Impurities in inelastic Maxwell models

Impurities in inelastic Maxwell moels Vicente Garzó Departamento e Física, Universia e Extremaura, E-671-Baajoz, Spain Abstract. Transport properties of impurities immerse in a granular gas unergoing homogenous cooling state are stuie. The results are obtaine from the Boltzmann-Lorentz kinetic equation for inelastic Maxwell moels in imensions. The kinetic equation is solve by means of the Chapman-Enskog metho up to first orer in the ensity graient of impurities. The mass an heat fluxes associate with impurities are etermine an the corresponing transport coefficients are ientifie. Both transport coefficients (iffusion an Dufour coefficients) are exactly obtaine in terms of the coefficients of restitution for the impurity-gas an gas-gas collisions as well as the ratios of mass an iameters. The results are compare with those obtaine for inelastic har spheres in the leaing Sonine approximation an by means of Monte Carlo simulations. The comparison shows goo agreement between both interaction moels, especially in the case of the iffusion coefficient. 1. INTRODUCTION Granular fluis are usually moelle by an iealize system of smooth har spheres with inelastic collisions. In the low-ensity limit, the Boltzmann equation has been conveniently moifie to account for the inelasticity of binary collisions an the Navier-Stokes transport coefficients have been compute by solving the corresponing kinetic equation by means of the Chapman-Enskog metho [1]. This task has been carrie out in the past few years in the case of a monocomponent gas [] as well as for multicomponent systems [3]. In both cases, as happens for elastic collisions, all the transport coefficients are given in terms of the solutions to linear integral equations which are solve approximately by taking the leaing terms in a Sonine polynomial expansion. In spite of this approximation, the kinetic theory results compare quite well with those obtaine from numerical solutions of the Boltzmann equation by using the irect simulation Monte Carlo (DSMC) metho [4]. The fact that the agreement between kinetic theory an DSMC results even occurs for strong issipation is a further testimony of the utility of the kinetic theory an the hyroynamic escriptions for granular fluis beyon the weak issipation limit. The main mathematical ifficulty in solving the Boltzmann equation for inelastic har spheres (IHS) comes from the form of the collision rate, which is proportional to the magnitue of the relative velocity of the two colliing particles. In the case of elastic collisions, an alternative to overcome this problem is to consier the repulsive Maxwell potential, for which the collision rate is inepenent of the relative velocity. This property simplifies the mathematical structure of the Boltzmann collision operator an allows one to get exact solutions of the Boltzmann equation for far from equilibrium situations [5]. In the case of inelastic gases, the so-calle inelastic Maxwell moels (IMM) [6] have been wiely use in the past few years as a toy moel to characterize the effect of the inelasticity of collisions on the properties of granular gases. The IMM share with elastic Maxwell molecules the property that the collision rate is velocity inepenent but their scattering rules are the same as for IHS. Most of the stuies performe in the context of IMM have been evote to homogeneous states, mainly to the analysis of the overpopulate high energy tails [7]. Much less is known for inhomogeneous situations, especially for multicomponent systems. One of the few stuies consiering mixtures of IMM has been carrie out by the author [8] in the uniform shear flow problem. The rheological properties (shear an normal stresses) of the mixture have been exactly evaluate in terms of issipation an parameters of the mixture an surprisingly, the results of IMM show an excellent agreement with those obtaine for IHS by means of Gra s approximation an Monte Carlo simulations. This fact stimulates the search for exact solutions for IMM which can be confronte with the results obtaine for IHS by using approximate analytical methos an computer simulations. The goal of this paper is to etermine the iffusion coefficient of impurities immerse in a gas of inelastic Maxwell molecules. The gas is in the homogeneous cooling state (HCS), namely, a homogeneous state where the temperature uniformly ecreases in time ue to collisional cooling. Diffusion is generate in the system by the presence of a 83

Report Documentation Page Form Approve OMB No. 74-188 Public reporting buren for the collection of information is estimate to average 1 hour per response, incluing the time for reviewing instructions, searching existing ata sources, gathering an maintaining the ata neee, an completing an reviewing the collection of information. Sen comments regaring this buren estimate or any other aspect of this collection of information, incluing suggestions for reucing this buren, to Washington Heaquarters Services, Directorate for Information Operations an Reports, 115 Jefferson Davis Highway, Suite 14, Arlington VA -43. Responents shoul be aware that notwithstaning any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it oes not isplay a currently vali OMB control number. 1. REPORT DATE 13 JUL 5 4. TITLE AND SUBTITLE Impurities in inelastic Maxwell moels. REPORT TYPE N/A 3. DATES COVERED - 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Departamento e Física, Universia e Extremaura, E-671-Baajoz, Spain 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 1. SPONSOR/MONITOR S ACRONYM(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Approve for public release, istribution unlimite 11. SPONSOR/MONITOR S REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES See also ADM179, International Symposium on Rarefie Gas Dynamics (4th) Hel in Monopoli (Bari), Italy on 1-16 July 4. 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU a. REPORT unclassifie b. ABSTRACT unclassifie c. THIS PAGE unclassifie 18. NUMBER OF PAGES 6 19a. NAME OF RESPONSIBLE PERSON Stanar Form 98 (Rev. 8-98) Prescribe by ANSI St Z39-18

weak concentration graient, which also inuces a heat flux. Both mass an heat fluxes efine the relevant transport coefficients of the problem: the iffusion coefficient an the Dufour coefficient. Here, I get explicit expressions for these coefficients by solving the corresponing Boltzmann-Lorentz equation for impurities by means of the Chapman- Enskog metho through the first orer in the concentration graient. These transport coefficients are given in terms of the coefficients of restitution for the impurity-gas an gas-gas collisions as well as the masses an iameters of impurities an gas particles. The epenence of these coefficients on the parameters of the system is illustrate an compare with known results erive for IHS [3, 9].. DESCRIPTION OF THE PROBLEM Let us consier a ilute gas of inelastic Maxwell molecules of mass m, iameter σ, an interparticle coefficient of restitution α. In the low-ensity regime, its velocity istribution function f (v) obeys the nonlinear Boltzmann equation (BE). We assume that the gas is in the HCS, namely, a homogeneous solution of the BE in which all the time epenence of f (v,t) is through the granular temperature T (t). In this case, the BE can be written as 1 ζ (v f ) = J[v f, f ], (1) v where the Boltzmann collision operator J[v f, f ] for IMM is J [v 1 f, f ] = ω v σ [ α 1 f (v nω 1,t) f (v,t) f (v 1,t) f (v,t) ]. () Here, n is the number ensity of the gas particles, ω is an effective collision frequency (to be chosen later), Ω = π / /Γ(/) is the total soli angle in imensions, an σ is a unit vector irecte along the centers of the two colliing spheres. Upon writing Eq. (1), I have taken into account the balance equation for the temperature T 1 t T = ζ, where the cooling rate ζ is given by [1] ζ (α) = 1 α ω. (3) The explicit form of the velocity istribution function f (v, t) is not explicitly known, although its velocity moments can be explicitly compute [1]. We assume now that some impurities of mass m an iameter σ are ae to the system. Given that its molar fraction is negligible, the state of the granular gas is not isturbe by the presence of impurities an so the velocity istribution function f (v, t) obeys the BE (1). Furthermore, collisions among impurities themselves can neglecte versus the impurity-gas collisions, which are characterize by a coefficient of restitution α. As sai in the Introuction, I want to etermine the transport properties of impurities generate in the system by the presence of a weak concentration graient x, x = n /n being the mole fraction of impurities. Uner these conitions, the velocity istribution function f (r,v,t) of impurities verifies the Boltzmann-Lorentz equation t f + v f = J[v f, f ], (4) where J [v 1 f, f ] = ω v σ [ α 1 f (r,v n Ω 1;t) f (r,v ;t) f (r,v 1 ;t) f (r,v ;t) ], (5) ω is an effective collision frequency (to be chosen also later) for impurity-gas collisions, an v 1 = v 1 m ( ) 1 + α 1 ( σ g 1 ) σ, v = v + m ( ) 1 + α 1 ( σ g 1 ) σ, (6) m + m m + m where g 1 = v 1 v is the relative velocity of the colliing pair. The partial temperature of impurities T is efine as n T = v m v f (v), (7) where n is the number ensity of impurities. In contrast to what happens for normal fluis, if impurities an gas particles are mechanically ifferent, the partial temperature T can iffer from the granular temperature T when 84

issipation is present. The failure of energy equipartition in granular fluis has even observe in real experiments [11]. Momentum an energy are not collisional invariants of the Boltzmann-Lorentz collision operator J[ f, f ]. Only the number ensity of impurities is conserve: where the mass flux j is efine as t n + j m =, (8) j = vm v f (v). (9) In the absence of iffusion, impurities are also in HCS an Eq. (4) becomes 1 ζ v (v f ) = J[v f, f ], (1) where use has been mae of the relation T 1 t T = ζ, where ζ is the cooling rate associate with the partial temperature T of impurities. Its expression is [8] ζ = ω [ µ(1 + α ) 1 µ ] (1 + α )(1 + θ), (11) where µ = m/(m + m ) an θ = m T /mt is the mean square velocity of gas particles relative to that of impurities. The balance equations for the temperatures T an T lea to the following time evolution equation for the temperature ratio γ T /T : γ 1 t γ = ζ ζ. (1) The fact that f epens on time only through T (t) necessarily implies that the temperature ratio γ must be inepenent of time, an so Eq. (1) gives the HCS conition ζ (t) = ζ (t). The impurity equilibrates to a common HCS with ifferent temperatures for the impurity an gas particles. This implies a breakown of the energy equipartition. In orer to get the explicit epenence of γ on the parameter space, one still nees to fix the parameters ω an ω. To optimize the agreement with the IHS results, the cooling rates ζ an ζ of IMM given by Eqs. (3) an (11), respectively, are ajuste to be the same as the ones given for IHS [1]. Since ζ an ζ are not exactly known for IHS, one can estimate them by taking their local equilibrium approximations. In this case, the collision frequencies ω an ω are given by ω = π Ω ν, ω = Ω π ( σ σ ) 1 ( ) 1 + θ 1/ ν, (13) θ where σ = (σ +σ )/ an ν = nσ 1 T /m is an effective collision frequency. With these choices, the epenence of the temperature ratio T /T on the parameters of the system given for IMM presents an excellent agreement with the one foun for IHS, even for strong issipation [13]. 3. TRANSPORT PROPERTIES OF IMPURITIES The aim of this Section is to get the transport properties (mass an heat fluxes) of impurities in the limit of small concentration graient x. These fluxes are etermine by solving the Boltzmann-Lorentz equation (4) from the Chapman-Enskog metho [1] aapte to issipative systems. Thus, I look for a normal solution in which all the space an time epenence of f is through the hyroynamic fiels. In our problem, this means that f (r,v,t) = f [v x (t),t (t)]. The normal solution is generate by expaning f in powers of x as f = f () + ε f (1) +, (14) where each factor ε correspons to the implicit factor x. The time erivative is also expane as t = () t +ε (1), where the action of the ifferent operators (k) t can be obtaine from the balance equation (8). They are given by () t x =, () t T = T ζ, (k) t t + x = j(k 1), (k) t T =, k 1, (15) m n 85

D(α) /D(1) 8 6 4 m /m=8 σ /σ=.5.6.7.8.9 1. α FIGURE 1. Plot of the reuce iffusion coefficient D(α)/D(1) as a function of the coefficient of restitution α = α in the three-imensional case for σ /σ = an m /m = 8. The soli line correspons to the exact result obtaine here for IMM while the ashe line is the result erive for IHS in the secon Sonine approximation.the symbols refer to Monte Carlo simulations for IHS. where j (k) = vm v f (k). (16) The zeroth-orer approximation f () is the solution of Eq. (1) but taking into account now the local epenence on the mole fraction x. Since f () is isotropic, it follows that the flux of impurities vanishes at this orer, i.e., j () an so (1) t x =. To first orer in ε, one has the kinetic equation () t f (1) J[ f (1) (, f ] = = (1) t ( f () x ) + v 1 f () =, ) v 1 x. (17) The secon equality follows from the balance equations (15) an the space epenence of f () through x. The mass flux j (1) an the contribution to the heat flux coming from impurities q (1) can be easily obtaine from Eq. (17). The heat flux is efine as q (1) = v m v v f (1). (18) To get these fluxes, one multiplies both sies of Eq. (17) by m v an 1 m v v an integrates over v. The result for the mass an heat fluxes is, respectively ( ) () t + ν j (1) = γ p x, (19) where p = nt is the hyrostatic pressure an ( ) () t + ν q q (1) = βj (1) pt ( + m c = 8 [ m ( + ) 4n T vv 4 f (v) 1 + c ) γ x, () ] ( + ). (1) 4 Upon writing Eqs. (19) an (), use has been mae of the collisional moments [13] vm vj[ f (1), f ] = ν j (1), () v m v vj[ f (1), f ] = ν q q (1) βj (1), (3) 86

L q (α) 14 1 1 8 m /m=8 σ =σ 6 4 m /m=.5.6.7.8.9 1. FIGURE. Plot of the reuce Dufour coefficient L q (α) as a function of the coefficient of restitution α = α in the threeimensional case for σ = σ, an two ifferent values of the mass ratio: m /m = an m /m = 8. The soli lines correspon to the exact results obtaine here for IMM while the ashe lines are the results erive for IHS in the first Sonine approximation. α where β = ν The mass an heat fluxes have the forms ν = Ω ( ) σ 1 ( ) 1 + θ 1/ µ ν, (4) π σ θ ν q = ν + {µ(1 + α )[ + 8 3µ(1 + α )] 3( + )}, (5) T m µ(1 + α )[ 3µ(1 + α ) + ]. (6) j (1) = m D x, (7) q (1) = T D q x, (8) where D is the iffusion coefficient an D q is a cross transport coefficient measuring energy transport ue to iffusion of impurities. Dimensional analysis requires that D T 1/ an D q T 1/. Consequently, () t j (1) = ζ T T j (1) = m ζ D x, () t q (1) = ζ T T q (1) = 3 ζ T D q x. (9) Inserting Eqs. (9) into Eqs. (19) an (), respectively, the transport coefficients D an D q can be easily ientifie from (7) an (8). The result is D = p γ m ν ν 1 ζ, (3) D q = n (m /m)β D + (1 + c )γ m ν 3 ζ. (31) νq Here, ν = ν /ν, β = βm/(t ν), ν q = ν q /ν, an ζ = ζ /ν. A useful way of characterizing the heat transport in a mixture ue to a concentration graient is through the Dufour coefficient. In our problem, the (imensionless) Dufour coefficient L q is efine by the relation J q q (1) + T j(1) = nt m m ν L q x. (3) The coefficient L q can be easily written in terms of the coefficients D an D q as L q = m ( ν T D q + ) p D. (33) 87

In the elastic case (α = α = 1), ζ =, T = T, an as expecte L q =. Figures 1 an show the epenence of the reuce coefficients D(α)/D(1) an L q (α), respectively, on the (common) coefficient of restitution α = α for ifferent systems. Here, D(1) is the elastic value of the iffusion coefficient. In the case of the iffusion coefficient, it is apparent that the results for IMM an IHS are practically inistinguishable over the whole range of values of α stuie. Moreover, both theories present an excellent agreement with Monte Carlo simulations. More iscrepancies between IMM an IHS are foun in the case of the Dufour coefficient L q, especially with increasing issipation. However, at a qualitative level, the IMM preictions compare well with the ones erive for IHS. 4. CONCLUDING REMARKS Diffusion of impurities in a ilute granular gas of IMM unergoing HCS has been analyze. The relevant transport coefficients of the problem (iffusion an Dufour coefficients) have been etermine from a Chapman-Enskog solution [1] of the Boltzmann-Lorentz equation up to first orer in the concentration graient. The expressions (3), (31), an (33) erive here for the iffusion D an Dufour L q coefficients are exact (within the context of IMM) an constitute the main goal of this paper. This contrasts with the previous results erive for IHS [3, 9], where the coefficients D an L q were obtaine by consiering the leaing terms in a Sonine polynomial expansion of the velocity istribution function of impurities. To make contact with the results reporte for IHS, the collision frequencies ω an ω (which can be seen as free parameters of the moel) nee to be chosen. Here, I have taken these frequencies to reprouce the cooling rates ζ an ζ of IHS in the local equilibrium approximation. With this choice, the comparison with known results for IHS shows that in general, the IMM preictions are reasonably goo for not too large issipation, especially for the iffusion coefficient which is relate to a first-egree velocity moment of the velocity istribution function. In the case of the Dufour coefficient, although the IMM capture qualitatively well the trens obtaine for IHS, the isagreement between both interaction potentials is more significant than the one foun for the iffusion coefficient D. In this sense, more iscrepancies between IMM an IHS are expecte when one consiers higher egree velocity moments. ACKNOWLEDGMENTS Partial support from the Ministerio e Eucación y Ciencia (Spain) through Grant No. FIS4-1339 is acknowlege. REFERENCES 1. Chapman, S., an Cowling, T. G., The Mathematical Theory of Nonuniform Gases (Cambrige University, Cambrige, 197).. Brey, J. J., Dufty, J. W., Kim, C. S., an Santos, A., Phys. Rev. E 58, 4638 4653 (1998). 3. Garzó, V., an Dufty, J. W., Phys. Fluis 14, 1476 149 (). 4. Bir G. A., Molecular Gas Dynamics an the Direct Simulation Monte Carlo of Gas Flows (Clarenon, Oxfor, 1994). 5. Garzó, V., an Santos, A., Kinetic Theory of Gases in Shear Flows. Nonlinear Transport (Kluwer Acaemic, Dorrecht, 3). 6. Bobylev, A. V., Carrillo, J. A., an Gamba, I., J. Stat. Phys. 98, 743 773 (); Ernst, M. H., an Brito, R., J. Stat. Phys. 19, 47 43 (). 7. See for instance, Ernst, M. H., an Brito, R., Phys. Rev. E 65, 431-1 4 (); Krapivsky, P. L., an Ben-Naim, E., Phys. Rev. E 66, 1139-1 1 (); Bobylev, A. V., an Cercignani, C., J. Stat. Phys. 11, 333 375 (3). 8. Garzó, V., J. Stat. Phys. 11, 657 683 (3). 9. Garzó, V., an Montanero, J. M., Phys. Rev. E 69, 131-1 13 (4). 1. Santos, A., Physica A 31, 336 466 (3). 11. Wilman, R. D., an Parker, D. J., Phys. Rev. Lett. 88, 6431-1 4 (); Feitosa, K., an Menon, N., Phys. Rev. Lett. 88, 19831-1 4 (). 1. Garzó, V., an Dufty, J. W., Phys. Rev. E 6, 576 5713 (1999). 13. Garzó, V., an Astillero, A., e-print con-mat/44386. 88