Abstract
The Boltzmann equation for inelastic Maxwell models is used to analyze nonlinear transport in a granular binary mixture in the steady simple shear flow. Two different transport processes are studied. First, the rheological properties (shear and normal stresses) are obtained by solving exactly the velocity moment equations. Second, the diffusion tensor of impurities immersed in a sheared inelastic Maxwell gas is explicitly determined from a perturbation solution through first order in the concentration gradient. The corresponding reference state of this expansion corresponds to the solution derived in the (pure) shear flow problem. All these transport coefficients are given in terms of the restitution coefficients and the parameters of the mixture (ratios of masses, concentration, and sizes). The results are compared with those obtained analytically for inelastic hard spheres in the first Sonine approximation and by means of Monte Carlo simulations. The comparison between the results obtained for both interaction models shows a good agreement over a wide range values of the parameter space.
Similar content being viewed by others
REFERENCES
J. J. Brey, J. W. Dufty, and A. Santos, Dissipative dynamics for hard spheres, J. Stat. Phys. 87:1051-1066 (1997); T. P. C. van Noije, M. H. Ernst, and R. Brito, Ring kinetic theory for an idealized granular gas, Physica A 251:266-283 (1998).
J. J. Brey, J. W. Dufty, C. S. Kim, and A. Santos, Hydrodynamics for granular flow at low density, Phys. Rev. E 58:4638-4653 (1998).
V. Garzó and J. W. Dufty, Dense fluid transport for inelastic hard spheres, Phys. Rev. E 59:5895-5911 (1999).
V. Garzó and J. M. Montanero, Transport coefficients of a heated granular gas, Physica A 313:336-356 (2002).
V. Garzó and J. W. Dufty, Hydrodynamics for a granular mixture at low density, Phys. Fluids 14:1476-1490 (2002).
J. Lutsko, J. J. Brey, and J. W. Dufty, Diffusion in a granular fluid. II. Simulation, Phys. Rev. E 65:051304(2002).
J. J. Brey, M. J. Ruiz-Montero, and D. Cubero, On the validity of linear hydrdodynamics for low-density granular flows described by the Boltzmann equation, Europhys. Lett. 48:359-364 (1999).
J. M. Montanero and V. Garzó, Shear viscosity for a heated granular binary mixture at low-density, Phys. Rev. E 67:021308(2003).
M. H. Ernst, Non-linear model-Boltzmann equations and exact solutions, Phys. Rep. 78:1-171 (1981).
C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, II, J. Rat. Mech. Anal. 5:55-128 (1956).
C. Truesdell and R. G. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas (Academic Press, New York, 1980).
A. Santos and V. Garzó, Exact non-linear transport from the Boltzmann equation, in Rarefied Gas Dynamics 19, J. Harvey and G. Lord, eds. (Oxford University Press, Oxford, 1995), pp. 13-22.
W. Loose and S. Hess, Velocity distribution function of a streaming gas via nonequilibrium molecular dynamics, Phys. Rev. Lett. 58:2443-2445 (1987); W. Loose, The constant-temperature constraint in the nonequilibrium molecular dynamics and the non-Newtonian viscosity coefficient of gases, Phys. Lett. A 128:39-44; J. Gómez Ordoñez, J. J. Brey, and A. Santos, Shear-rate dependence of the viscosity for dilute gases, Phys. Rev. A 39:3038-3040 (1989).
E. Ben-Naim and P. L. Krapivsky, Multiscaling in inelastic collisions, Phys. Rev. E 61:R5-R8 (2000).
A. V. Bobylev, J. A. Carrillo, and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Stat. Phys. 98:743-773 (2000).
J. A. Carrillo, C. Cercignani, and I. M. Gamba, Steady states of a Boltzmann equation for driven granular media, Phys. Rev. E 62:7700-7707 (2000).
M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, J. Stat. Phys. 109:407-432 (2002).
A. Baldasarri, U. M. B. Marconi, and A. Puglisi, Influence of correlations of the velocity statistics of scalar granular gases, Europhys. Lett. 58:14-20 (2002).
P. L. Krapivsky and E. Ben-Naim, Scaling, multiscaling, and nontrivial exponents in inelastic collision processes, Phys. Rev. E 66:011309(2002).
M. H. Ernst and R. Brito, Driven inelastic Maxwell models with high energy tails, Phys. Rev. E 65:040301(R)(2002).
A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys. 110:333-375 (2003).
A. V. Bobylev, C. Cercignani, and G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, J. Stat. Phys. 111:403-416 (2003).
A. Santos, Transport coefficients of d-dimensional inelastic Maxwell models, Physica A 321:442-466 (2003).
H. Hayakawa, Hydrodynamics for inelastic Maxwell model, cond-mat/0209630.
C. Cercignani, Shear flow of a granular fluid, J. Stat. Phys. 102:1407-1415 (2001).
U. M. B. Marconi and A. Puglisi, Mean-field model of free-cooling inelastic mixtures, Phys. Rev. E 65:051305(2002); Steady state properties of a mean field model of driven inelastic mixtures, Phys. Rev. E 66:011301(2002).
E. Ben-Naim and P. L. Krapivsky, Impurity in a granular fluid, Eur. Phys. J. E 8:507-515 (2002).
J. M. Montanero and V. Garzó, Rheological properties in a low-density granular mixture, Phys. A 310:17-38 (2002).
V. Garzó, Tracer diffusion in granular shear flows, Phys. Rev. E 66:021308(2002).
J. M. Montanero and V. Garzó, Energy nonequipartition in a sheared granular mixture, Mol. Sim. (to be published) and cond-mat/0204205.
A. V. Bobylev and C. Cercignani, Moment equations for a granular material in a thermal bath, J. Stat. Phys. 106:547-567 (2002).
R. Clelland and C. Hrenya, Simulations of a binary-sized mixture of inelastic grains in rapid shear flow, Phys. Rev. E 65:031301(2002).
R. D. Wildman and D. J. Parker, Coexistence of two granular temperatures in binary vibrofluidized beds, Phys. Rev. Lett. 88:064301(2002); K. Feitosa and N. Menon, Breakdown of energy equipartition in a 2D binary vibrated granular gas, Phys. Rev. Lett. 88:198301(2002).
V. Garzó and J. W. Dufty, Homogeneous cooling state for a granular mixture, Phys. Rev. E 60:5706-5713 (1999); J. M. Montanero and V. Garzó, Monte Carlo simulation of the homogeneous cooling state for a granular mixture, Granular Matter 4:17-24 (2002).
J. W. Dufty and V. Garzó, Mobility and diffusion in granular fluids, J. Stat. Phys. 105:723-744 (2001).
S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, Cambridge, 1970).
V. Garzó, A. Santos, and J. J. Brey, Influence of nonconservative external forces on self-diffusion in dilute gases. Physica A 163:651-671 (1990); V. Garzó and M. López de Haro, Tracer diffusion in shear flow, Phys. Rev. A 44:1397-1400 (1991); Kinetic models for uniform shear flow, Phys. Fluids A 4:1057-1069 (1992); C. Marín and V. Garzó, Mutual diffusion in a binary mixture under shear flow, Phys. Rev. E 57:507-513 (1998).
C. S. Campbell, Self-diffusion in granular shear flows, J. Fluid Mech. 348:85-101) (1997).
A. V. Bobylev, M. Groppi, and G. Spiga, Approximate solutions to the problem of stationary shear flow of granular material, Eur. J. Mech. B Fluids 21:91-103 (2002).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Garzó, V. Nonlinear Transport in Inelastic Maxwell Mixtures Under Simple Shear Flow. Journal of Statistical Physics 112, 657–683 (2003). https://doi.org/10.1023/A:1023828109434
Issue Date:
DOI: https://doi.org/10.1023/A:1023828109434