Hydrodynamic equations for granular gases driven by the Fokker-Planck
operator are derived. Transport coefficients appeared in Navier-Stokes order
change from the values of a free cooling state to those of a steady state.Comment: 5 pages, 3 figure

A. Goldstein

A. Santos

F. Gerl

G. Kuwabara

H. Hayakawa

H. Kuninaka

H.-D. Kim

Hisao Hayakawa

J.A. Carrillo

J.J. Brey

J.M. Montanero

K. Ichiki

L. Labous

N.V. Brilliantov

R. Pagnani

T. Biben

T. Kawaguchi

T.P.J. van Noije

V. Garzo

W.A.M. Morgado

English

arXiv

Hayakawa, H

Kyoto University Research Information Repository

Title Hydrodynamics of driven granular gasesAuthor(s)Hayakawa, HCitationPHYSICAL REVIEW E (2003), 68(3)Issue Date2003-09URL http://hdl.handle.net/2433/50477RightCopyright 2003 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityHydrodynamics of driven granular gasesHisao HayakawaDepartment of Physics, Yoshida-South Campus, Kyoto University, Kyoto 606-8501, Japan~Received 8 April 2003; published 12 September 2003!Hydrodynamic equations for granular gases driven by the Fokker-Planck operator are derived. Transportcoefficients appeared in Navier-Stokes order change from the values of a free-cooling state to those of a steadystate. The mismatch between the granular temperature and the constant temperature of the heat bath producesa nontrivial steady state.DOI: 10.1103/PhysRevE.68.031304 PACS number~s!: 81.05.Rm, 47.20.2k, 05.20.DdI. INTRODUCTIONThe gas kinetic theory of elastic particles played impor-tant roles in history of statistical mechanics @1,2#. Whenthere are inelastic interactions among particles, the behaviorof collections of particles is completely different from that ofelastic particles: There are no equilibrium states and any spa-tially homogeneous states are no longer stable. Such a col-lection of particles having inelastic interactions is called thegranular gas, whose physical realization can be observed inrings of planets, small planets, suspended particles in fluid-ized beds, aerosols, and rapid granular flows, etc. @3#.A typical example of granular gases can be found in aero-sols or suspensions, etc., in which the buoyancy is balancedwith gravity @4,5#. In dense suspensions the hydrodynamicinteraction among particles are important @6#, but effects ofthe air may be regarded as a thermostat driven by the Lange-vin forces in dilute suspensions. Most researches for granulargases are interested in undriven systems which are difficultto be achieved in actual experiences. We believe that moresystematic studies for driven granular gases are required.Montanero and Santos analyzed statistical properties of thehomogeneous state and hydrodynamics of granular gases, thewhite noise thermostat, and the Gaussian thermostat @7,8#.Carrillo et al. @9# and Pagnani et al. @10# analyzed granulargases in driven systems by the Langevin force which in-cludes both the white noise and the friction force in propor-tion to the velocity of particles, but they are not interested inhydrodynamics of such a granular fluid. Since we believethat the model driven by such a thermostat, we call theLangevin thermostat, can describe physical situations of di-lute suspensions, we need to clarify the properties of thehydrodynamics of such a system. The main purpose of thispaper is to investigate the effects of Langevin thermostat fortransport coefficientsThe organization of this paper is as follows. In the follow-ing section, we introduce our model and the framework ofthe Chapman-Enskog method for the analysis of granulargases. In Sec. III, we calculate the valuables such as thegranular temperature and the fourth cumulant in a homoge-neous state. In Sec. IV, we obtain the transport coefficientssuch as the viscosity and the heat conductivity. In Sec. V, wediscuss and summarize our results.II. FRAMEWORKWe consider rarefied gases of smooth identical particleswith the mass m, the velocity v, and the diameter s . Thedistribution function f (r,v,t) in our system obeys the heatedinelastic Boltzmann equation~] t1v ! f 5J@ f , f #1LFP f , ~1!where J@ f , f # represents the collisional integral given by@11,12#J@ f ,h#5sd21E dsˆ E dv1Q~gsˆ !gsˆ ~e22b2121 !3 f ~r,v,t !h~r,v1 ,t !, ~2!where Q(x) is the Heaviside function, g5v2v1, and sˆ isthe unit vector along the line connecting centers of mass ofcontacting particles. The operator b21 is the inverse of thecollisional operator b which are defined asbg5g2~11e !~gsˆ !sˆ , ~3!b21g5g211ee~gsˆ !sˆ , ~4!where e is the coefficient of restitution which is ranged 0,e<1. Here we assume that e is a constant for the simpli-fication of our argument, though the actual coefficient ofrestitution depends on the impact velocity @13–15#. The ef-fects of the impact velocity dependence of e to macroscopichydrodynamics can be seen in Ref. @16#. It should be notedthat the oblique impacts have important contributions in ac-tual inelastic collisions @17,18#. We, however, assume thatthe effects of inelastic oblique collision can be neglected,which may be justified when particles are smooth hard-coreparticles.The Fokker-Planck operator LFP in Eq. ~1! represents thedriven force coming from the Langevin force asLFP5g0]]vFV1 TBm]]vG , ~5!where the first term and the second term represent the fric-tional force and the thermal activation, respectively. In gen-eral, the temperature of the heat bath TB is different fromgranular temperature T. Here, we assume that TB is a con-stant throughout the argument in this paper. This assumptionis justified when the influence of granular particles to theheat bath is negligible. If there is no contribution from thePHYSICAL REVIEW E 68, 031304 ~2003!1063-651X/2003/68~3!/031304~6!/$20.00 ©2003 The American Physical Society68 031304-1collisional integral, the distribution function is relaxed to anequilibrium state as f→ f eq}exp@2mV2/2TB# . We note thatthe Fokker-Planck operator ~5! is different from white noisethermostat Lwhite5(g0T/m)(]2/]v2) even when we neglectthe contribution the friction term. Actually, our LFP is re-duced to g0TB /m(]2/]v2) in such a situation with a constanttemperature TB . We also note that the frictional force in thefirst term of Eq. ~5! has the inverse sign of the Gaussianthermostat @7# LGauss52g0(]/]v)V which is also used insimulation of molecular hydrodynamics @19#.Hydrodynamic variables which characterize the macro-scopic behavior of the gas are the number density, the veloc-ity field, and the granular temperature defined byn~r,t !5E dvf ~r,v,t !, ~6!n~r,t !u~r,t !5E dv vf ~r,v,t !, ~7!d2 n~r,t !T~r,t !5E dv12 mV2 f ~r,v,t !, ~8!where V[v2u. The integral of the collisional invariancemultiplied by J@ f , f # over V is zero. Since the loss of kineticenergy in each collision is given byDE5212e24 m~gsˆ !2, ~9!the following relation holds:E dv12 mV2J~ f , f !52 nd2 Tz@ f , f # . ~10!Here, the cooling rate z in Eq. ~10! can be evaluated approxi-mately @12#z.z (0)512e24d n0~d12 !S 11 316 a2D , ~11!where n05p21/2nsd21(T/m)1/24Vd /(d12), with Vd52pd/2/G(d/2). a2 is the fourth cumulant defined bya2[dd12^V4&^V2&221, ~12!^Vk&[1nE dVVk f ~V,t !, ~13!which will be determined later.The balance equations for hydrodynamic variables areDtn1nu50, ~14!Dtui1~mn !21 jPi j50, ~15!DtT12dn ~Pi j jui1q!1Tz52g0~TB2T !, ~16!where Dt5] t1u . The pressure tensor Pi j and the heatflux q are, respectively, defined byPi j5mE dvViV j f ~r,v,t !, ~17!q5m2 E dvV2Vf ~r,v,t !. ~18!We adopt the Chapman-Enskog method @2#, where spaceand time dependences appear through hydrodynamic vari-ables. The expansion parameter is regarded as the magnitudeof spatial inhomogeneity. Thus, we expand f around the ho-mogeneous solution f (0) as f 5 f (0)1« f (1)1«2 f (2)1 , thetime derivative is also expanded as ] t5] t(0)1«] t(1)1 .Here we apply the Chapman-Enskog method for dilutegranular gases developed by Brey et al. @20,21# and Santos@8# to driven systems. To remove the ambiguity of the distri-bution function, we impose the solubility conditions in whichhydrodynamic variables are unchanged from the evaluationby f (0)(v,t).III. HOMOGENEOUS STATESAs the first step of Chapman-Enskog method, we needalso to obtain the homogeneous solution of the inelastic Bolt-zmann equation. We usually assume the scaling formf ~v,t !5nv0~ t !2d f˜~c,t!, c5V/v0~ t !, ~19!with dt5vEdt and v0(t)5A2T/m . The fourth cumulant in-troduced in Eq. ~12! for the scaling function is related to^c4&[*dcc4 f˜(c) as ^c4&5d(d12)(a211)/4. Here vE isEnskog’s collision frequency, given byvE5d124 n05A2pVdnsd21v0 . ~20!For the calculation we need to obtainmk[2E dcckJ@ f˜ , f˜# . ~21!The cumulants and mk can be evaluated by an approximateexpansion of Sonine polynomials @12#. For example, m2 isevaluated asm2.12 ~12e2!VdA2pS 11 316 a2D . ~22!m4 is estimated as @12#m4.A2pVd~A11a2HA2!, ~23!withA1512e24 S d1 32 1e2D ,HISAO HAYAKAWA PHYSICAL REVIEW E 68, 031304 ~2003!031304-2A25 3128 ~12e2!~10d139110e2!1 14 ~11e !~d21 !.~24!It should be noted that these evaluations are based on twoapproximations: ~i! the truncation of the first Sonine expan-sion and ~ii! the linearization of a2. If we adopt the firstassumption, the linearization of a2 gives a nice evaluation@22#. We also indicate that the direct comparison of the trans-port coefficients obtained by the Monte Carlo simulation andthe linearized approximation gives good agreement in free-cooling states @21#. However, nobody knows how to con-verge the Sonine expansion for driven granular gases. Thus,we may need to check the convergence of the Sonine expan-sion as in the case of elastic particles @23#.Now, let us discuss the time evolution of temperaturefield. Eq. ~16! becomes] t(0)u52g02~2g01z!u ~25!in the homogeneous state with the Langevin thermostat,where u[T/TB . Assuming u5u (0)1a2u (1)1O(a22), Eq.~25! is reduced to]tu(0)52$gˆ 2~gˆ 1zˆ !u (0)% ~26!in the lowest order, where gˆ 5g0 /vE and zˆ 5(12e2)/2d .This equation has the solutionu (0)5u‘1~u(0)~0 !2u‘!e22(gˆ 1zˆ )t, ~27!with u‘[gˆ /(gˆ 1zˆ ).In the scaling limit, the inelastic Boltzmann equation ~1!is reduced toVdA2p]t f˜~c,t!5J˜ @ f˜ , f˜#1S gˆu 2 m2d D ]]c @cf˜~c,t!#1gˆ2u]2]c2f˜~c,t!, ~28!where J@ f , f #5n2(s/v0(t))d21J˜ @ f˜ , f˜# . From the equationfor ^c4&5d(d12)(a211)/4 we obtain the equation of thelowest order of a2:]ta254zˆ 2Aˆ 11a2F194 zˆ 24gˆu 2Aˆ 2G1 34zˆ a22 ~29!.4zˆ 2Aˆ 11a2F194 zˆ 24 gˆu (0) 2Aˆ 2G , ~30!with Aˆ 158A1 /d(d12) and Aˆ 258A2 /d(d12). The solu-tion of Eq. ~30! is given bya2~t!.a2‘1@a2~0 !2a2‘#3expF S 194 zˆ 2Aˆ 2D t24gˆ E0t dt8u (0)~t8!G , ~31!wherea2‘5Aˆ 124zˆ194 zˆ 24~gˆ 1zˆ !2Aˆ 2. ~32!Although we present the result of linearization of a2, it ispossible to obtain the exact steady values of u and a2 forEqs. ~25! and ~29!. The result is presented in the Appendix,but the differences between the exact values and the resultfrom linearized approximation are invisible ~Fig. 1!. We alsocompare a2 in Eq. ~31!and u in Eq. ~27! with the numericalsolution of Eqs. ~29! and ~25! to confirm the validity of thelinearization of a2.IV. THE DETERMINATION OF THE TRANSPORTCOEFFICIENTSHere, we explicitly obtain the result of the transport coef-ficients for heated granular gases. The solution f (0) is isotro-pic so that the zeroth-order pressure and the heat flux aregiven byPi j(0)5pd i j , q(0)50, ~33!where p5nT is the hydrostatic pressure.The first-order equation of the Boltzmann equation be-comes~] t(0)1L2LFP! f (1)52~] t(1)1v ! f (0)52~Dt(1)1V ! f (0), ~34!with Dt(1)5] t(1)1u . Here the linear operator L in Eq.~34! is defined byL f (1)52J@ f (0), f (1)#2J@ f (1), f (0)# . ~35!FIG. 1. Steady a2 as a function of e for d53 and gˆ 50.1. Thesolid line and the solid circles represent the result of linearizedapproximation and the exact one, respectively.HYDRODYNAMICS OF DRIVEN GRANULAR GASES PHYSICAL REVIEW E 68, 031304 ~2003!031304-3Multiplying both sides of Eq. ~34! by mViV j and integrat-ing over V, we obtain~] t(0)1n!Pi j(1)1P i j(1)52pD i jklkul , ~36!whereP i j(1)[2mE dVViV jLFP f (1), ~37!D i jkl[d ikd j l1d ild jk22d d i jdkl . ~38!The solution of Eq. ~36! can be written asPi j(1)52hD i jklkul , ~39!where h is the viscosity.It is possible to obtain n in Eq. ~36! through the relationmE dVViV jL f (1)~V!5nPi j(1). ~40!The evaluation of n is independent of the existence of thethermostat. We have evaluated n as @12#nh*[nn0.34d S 12e1 23 d D ~11e !S 12 132 a2D . ~41!With the aid of] t(0)Pi j(1)5S g0u 2g02 z2 D Pi j(1), ~42!Eqs. ~36!–~42! lead toh*[hh052g*11g*~11u21!1nh*2z*/2, ~43!where g*5g0/n0 , nh*5n/n0, and z*5z/n0 . h0(gˆ ) is theviscosity for e51, which is different from the value he ofthe elastic gas as he5h0(2gˆ *11). We note he55ApmT/(16Aps2) for d53 in the lowest-order Sonineapproximation @2#. The steady value of h* is obtained whenwe substitute u‘ and a2‘ into Eq. ~43!.Let us consider the heat flux. Multiplying both sides ofEq. ~34! by mV2V/2 and integrating over V we obtain~] t(0)1n8!q(1)1Q(1)52 d122 ~112a2!pmT2d122 a2T2mn , ~44!whereQ(1)52 m2 E dVV2VLFP f (1). ~45!Here, n8 has already been calculated as @21#nk*[n8n0.11ed Fd212 1 316 ~d18 !~12e !1415d23~42d !e512 a2G . ~46!The heat flux is described byq(1)52kT2mn , ~47!where h and k are the shear viscosity and the thermal con-ductivity, respectively. The other transport coefficient m ap-pears only in granular gases. Through the substitution of Eqs.~47! and ~46! with the result of Q(1) into Eq. ~44! we canobtain k and m .From the scaling form ~19! the following relations shouldbe satisfied:] t(0)q(1)5~2z13g02g0 /u!kT1H mF32 z13g0~12u21!G1 kzTn Jn . ~48!On the other hand, P (1) and Q(1) have the relationsP i j(1)52g0Pi j(1), Q(1)53g0q(1). ~49!Thus, k and m can be obtained from Eqs. ~48!, ~49!, and ~44!ask5d122112a2n822z1g0 /unTm, ~50!m5kzTn1d122 a2T2mn8232 z13g0 /u. ~51!Thus, k*5k/k0 which is the heat conductivity normalizedby its value of e51, k0(gˆ ), is given byk*5S d21d 1g*D 112a2nl*22z*1g*/u , ~52!m*5k*112a2a2~nk*1g*/u!nk*232 z*13g*/u. ~53!Their steady values are evaluated replacing u and a2 by theirsteady values.Figure 2 shows the time evolution of h*, k*, and m* fore50.9, d53, and gˆ 50.1. Figure 3 shows the steady valuesof h*, k*, and m* as functions of e for d53 and gˆ 50.1.As we can see from the figures, m* can be comparable withothers. This situation has not been realized for free-coolingsystems.HISAO HAYAKAWA PHYSICAL REVIEW E 68, 031304 ~2003!031304-4V. DISCUSSION AND CONCLUSIONIt is remarkable that m* is not proportional to a2 andkeeps positive for the region of a2,0 even in the steadylimit. This result is contradicted with the framework by San-tos @8#. To clarify the reason, let us calculate ] t(0)q(1) in thelong time limit:] t(0)q(1)→2k] t(0)T2m] t(0)n52k@2g0TB2~2g01z!T#5k~2g01z!TÞ0. ~54!Thus, ] t(0)q(1) cannot be zero even in the long time limit.This paradoxical situation comes from the following. In oursystem, the system is heated by the constant TB uniformly,but the temperature field T contains spatial fluctuations.Thus, the mismatch of the two temperature fields survives inthe long time limit. In other words, we cannot neglect thetime derivative in the scaled inelastic Boltzmann equation~28! to reproduce the relaxation to the steady state.After submission of the paper, the author has realized theexistence important related papers @24–26#. In particular, thelatter two papers discussed mixtures of elastic particles andgranular particles. Their model reduces to our model whenthe density ratio and the mass ratio of two kinds of particlesare large. They also include the feedback effect from granu-lar particles to the elastic particles in the heat bath. Atpresent, they do not discuss the spatial fluctuations but dis-cuss only homogeneous states. Thus, the study of such amodel will be fruitful. On the other hand, Garzo and Mon-tanero @24# determined the transport coefficients in a heatedsystem by the white noise thermostat, but their result is lessrelated to ours, because the white noise thermostat is heatednot by TB but by the granular temparture.We have derived hydrodynamic equations based on a sys-tematic Chapman-Enskog method for dilute granular gasesdriven by the Langevin thermostat in this paper. We havedetermined all the transport coefficients h , k , and m appearin Navier-Stokes order as a function of the restitution coef-ficient e.The result is based on the linearized approximation of a2in the first-order truncation of Sonine expansion. Althoughwe believe that this approximation gives a nice evaluation,nobody knows its theoretical background and quantitativevalidity. In particular, Pagnani et al. @10# have reported thatdeviation of f from the Gaussian in driven granular gases islarge. So we may need to check the convergence of the So-nine expansion and to compare the theoretical predictionwith simulations. We also need to look for the possibility toapply our result to explain the data in actual experiments forsuspensions.ACKNOWLEDGMENTSThe author would like to thank A. Santos for fruitful dis-cussion. This study was partially supported by the InamoriFoundation.APPENDIX A: STEADY VALUES OF a2 AND uAs mentioned in the text, it is possible to obtain exactsteady solutions of Eqs. ~25! and ~29!. The result is so com-plicated and the difference between them and the linearizedsolutions is small so that we do not use the exact form forlater discussion. Here we present the exact solutionsa2‘516~123e212e4!Q1 , ~A1!where Q1573232e275e2230e4164d2gˆ 18d(714e23e2116gˆ ), andu‘5dgˆ Q1Q2164d3gˆ 21Q32d~11e !Q4, ~A2!where Q252(11e)2(19246e133e2212e316e4), Q358d2gˆ (1114e27e2116gˆ ), and Q352281e2(28230gˆ )2137gˆ 16e3(5gˆ 22)1e(121169gˆ ).@1# L. Boltzmann, Lecture Notes on Gas Theory ~Dover, NewYork, 1995!.@2# S. Chapman and T.G. Cowling, The Mathematical Theory ofNonuniform Gases, 3rd ed. ~Cambridge University Press,FIG. 2. The time evolution of h* ~solid line!, k* ~dashed line!,and m* ~dot-dashed line! for e50.9, d53, gˆ 50.1, and u(0)52.FIG. 3. The steady values of h* ~solid line!, k* ~dashed line!,and m* ~dot-dashed line! for e50.9, d53, and gˆ 50.1.HYDRODYNAMICS OF DRIVEN GRANULAR GASES PHYSICAL REVIEW E 68, 031304 ~2003!031304-5Cambridge, 1970!.@3# Granular Gases, edited by T. Po¨schel and S. Luding ~Springer,Berlin, 2000!.@4# S.K. Friedlander, Smoke, Dust and Haze: Fundamentals ofAerosol Behavior ~Wiley, New York, 1977!.@5# T. Kawaguchi, T. Tanaka, and Y. Tsuji, Powder Technol. 96,129 ~1998!; T. Kawaguchi, M. Sakamoto, T. Tanaka, and Y.Tsuji, ibid. 109, 3 ~2000!.@6# K. Ichiki and H. Hayakawa, Phys. Rev. E 52, 658 ~1995!.@7# J.M. Montanero and A. Santos, Granular Matter 1, 57 ~1998!.@8# A. Santos, Physica A 321, 442 ~2003!.@9# J.A. Carrillo, C. Cercignani, and I.M. Gamba, Phys. Rev. E 62,7700 ~2000!.@10# R. Pagnani, U.M. Bettolo Marconi, and A. Puglish, Phys. Rev.E 66, 051304 ~2002!.@11# A. Goldstein and M. Shapiro, J. Fluid Mech. 282, 75 ~1995!.@12# T.P.J. van Noije and M.H. Ernst, Granular Matter 1, 57 ~1998!.@13# W.J. Stronge, Impact Mechanics ~Cambridge University Press,Cambridge, 2000!.@14# K.L. Johnson, Contact Mechanics ~Cambridge UniversityPress, Cambridge, 1985!.@15# G. Kuwabara and K. Kono, Jpn. J. Appl. Phys., Part 1 26, 1230~1987!; N.V. Brilliantov, F. Spahn, J.-M. Hertzsch, and T.Po¨schel, Phys. Rev. E 53, 5382 ~1996!; W.A.M. Morgado andI. Oppenheim, ibid. 55, 1940 ~1997!; F. Gerl and A. Zippelius,ibid. 59, 2361 ~1999!; H. Hayakawa and H. Kuninaka, Chem.Eng. Sci. 57, 239 ~2002!.@16# N.V. Brilliantov and T. Po¨schel, Philos. Trans. R. Soc. London,Ser. A 360, 415 ~2002!.@17# L. Labous, A.D. Rosato, and R.N. Dave, Phys. Rev. E 56, 5717~1997!.@18# H. Kuninaka and H. Hayakawa, J. Phys. Soc. Jpn. 72, 1655~2003!.@19# W.G. Hoover, Computational Statistical Mechanics ~Elservier,Amsterdam, 1991!.@20# J.J. Brey, J.W. Dufty, C.S. Kim, and A. Santos, Phys. Rev. E58, 4638 ~1998!.@21# J.J. Brey and D. Cubero, Granular Gases ~Ref. @3#!.@22# N.V. Brilliantov and T. Po¨schel, Phys. Rev. E 61, 2809 ~2000!.@23# H.-D. Kim and H. Hayakawa, J. Phys. Soc. Jpn. 72, 1904~2003!.@24# V. Garzo and J.M. Montanero, Physica A 313, 336 ~2003!.@25# T. Biben, Ph.A. Martin, and J. Piasecki, Physica A 310, 308~2003!.@26# A. Santos, Phys. Rev. E 67, 051101 ~2003!.HISAO HAYAKAWA PHYSICAL REVIEW E 68, 031304 ~2003!031304-6

Hydrodynamics of driven granular gases

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Hydrodynamics of driven granular gases

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