Granular mixtures modeled as elastic hard spheres subject to a drag force

TL;DR

This paper introduces a simplified elastic hard sphere model with a drag force to mimic inelastic granular gas behavior, enabling easier calculation of transport coefficients while maintaining good accuracy compared to the inelastic Boltzmann equation.

Contribution

It proposes a new elastic sphere model with a drag force to replicate granular gas dissipation effects, simplifying analysis of granular mixtures.

Findings

Transport coefficients match inelastic Boltzmann results for mass flux.

Good agreement in thermal conductivity coefficients.

Model provides a practical extension of elastic kinetic models for granular gases.

Abstract

Granular gaseous mixtures under rapid flow conditions are usually modeled by a multicomponent system of smooth inelastic hard spheres with constant coefficients of normal restitution. In the low density regime an adequate framework is provided by the set of coupled inelastic Boltzmann equations. Due to the intricacy of the inelastic Boltzmann collision operator, in this paper we propose a simpler model of elastic hard spheres subject to the action of an effective drag force, which mimics the effect of dissipation present in the original granular gas. The Navier--Stokes transport coefficients for a binary mixture are obtained from the model by application of the Chapman--Enskog method. The three coefficients associated with the mass flux are the same as those obtained from the inelastic Boltzmann equation, while the remaining four transport coefficients show a general good agreement, especially in the case of the thermal conductivity. Finally, the approximate decomposition of the inelastic Boltzmann collision operator is exploited to construct a model kinetic equation for granular mixtures as a direct extension of a known kinetic model for elastic collisions.