of particles flowing in and out of the elementary phavolumedr dv at r andv in time intervaldt, it is given by

S ]

]t1v•“1F•“vD f ~v,r ;t !dr dv dt. ~20!

HereF is the external force per mass and“v5]/]v. This isthe change in number density due to the kinematic flowparticles in the elementary phase volume mentioned. Iimportant to note that the distribution function is taken touniform in this elementary phase volume, and this phaseume must be clearly larger thanGc. This means that thedistribution function is coarse grained in a scale comparato or larger than that ofDc . The kinematic change in numbedensity must be balanced by the change in number dendue to collisions of particles within the phase volume. If wdenote the collisional rate of change in number densityR8@ f #, then the number density change indr dv in time du-rationdt is

R8@ f #dr dv dt. ~21!

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On equating Eqs.~20! and ~21!, there follows the evolutionequation forf ~v,r ;t!:

S ]

]t1v•“1F•“vD f ~v,r ;t !5R8@ f #. ~22!

We now calculate the collisional contribution. It is useful fthis purpose to recognize that the collision integral in tkinetic ~e.g., Boltzmann! equation consists of two basic components. One is the collision dynamical part, which accoufor the collision event that is imagined to occur betweisolated particles~e.g., a pair of particles!. This collisionevent is governed by Newtonian laws of motion for a feparticles and, in the present theory, two particles. The otis the statistical part that gives the statistical weight foroccurrence of particles having the values of the dynamvariables predicted by the Newtonian laws. It is importantkeep a clear distinction between these two factors in makup the collision integral in the kinetic equation. The statiscal weighting factor is not altering the collision dynamiitself governed by the Newtonian mechanical laws, butstatistical probability of the mechanically predicted collisioevent deviates from the classical, Boltzmann kinetic-theform, which is usually taken in the kinetic-theory treatmen@20#.

First, let us define by vectork the unit vector along theapse line of the two hard spheres in contact that is parallethe vector connecting the centers of mass of the two pticles. The relative velocity vector of the particles is denotby g125v12v2, wherev1 andv2 are the velocities of the twoparticles 1 and 2 in collision. Then the volume swept by ttwo particles in collision in timedt is

s2~g12•k!dt.

We will henceforth drop the subscript 1 from the quantitifor particle 1. The initial velocities of the colliding pair oparticles are denoted byv andv2, wherev[v1. We also takeinto account the aforementioned coarse graining of distrition functions so that the distribution functions are assumto be uniform in the collision volume. The population of thcolliding pair of particles is f ~v,r ;t! f ~v2,r ;t!x~r ,r2ks!,wherex~r ,r2ks! is the spatial correlation function for thpair of granular particles in collision. In the case of a granlar assembly that has a large excluded volume relative toactual volume of the system and thus a relatively larelementary-phase volume occupied by a particle, the polation of the particles in the final state of collisionthat isstatistically realizableis

Cg*[@12Dc f * ~v,r ;t !#@12Dc f * ~v2 ,r ;t !#. ~23!

We elaborate on the presence of theCg* factor in the colli-sional rate of change. In the case of the collision of an ilated pair of granular particles, the collision process is deministically described by the Newtonian laws of motion athe particles end up in the final state as predicted byclassical mechanical laws. However, in a congested assbly of granular particles that are constantly in collision, somcollisional events are not statistically realizable, althouthey are dynamically dictated by the Newtonian laws of mtion, if the basic phase volumes into which the particles

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destined to go are already occupied by other particles inassembly. At this point, it is useful to recall the comparisof a bead and a sack of beads made by Jaeger, NagelBehringer @19#, which was mentioned earlier. SincDc f * ~v,r ;t! gives, on the average, the fraction of particlesbe found inDc , which are already at the state predictedthe collision of an isolated pair by the Newtonian laws athus precludes other particles coming intoDc , the factor@12Dc f * ~v,r ;t!# gives the fraction of particles that are stattically allowed in the said state on completion of the saevent of collision between the pair of particles 1 and 2. Sinthe other particle must be found atr2ks if one is atr , theother factor for particle 2 is similarly@12Dc f * ~v2,r2ks;t!#,but this factor is taken in the form@12Dc f * ~v2,r ;t!# for theconsistency with the coarse graining of the distribution fution over the collision volume. This factor, of course, hasphysical meaning similar to the factor@12Dc f * ~v,r ;t!#.Therefore, the factorCg* represents, on the average, tphysical reality that not all collisional events in a congesassembly of granular particles can be realized, at the sttical level of description, in states as predicted by the Netonian laws of motion for an isolated pair of particles athat there is an additional statistical constraint manifestitself because of the interactions of mutual exclusion amthe particles. This exclusion effect even appears in thementum space in the form of a momentum correlation fution, whichCg* in effect represents, since the statistical evlution of the system is considered in the phase space.factorCg* , a momentum correlation function, is absent in tordinary classical fluids sinceDc is O~\3! for ordinary fluidsand hence, for example,Dc f * ~v,r ;t!;nDc5O(n\3), whichis small at normal states. The factorCg* may also be com-pared to the friction force on contact that is indispensaused in molecular-dynamics simulations@17# of granularmatter. As will be seen later, the end effects of the momtum correlation factors such asCg* and the friction forcesused in molecular-dynamics simulations are the same inthey tend to keep the particles in the gravitational field piup. We remark that this momentum-correlation effect istaken into account in the existing kinetic theories of granufluids cited earlier.

In the configuration space conjugate to the momentspace, the exclusion effect is relatively easy to comprehena spatial correlation function is made use of, and for tpurpose we insertx~r ,r2ks! in the collisional rate. This isthe probability of finding a pair of particles separatedks—namely, a pair correlation function—at the instantcollision and the excluded-volume effect is already built init. Since the spatial correlation function must be a functionthe relative distance between the particles, we find

x~r ,r6ks!5x~s!,

which is the pair correlation function evaluated at the contpoint of the hard granular particles regardless of the mmenta of the particles involved. Thisx~s! is also in confor-mation with the coarse graining of distribution functioover the collision volume. This distribution function can bin principle, determined within the framework of kinettheory if its own kinetic equation is formulated and solve

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but in this work we will use the equilibrium-pair correlatiofunction determinedad hoc from elsewhere, e.g., the equlibrium theory.

Taking the aforementioned factors into account, we obtthe forward collisional rate of change in the number of pticles in the volume elementdk dv dr dv2 and the collisionvolume

~R8@Dc f #dr dv dt! forward

5s2m3E dv2E dk~g12•k!x~r ,r2ks! f ~v,r ;t ! f ~v2,r ;t !

3@12Dc f * ~v,r ;t !#@12Dc f * ~v2 ,r ;t !#dv dr dt, ~24!

where the asterisk denotes the postcollision value. Notex~r ,r2ks!5x~s! in Eq. ~24!.

It is possible to calculate the collisional rate for the rverse collision in a manner similar to Eq.~24!, and we obtain

~R8@Dc f #dr dv dt!reverse

5s2m3E dv2E dk~g12•k!x~r ,r1ks! f * ~v,r ;t ! f * ~v2 ,r ;t !

3@12Dc f ~v,r ;t !#@12Dc f ~v2 ,r ;t !#dv dr dt, ~25!

for which we have used the Liouville theorem for the phavolume. Recallx~r , r1ks!5x~s! in Eq. ~25!. The restitutivecoefficient, which appears in the existing kinetic theor@10–14# for granular matter, is set equal to unity in thecollision terms, since the granules are assumed to be elaWe reiterate that despite the fact that the collisions are etic, the assembly is still dissipative because of random cosions between the particles. The net coarse-grained csional rate of change in number is given by combining~24!and ~25!:

R8@Dc f #5s2m3x~s!E dv2E dk~g12•k!

3$ f * ~v,r ;t;t ! f * ~v2 ,r !@12Dc f ~v,r ;t !#

3@12Dc f ~v2 ,r ;t !#2 f ~v,r ;t ! f ~v2 ,r ;t !

3@12Dc f * ~v,r ;t !#@12Dc f * ~v2 ,r ;t !#%.

~26!

This collision term will be used in this work. We remark tha fine-grained form of the collision integral does not give rto a satisfactory kinetic equation and macroscopic continuequations.

With the definition of a new distribution function

f5Dc f , ~27!

the kinetic equation forf ~r , v, t! may be written in the form

S ]

]t1v•“1F•“vD f ~v,r ;t !5R@ f #, ~28!

whereR@ f #5Dc21R8@Dc f #. The kinetic equation~28! with

the collision integral defined by Eq.~26! is the basis of the

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kinetic theory of fluidized, elastic granular matter presenbelow. To be more precise, this kinetic equation mustaccompanied by a kinetic equation for configuration corretion functionx~s! appearing in the collision integral, but ththeory becomes much more complex if such a kinetic eqtion is added. Since we are interested in the essential feathat captures the kinetic evolution of nonequilibrium prcesses in fluidized, granular matter, we will confine the dcussion to the level of precision by which we not only nglect inelastic collisions and set the restitutive coefficieequal to unity, but also take the equilibrium pair correlatifunction for x~s!. For a more complete theory, we deferfuture studies that more properly treatx~s! and take intoaccount the internal degrees of freedom for the granu~e.g., inelasticity of granules!.

IV. CONSERVATION LAWS, THE H THEOREM,AND EQUILIBRIUM SOLUTIONOF THE KINETIC EQUATION

A. Conservation laws

Mean values for macroscopic observables are statisticcalculated by the formula

A~r ,t !5Dc21m3E dv A~r ,v! f ~v,r ;t ![^A~r ,v! f ~v,r ;t !&.

~29!

With this definition for macroscopic variables, the kineequation~28! yields the balance equations for mass, momtum, and energy as well as other evolution~constitutive!equations. They can be readily derived by using the wknown procedure in kinetic theory. Since the procedurewell documented@20# and no particular consideration is required for the present problem, we will simply give the stistical definitions for the quantities involved and will referthe explicit forms for the balance equations in Ref.@20#.

The mass densityr, momentum densityru, and internalenergy densityrE are defined, respectively, by the statisticformulas

r5^mf&, ~30!

ru5^mvf &, ~31!

E5rE5^ 12mC2f &, ~32!

whereC5v2u denotes the peculiar velocity. These defintions and the kinetic equation give rise to the balance eqtions for the conserved variables, if the stress tensorP andthe heat fluxQ are defined, respectively, by the statisticformulas

P5^mCCf &, ~33!

Q5^ 12mC2Cf &. ~34!

The stress tensorP and the heat fluxQ, in turn, obey theirown evolution equations which can be derived from the stistical formulas with the help of the kinetic equation. Suevolution equations are in fact the constitutive equationsthe stress and the heat flux. The derivations of the bala

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-

rce

equations indicate that the conventional continuum mechcal equations hold for fluidized granular matter, if the kineequation~28! is assumed. They are in the same forms@20# asthose for ordinary molecular fluids. It implies that granulmatter, when sufficiently fluidized, acts as if it is a cotinuum matter obeying the conventional mass, momentand energy conservation laws.

B. H theorem and equilibrium solution of the kinetic equation

The stability of the equilibrium solution of the kinetiequation~28! can be examined in terms of a Lyapunov funtion, which in the case of kinetic equations appears asBoltzmannH function, that is, the Boltzmann entropyS. It isdefined by the statistical formula

S[rS52kB^ f ~r ,v;t !ln f1@12 f ~r ,v;t !# ln~12 f !&.~35!

The derivative of this Boltzmann entropy satisfies theequality

dS

dt>0, ~36!

where the equality holds only at equilibrium. This inequalcan be proved by the well-known procedure of the Bolmann kinetic theory. We remark that the statistical formS in ~35! is strictly dictated by the collision term~26! of thekinetic equation~28!. It is not arbitrarily chosen at all. If theform for S is not taken exactly as in~35!, theH theorem,namely, the inequality~36!, cannot be proved, given the colision term ~26! for the kinetic equation.

If the kinetic equation is used to calculate the local foof the inequality~36!, the Boltzmann entropy balance eqution follows:

rdtS52“•Js~r ,t !1sent~r ,t !, ~37!

where the statistical formulas for the Boltzmann entropy flJs and the Boltzmann entropy productionsent are respec-tively given by

Js52kB^C$ f ~r ,v;t !lnf1@12 f ~r ,v;t !# ln~12 f !%&,~38!

sent5kB^ ln~ f2121!R@ f #&>0. ~39!

The positivity ofsent is a local representation of theH theo-rem andsent vanishes at equilibrium.

Thus, the equilibrium solutionf e of the kinetic equation isgiven by

^ ln~ f e2121!R@ f e#&50. ~40!

On applying the procedure@20# of the kinetic theory of or-dinary fluids, the solution of this equation can be easily otained in a well-known form. That is, if the Hamiltonian ofparticle subjected to the gravitational field is denoted by

H5 12mC21mgz, ~41!

then the equilibrium solution is given by

ly

aa-itonnalaicl

iti

cted

oreas

dawthtuidteeisoarktuotmpthdveemannnolaanicidii-

icso-

icse ofstheerepyh

his-

terturerate

the

terra-ra-t in

iestogy isadyofthe

d,

n-d,uskengy begyingy ofbs

laraw,aricalicalthelar


f e5@expb~H2me!11#21, ~42!

whereb is a constant and the parameterme is closely relatedto the normalization factor off e . These parameters generaldepend onr . The gravitational field is taken parallel to thezaxis. The equilibrium solution has exactly the same formthe distribution function for Fermi-Dirac particles. The reson for this is the excluded volume effect that manifestsself even in the momentum space by statistically putting cstraints on the population of particles undergoing collisiotransitions and thus resulting in strong momentum corretions for the collision pairs in the congested granular partassembly. The distribution functionf e is normalized to den-sity:

n5^@expb~H2me!11#21&. ~43!

In the case of granular matter, the meaning of parameterb ismore subtle than the case of Fermi-Dirac particles, sincenot clear whether there is a thermodynamic structure admsible from the viewpoint of the thermodynamic laws. In fathe meaning of temperature must be altered in the casgranular matter. The reason for this statement is discussethe following.

V. THERMODYNAMIC ANALOGYFOR GRANULAR MATTER

In the case of molecular fluids, the thermodynamic theof processes@21# is well founded on the notion of cycles, thCarnot cycle being the prototype. As is well known, it wthe Carnot cycle and its analysis@22,23# that gave rise to themathematical structure of thermodynamics. The thermonamics of fluidized granular matter, however, presentsunfamiliar and vexing situation, since it is not obvious hoto construct a cycle with it as a working substance. Incase of granular matter, the notions of heat and temperaare not as well clarified or established as for molecular fluand it is not clear if one can simply apply to granular matthe thermodynamic theory valid for molecular fluids. To sthis point more clearly, let us examine whether or not itpossible to construct a cycle similar to the Carnot cyclethe conventional notions of heat and temperature, whichused for the Carnot cycle with a molecular fluid as the woing substance. In the case of the latter fluid, the temperaof the fluid rises and the volume expands on absorptionheat by the system, whereas injecting the same amounheat into granular matter as supplied to the mass of alecular fluid will not result in a noticeable change in it excefor, perhaps, a slight expansion of the volume due tothermal expansion of the granules, but the change is notto random translational motions of the granules themselwhich are induced by the supplied heat and the raised tperature. Therefore, we see that the granular matter is trlationally cold and random translational motions of the graules do not get excited by heat. This indicates that it ispossible to construct a thermodynamic theory of granumatter on the basis of the conventional notions of heattemperature, even if the matter is fluidized by a mechanexternal force, in the same manner as for molecular fluHowever, we might be able to resort to a thermodynamanalogy to acquire from the equilibrium solution of the k

s

--l-e

iss-,ofin

y

y-n

eres,re

nre-refofo-teues,-s--trdals.c

netic equation a mathematical structure of thermodynamfor fluidized granular matter, that resembles that of thermdynamics of molecular fluids.

An example of such a thermodynamic analogy in physand in the description of natural phenomena is the casradiation@24#. It is helpful to recall that the thermodynamicof radiation was constructed on the basis of an analogy tothermodynamics of matter under the assumption that thexists an equilibrium Gibbs relation for the radiation entroand with the help of the black body in equilibrium witradiation and the Stefan-Boltzmann law@25# Erad5asbT

4,whereT is the temperature of the black body. We use tprocedure@24# as a guide for constructing a thermodynamicslike phenomenological theory of fluidized granular matunder consideration and acquire the meaning of temperafrom the experimental observation of the energy-shearrelation @2#.

Anticipating the result presented later, we may set incanonical form~42!

b51/kBT. ~44!

Here T is the translational temperature of granular matthat is mechanically fluidized, but not the internal tempeture of the granules at rest. We remark that this ‘‘tempeture’’ is not the conventional temperature we talked abouthe case of ordinary molecular fluids.

Bagnold@2# observed on the basis of experimental studthat the mean velocity of granular matter is proportionalthe shear rate and, consequently, the mean internal enerrelated to the shear rate squared. In fact, we have alreavailed ourselves of this result in Sec. II, where the notionbasic phase volume is introduced. Thus, we may writeinternal energy of fluidized granular matter in the form

E5E0@11cg~g!g2#, ~45!

wherec is a constant with the dimension of time squareg~g! is a dimensionless function of shear rateg which tendsto unity asg vanishes, andE0 is the energy of the system ia stationary configuration atg50 ~e.g., a stationary metastable configuration!. Within the range of shear rate studieBagnold foundg~g! is independent of the shear rate and thcan be set equal to unity. Following Bagnold, we will tag~g!51 in this work. However, this assumption concernig~g! is not mandatory in the subsequent analysis and maeasily removed without altering the thermodynamic analopresented for granular matter. We now state the followassumption about the existence of the Clausius entropfluidized granular matter that obeys the equilibrium Gibrelation

dSe5T21~dE1pdV!. ~46!

Here p is the pressure andV is the conjugate volume. Weemphasize that this relation will be established for granumatter not on the basis of the thermodynamic second lwhich was originally stated only for cycles with a moleculfluid as the working substance, but by means of statistmechanics under the thermodynamic-statistical mechancorrespondence used in the ensemble theory; this timeprocedure is only reversed from that for ordinary molecu

asi

hor-

omn

o

usid

edu

onethc

cserflu

aethrcnit

ls

ro-

he

ere

it

t,

ics,t beits

r


fluids. We elaborate on this in the following. We remark thwe are not considering a system at rest, which was conered by Edwards and Oakeshott@7# for the thermodynamicformalism for a granular assembly.

It is now asserted that as in the Gibbs ensemble met@26# for ordinary molecular fluids, there is the following corespondence@27# between thermodynamicSe in Eq. ~46! andthe equilibrium Boltzmann entropy computed from Eq.~35!with the equilibrium canonical form~42!

Seust⇒Seu th , ~47!

where the subscripts st and th mean the statistically cputed and the thermodynamic entropy, respectively, asimilarly, there hold the correspondences forE andp

Eust⇒Eu th , pust⇒pu th . ~48!

We use Eqs.~45!–~48! to fix the meaning of parameterb andthus the meaning ofT therewith. First, the meaning ofb inthe distribution function is fixed by comparing Eq.~45! withthe statistically computed mean energy. Then the correspdence~48! is used to endowT in Eq. ~46! with the ‘‘thermo-dynamic’’ meaning. It must be noted that this process is jthe reverse of the one used for ordinary molecular fluwhere the meaning ofb is fixed in terms ofT that is foundedon the thermodynamic second law which provides the thmodynamic temperature scale. This reversal of the proceis forced upon us because Eq.~46! is not given by the secondlaw on the basis of analysis of a reversible cycle conventially made in thermodynamics of molecular fluids. This timit is the statistical theory, constructed by an analogy tomolecular statistical theory which is known to yield a struture of thermodynamics, and the correspondences, Eqs.~47!and~48!, that is enabling us to construct a thermodynamilike mathematical structure for fluidized, granular mattwhich is analogous to the thermodynamics of molecularids.

VI. DETERMINATION OF PARAMETERS b AND me

It is necessary, for the stated aim, to calculate the normization condition forf e and the mean internal energy for thgranular matter subjected to the gravitational force whengranules have acquired kinetic energies by an external foSince the integrals involved do not permit exact evaluatiothey will be calculated to an approximation, first, in the limwhere the parameterb is large andme2mgz.0. In this limitthe meaning ofb is elucidated. With the meaning ofb thusestablished, we then consider the case whereb is moderatein value. It is necessary to evaluate the following integra

n~z!52p

DcS 2mb D 3/2E

0

`

dtt1/2

exp~ t2a!11, ~49!

E52p

DcS 2mb D 3/2b21E

0

`

dtt3/2

exp~ t2a!11, ~50!

where withb5bmg

a5b~me2mgz![me2bz. ~51!

td-

d

-d,

n-

ts

r-re

-,e-

-,-

l-

ee.s,

:

These integrals can be treated by means of well-known pcedures@28#. We summarize the results below.

A. The case ofa positive and very large

The normalization integral for this case is given by tformula

n~z!54p

3DcS 2mb D 3/2a3/2F11

p2

8a2 17p4

640a4 1••• G . ~52!

By using this result, we find to the lowest order

a5me2bz.S 3Dcn~z!

4p D 2/3 b

2m. ~53!

The mean internal energy is given by the formula

E54p

5DcS 2mb D 3/2b21a5/2F11

5p2

8a227p4

384a4 1••• G[

4p

5DcS 2mb D 3/2b21a5/2@11a22K~a!#. ~54!

On truncation of the series, Eq.~54! may be approximated by

E.4p

5DcS 2mb D 3/2b21a5/2S 11

5p2

8a2D . ~55!

The front factor in Eq.~54! is independent ofb if Eq. ~53! isused. With the definitions

E053n~z!5/3

10m S 3Dc

4p D 2/3 ~56!

and

u5A2/5~pkBm!21S 3Dcn~z!

4p D 2/3, ~57!

we obtain the internal energy in the form

E5E0~z!F11S Tu D 2G , ~58!

for which Eq.~44! is used. By using Eq.~45! and the corre-spondence~48!, we now identifyT:

T5cug. ~59!

This identification ofT with g ~more precisely, the absolutvalue of the shear rate! can be made more precise if a moprecise phenomenological expression forE exists than Eq.~45! and if Eq.~54! is used without truncation. However,would not change the basic conclusion thatT is directly pro-portional tog. Therefore, we will be content with the resulEq. ~59!, obtained here. We remark that the constantc in~59! cannot be calculated by means of statistical mechanbut is a parameter like the Boltzmann constant that muschosen to correlate the mechanics of granular matter withthermodynamics.

Since f e , on substitution into the statistical formula fothe Boltzmann entropy, yields the differential form forSe

c

rem

pe

-

or-

l and

slare weyrd

erepa-or.


dSe5kBb~dE1pdV! ~60!

if p is defined statistically by the formula

p52pkBTDc23/2E

0

`

dt t1/2ln~11ea2t!. ~61!

On making use of the correspondence~47!, we find

Tu th⇔~kBb!21ust, ~62!

which by virtue of Eq.~44! implies that the thermodynamitemperature must be identified withT as in Eq.~59!, namely,

Tu th5cug. ~63!

In this manner the statistical theory gives rise to an interptation of thermodynamic temperature and thus to a therdynamic analogy, as characterized by Eq.~60! for the granu-lar matter under consideration. We note the relation

p5 23E~z!. ~64!

This relationship can be used to compute the density dedence ofp. If Eq. ~54! is used in this expression forp, theequation of state is given by

p58p

15DcS 2mb D 3/2b21a5/2@11a22K~a!#

' 23E0~z!F11S Tu D 2G . ~65!

This is the equation of state in the limit ofa→`.

B. The case ofa moderate in value

The density is given by the expansion

n~z!5n0(l50

`al

~11e2a! l11 , ~66!

where

n052

psA3Ug

3~mAv!3S 2pm

b D 3/2, ~67!

al5 (k50

l S lkD ~21!k

~11k!3/2.0. ~68!

It is possible to solve Eq.~66! for a to calculate the normalization factor. The result is

a5 lnFn~z!DcS b

2pmD 3/2G1 (k>1

ckjk, ~69!

where

j5n~z!Dc~2pmkBT!23/2, ~70!

-o-

n-

ck51

k! H dk

djkln (

j51

`j j21

j ! F dj21

dl j21 w j~l!Gl50

Jj50

,

~71!

with w~l! defined by

w215d

dl (l51

`~21! l21l l

l 5/2.

The leading examples forck are

c15223/2, c253

2 S 1822

35/2D , ~72!

etc. If al(11e2a)2 l,1, Eq. ~66! may be approximated by

n~z!.n0

11e2a . ~73!

By applying for the same procedure as for Eq.~69!, weobtain the internal energy in the form

E5 32n~z!b21S 11 (

k>1

k

k11ckj

kD , ~74!

which implies that the equation of state is given by the fmula

p5n~z!b21S 11 (k>1

k

k11ckj

kD . ~75!

These results suggest that the granular matter is not ideathe virial coefficients are given by

Bk5k

k11ckL

3k, ~76!

where

L5Dc1/3S b

2pmD 1/2. ~77!

These coefficientsBk stem from the momentum correlationinduced by the excluded volume inherent in the granuparticles in a congested fluidized assembly. Perhaps hermay make anad hoccorrection to the equation of state badding the contribution from the spatial correlation of haspheres:

p5n~z!b21S 11 (k>1

Bknk1

2ps3n

3x~s! D . ~78!

Note that the last term in this equation is the hard-sphcontribution to the equation of state that arises from the stial correlation of particles and the trace of the virial tens

It is possible to resume the series in Eq.~75! into a formsimilar to the Carnahan-Starling form@29# by applying thePadeapproximant method as used in Ref.@30# and obtain theformula for the equation of state

s

thth

tethnsceu

leu--fiaaor

ed-docs-nfln

tanlynonponee,w

with

s-

nde

anto


p5n~z!b21H 11j@ 1

2 c11116 ~ 32

3 c223c1!j#

~12 18 j!3

J5n~z!b21F11

j~0.17720.070j!

~12 18 j!3

G . ~79!

This equation of state will be useful for calculating the tranport coefficients later in this work.

VII. COMPARISON WITH EXPERIMENT

A. Distribution of unperturbed granular matter

Having identified the parameters inf e in terms of phe-nomenological quantities, we are now ready to examinedistribution function in more detail and in comparison wiavailable experimental observations. SinceT tends to zeroaccording to the relation~59! as g vanishes, the limit ofvanishing shear rate corresponds to the absolute zero ofperature for molecular fluids. Thus, we may say thatgranular matter is absolutely cold with respect to the tralational motion when it is not perturbed by an external forsuch as shearing force or tapping. In this limit the distribtion function becomes a step function:

f e5 H 1 if a>00 if a,0. ~80!

The value ofme determines the positionz where the distri-bution function vanishes. This is the situation where granuare statically piled up in the gravitational field. The distribtion function in Eq.~80!, together with the density distribution already calculated in Eq.~52!, indicates the behavior othe granular matter under the gravitational field in the limof vanishing perturbation. Their behavior is physically resonable and in agreement with experimental observationeveryday experience with granular matter. In a recent wof Goldhirsh and Tan@9#, which was not in print before theoriginal version of this work was submitted, a reducvelocity—in fact, radial speed—distribution function computed by a simulation method for a sheared granular fluishown to deviate from the normal Gaussian form. The velity distribution function f e obtained in the present work iconsistent with their finding, sincef e and any reduced distribution function obtained from it is certainly not GaussiaThis aspect and other macroscopic properties of granularids predicted by the present theory will be discussed iseparate work in the future.

B. Density distribution at a nonvanishing perturbation

If a granular assembly is vertically shaken or tapped afinite amplitude and frequency as was done by ClementRajchenbach@16#, the particles in the assembly collectivemove more or less vertically, although their velocities areuniform but have a distribution. Such a velocity distributiois given byf e . There are cases where some velocity comnents vanish owing to the setup of the experiment in haFor example, if thex andy components of the velocity arvirtually absent, as in some experiments, and furthermorthez component of the velocity in the up phase or the do

-

e

m-e-

-

s

t-ndk

is-

.u-a

ad

t

-d.

ifn

phase has a narrow distribution, as seems to be the casethe experiment by Clement and Rajchenbach@16#, then thedistribution of the particles can be given by the density ditribution function defined by the dimensionless densityr(z):

r~z!5ps3

DcS 2pm

b D 3/2E2`

`

dqE2`

`

ds

3E2`

`

dtd~q!d~s!d~ t2t0!

exp@q21s21~ t2t0!21bz2me#11

,

~81!

whereq5Amb/2vx , s5Amb/2vy , and t5Amb/2vz witht05Amb/2pu ~u5the collective velocity of granules!. Thisgives rise to the density distribution

r~z!5r0

exp~bz2me!11, ~82!

where

r05ps3

DcS 2pm

b D 3/2. ~83!

By choosingb50.5 mm21, r050.91, andme515, which im-pliesmb550 m22 s2, we show that the distribution in Eq.~82! fits well the experimental data on thez dependence ofdensity obtained by Clement and Rajchenbach@16# as shownin Fig. 1. The experimental values forA andv are, respec-tively, A52.5 mm andv520 s21. We have not made com-parison with the simulation data of Gallas, Herrmann, aSokolowski @17#, since the simulation data agree with thexperiment.

VIII. THEORY OF TRANSPORT PROCESSESIN GRANULAR MATTER

The theory of transport processes in granular matter cbe developed in a completely parallel manner analogous

FIG. 1. Reduced density distribution with respect toz. The opencircles are the data by Clement and Rajchenbach@16#. Here thestatic packing density is takenr050.91. Note thatr0 is dimension-less.

laorco

r

th

t

se

ls

las-ete-

on

a-

ty,

the

te-

ityde-

e-lade

s a

atisanffi-


the procedures developed for ordinary fluids in Ref.@20#. Wetherefore will simply present the relevant results for granumatter. We will confine our discussion to linear transpprocesses in this paper. It is possible to show that the visity h and thermal conductivityk are given by the formulas

h52p2bg

R~11! , ~84!

k5~CpTp!2bg

R~33! , ~85!

where g5(m/2kBT)1/2/n2s2, Cp55kB/2m, and R( i i )

( i51,3) are collision bracket integrals defined by the fomula

R~ i i !5 14 @~h1

~ i !1h2~ i !2h1

~ i !*2h2~ i !* !

3~h1~ i !1h2

~ i !2h1~ i !*2h2

~ i !* !#12. ~86!

Here various symbols are defined as

h1~1!5m@CC2 1

3C2d#1 , ~87!

h1~3!5m@ 1

2C2C2CpTC#1 , ~88!

with the subscript 1 and 2 referring to particles andsquare brackets in Eq.~86! stands for the integral

@AB#125bgDc22m6E dv1E dv2E dk~g12•k! f e~v1! f e~v2!

3@12 f e~v1!#@12 f e~v2!#AB. ~89!

These collision bracket integrals are explicitly evaluatedan approximation and presented below.

We will calculate the collision bracket integral in the cawherea is moderate in value. By using the expansion

f e~v!@12 f e~v!#5(l50

`d

da F ~11e2a!21e2tS 12e2t

11e2aD l G ,~90!

wheret is the reduced kinetic energyt i512mCi

2b ( i51,2)anda is as defined before. Let us now define the integra

@AB#12~c!5bgn2Smb

2p D 3E dv1E dv2E dk~g12•k!

3exp~2t12t2!AB, ~91!

@AB#12~ l1l2!

5bgn2Smb

2p D 3E dv1E dv2E dk~g12•k!

3exp~2t12t2!~12e2t1! l1~12e2t2! l2AB.

~92!

Then [AB] 12 can be split into two terms as follows:

rts-

-

e

o

@AB#125@2nL3~11cosha!#22@AB#12~c!

1 (l151

`

(l251

`

~ l 111!~ l 211!

3@n2L6~11e2a! l11 l214#21@AB#12~ l1l2! . ~93!

The second term corresponds to the correction to the ‘‘csical’’ contribution for which the distribution functions artaken to be Maxwellian. The classical collision bracket ingrals [AB] 12

(c) are available in the literature@15#. We will becontent with the lowest-order approximation for the collisibracket integrals. Thus, we take

@AB#12'@2nL3~11cosha!#22@AB#12~c! . ~94!

For the value ofa the result in Eq.~69! must be used withthe last term neglected for consistency with the approximtion made for Eq.~94!. We use this result and Eq.~84! tocalculate the viscosity of granular matter. For the viscosi

A5B5m@CC#~2!5m~CC2 13C

2d!, ~95!

with d denoting the unit second-rank tensor. Calculatingfront factor involvinga in Eq. ~94! with the aforementionedapproximation neglecting thej-dependent term in Eq.~69!and using the literature result for the collision bracket ingral [AB] 12

(c) for hard spheres@15#, we obtain from Eq.~84!the viscosity in the form

h55

16s2 SmkBT

p D 1/2F11L3n~0.17720.070L3n!

~12 18 L3n!3

G2,~96!

for which we have used the equation of state, Eq.~79!. If theequation of state given in Eq.~78! is used for viscosity, thenit is given by the formula

h55

16s2 SmkBT

p D 1/23F11

L3n~0.17720.070L3n!

~12 18L3n!3

12ps3n

3x~s!G2.

~97!

It must be noted that in contrast to the Newtonian viscosof molecular fluids the viscosity formulas presented herepend on the shear rateg on account of theg dependence ofT; see Eqs.~59! and ~63!. Since the quantity in the squarbracket in Eq.~96! is proportional to the compressibility factor that reaches unity asg increases according to the formu~75!—the equation of state before resumming by a Pa´approximant—h approximately increases withg like Ag.That is, the granular fluid is seen to be dilatant and this iqualitatively correct behavior as is well known@1# of granu-lar matter.

The viscosity formula~96! is plotted in Fig. 2. It shows arather flat viscosity that slowly reaches a minimumj5L3n'5.7 before it rapidly increases. This behaviorreminiscent of the experimental data by Hanes and Inm@31#. This minimum appears because the third virial coe

s

ed

uedninnuiasnedkeri-

y

is-iser-ter,micisen-dy-inicstheu-calhenasin

res-or-

i-m-thelynu-sityhet asvelyasininnd,thel oftterdies

ci-

e


cient B2 is negative owing to the momentum correlationWe note that if Eq.~78! is used instead of Eq.~75! for theequation of state, the minimum will be practically washout, since the term containingx~s! does not exhibit a mini-mum and is dominant.

IX. CONCLUDING REMARKS

In this paper, we have formulated a kinetic theory of flidized granular matter by taking into account the excludvolume effects and significant momentum correlatiopresent in congested elastic granular fluids. These two kof effects combine to give rise to an equilibrium distributiofunction which has a mathematical form similar to the eqlibrium distribution function of a Fermi gas, which also hthe excluded-volume effects and momentum correlatioWe believe thatthe momentum correlations in a congestassembly of particles with an excluded volume is thefeature that controls the kinetic evolution of statistical distbution in a fluidized granular system. Another significantpoint in the present theory is the assertion that in analog

FIG. 2. Viscosity vs reduced densityj5nL3 ~a scaled packingfraction!. The inset is the blowup of the minimum region. ThChapman-Cowling formula for hard spheres is used forn0:n05(5/16s2)(mkBT/p)

1/2.

ys

ld,

.

--sds

-

s.

y

to

the statistical mechanics of ordinary molecular fluids the dtribution function obeying the kinetic equation postulatedthe thermodynamic branch of solution, which yields a thmodynamic theory of processes in fluidized granular matand this assertion enables us to construct a thermodynatheory from the statistical theory. Note that this procedurejust the opposite of the one used in the conventionalsemble theory of statistical mechanics, where the thermonamic branch of the distribution function is constructedcorrespondence with the phenomenological thermodynamestablished from the thermodynamic laws by means ofClausius inequality. This thermodynamic formalism for flidized granular fluid, with the help of the attendant statistitheory, elucidates the meaning of granular temperature wthe system is subject to an external perturbation suchshearing. This theory also provides the equation of stateterms of the granular temperature so determined. The psure increases with the density, which is a qualitatively crect behavior.

The density distribution function obtained from the knetic equation is in agreement with experiment and the coputer simulation result on a vibrated granular matter andviscosity computed in the theory also exhibits qualitativecorrect behavior when compared with the viscosity of gralar matter in that it remains almost constant over a deninterval and then steeply increases with the density. Tpresent theory also predicts that granular matter is dilatanexperimentally known. Such behaviors would not haarisen if the equilibrium distribution function was simpgiven by the Boltzmann distribution function that arisesthe equilibrium solution of the kinetic equations appearingRefs.@10–14#. Therefore, they may be taken as evidencesupport of the kinetic equation used in the present work ain particular, the collision integral therein. Nevertheless,theory presented here is only a first step toward the goaunderstanding kinetic processes in fluidized granular maand we hope that the present work stimulates further stuin kinetic theory of granular matter.

ACKNOWLEDGMENT

This work has been supported in part by the Natural Sences and Engineering Research Council of Canada.

, J.

f-

A

@1# O. Reynolds, Philos. Mag.20, 469 ~1885!.@2# R. A. Bagnold, Proc. R. Soc. London, Ser. A225, 49 ~1954!;

295, 219 ~1966!.@3# S. Ogawa, A. Umemura, and N. Oshima, J. Appl. Math. Ph

~ZAMP! 31, 483 ~1980!.@4# Advances in the Flow of Granular Materials, edited by M.

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Kinetic theory of fluidized granular matter

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