Hydrodynamic equations for granular mixtures

TL;DR

This paper derives hydrodynamic equations for granular mixtures of inelastic hard spheres using kinetic theory, analyzing stability and inhomogeneity development in granular flows.

Contribution

It provides explicit derivations of hydrodynamic equations and transport coefficients for binary granular mixtures near the homogeneous cooling state.

Findings

Homogeneous state is unstable to long-wavelength perturbations.

Dispersion relations for linearized hydrodynamics are obtained.

Conditions for stability depend on wave vector, dissipation, and mixture parameters.

Abstract

Many features of granular media can be modeled by a fluid of hard spheres with inelastic collisions. Under rapid flow conditions, the macroscopic behavior of grains can be described through hydrodynamic equations accounting for dissipation among the interacting particles. A basis for the derivation of hydrodynamic equations and explicit expressions appearing in them is provided by the Boltzmann kinetic theory conveniently modified to account for inelastic binary collisions. The goal of this review is to derive the hydrodynamic equations for a binary mixture of smooth inelastic hard spheres. A normal solution to the Boltzmann equation is obtained via the Chapman-Enskog method for states near the {\em local} homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the set of transport coefficients are determined in terms of the restitution coefficients and the ratios of mass, concentration, and particle sizes. As an example of their application, the dispersion relations for the hydrodynamic equations linearized about a homogeneous state are obtained and the conditions for stability are identified as functions of the wave vector, the dissipation, and the parameters of the mixture. The analysis shows that the homogeneous reference state is unstable to long enough wavelength perturbations and consequently becomes inhomogeneous for long times. The relationship of this instability to the validity of hydrodynamics is discussed.