Mass transport of driven inelastic Maxwell mixtures

Abstract

Mass transport of a driven granular binary mixture is analyzed from the inelastic Boltzmann kinetic equation for inelastic Maxwell models (IMM). The mixture is driven by a thermostat constituted by two terms: a stochastic force and a drag force proportional to the particle velocity. The combined action of both forces attempts to mimic the interaction of solid particles with the interstitial surrounding gas. As with ordinary gases, the use of IMM allows us to exactly evaluate the velocity moments of the Boltzmann collision operator and so, it opens up the possibility of obtaining the exact forms of the Navier--Stokes transport coefficients of the granular mixture. In this work, the diffusion coefficients associated with the mass flux are explicitly determined in terms of the parameters of the mixture. As a first step, the steady homogeneous state reached by the system when the energy lost by collisions is compensated for by the energy injected by the thermostat is addressed. In this steady state, the ratio of kinetic temperatures are determined and compared against molecular dynamics simulations for inelastic hard spheres (IHS). The comparison shows an excellent agreement, even for strong inelasticity and/or disparity in masses and diameters. As a second step, the set of kinetic equations for the mixture is solved by means of the Chapman-Enskog method for states near homogeneous steady states. In the first-order approximation, the mass flux is obtained and the corresponding diffusion transport coefficients identified. The results show that the predictions for IMM obtained in this work coincide with those previously derived for IHS in the first-Sonine approximation when the non-Gaussian corrections to the zeroth-order approximation are neglected.