Kinetic Theory for Binary Granular Mixtures at Low Density
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any features of granular media can be modeled as a fluid of hard spheres with inelastic collisions. Under rapid flow conditions, the macroscopic behavior of grains can be described through hydrodynamic equations. At low density, a fundamental basis for the derivation of the hydrodynamic equations and explicit expressions for the transport coefficients appearing in them is provided by the Boltzmann kinetic theory conveniently modified to account for inelastic binary collisions. The goal of this chapter is to give an overview of the recent advances made for binary granular gases by using kinetic theory tools. Some of the results presented here cover aspects such as transport properties, energy nonequipartition, instabilities, segregation or mixing, non-Newtonian behavior, etc. In addition, comparison of the analytical results with those obtained from Monte Carlo and molecular dynamics simulations is also carried out, showing the reliability of kinetic theory to describe granular flows even for strong dissipation.
10.1 Introduction Granular systems have attracted the attention of the physics community in the past few years, in part because the behavior of these systems under many conditions exhibit a great similarity to ordinary fluids [1]. These conditions include rapid, dilute flows where the dominant transfer of momentum and energy is through binary collisions of the grains. The main difference from ordinary fluids is the absence of energy conservation, leading to both obvious and subtle modifications of the usual macroscopic balance equations as well as the constitutive equations for the irreversible fluxes. However, in spite of the utility of hydrodynamics to describe rapid granular flows, there are still some open questions about its domain of validity and the associated constitutive equations appearing in the hydrodynamic equations [2]. To isolate the effects of such collisional dissipation from other important properties of granular media, an idealized microscopic model system is
Garz´ o, V.: Kinetic Theory for Binary Granular Mixtures at Low Density. Lect. Notes Phys. 753, 493–540 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-78767-9 10
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V. Garz´ o
usually considered: a system composed by smooth hard spheres with inelastic collisions. As in the elastic case, the collisions are specified in terms of the change in relative velocity at contact but with a decrease in the magnitude of the normal component measured by a positive coefficient of restitution α ≤ 1. This parameter distinguishes the granular fluid (α < 1) from the ordinary fluid (α = 1). Given that the hard sphere system with elastic collisions has been widely studied for both equilibrium and nonequilibrium statistical mechanics [3], it is tempting to apply the same methods for the case of inelastic collisions. However, some care is warranted in translating properties of ordinary fluids to granular fluids. In this presentation, a kinetic theory description based on the Boltzmann kinetic equation (which applies at sufficiently low density) will
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