Kinetic theory of granular particles immersed in a molecular gas

TL;DR

This paper develops a kinetic theory framework for dilute inelastic granular gases immersed in a molecular gas, deriving transport coefficients and analyzing stability, with results validated against simulations.

Contribution

It introduces a model treating the molecular gas as a thermal bath and derives new transport coefficients for the granular gas considering inelastic collisions.

Findings

Transport coefficients depend on the coefficient of restitution differently than in dry granular gases.

Homogeneous steady state is linearly stable under perturbations.

Results agree well with Monte Carlo simulations and previous Fokker-Planck analyses.

Abstract

The transport coefficients of a dilute gas of inelastic hard spheres immersed in a molecular gas are determined. We assume that the number density of the granular gas is much smaller than that of the surrounding molecular gas, so that the latter is not affected by the presence of solid particles. In this situation, the molecular gas may be treated as a thermostat (or bath) of elastic hard spheres at a fixed temperature. This system (granular gas thermostated by a bath of elastic hard spheres) can be considered as a reliable model for describing the dynamic properties of particle-laden suspensions. The Boltzmann kinetic equation is the starting point of the present work. First step is to characterise the reference state in the perturbation scheme, namely the homogeneous state. Theoretical results for the granular temperature and kurtosis obtained in the homogeneous steady state are compared against Monte Carlo simulations showing a good agreement. Then, the Chapman-Enskog method is employed to solve the Boltzmann equation to first order in spatial gradients. As expected, the Navier-Stokes-Fourier transport coefficients of the granular gas are given in terms of the solutions of a coupled set of linear integral equations which are approximately solved by considering the leading terms in a Sonine polynomial expansion. Our results show that the dependence of the transport coefficients on the coefficient of restitution is quite different from that found when the influence of the interstitial molecular gas is neglected (dry granular gas). When the granular particles are much more heavier than the gas particles (Brownian limit) the expressions of the transport coefficients are consistent with those previously derived from the Fokker-Planck equation. Finally, a linear stability analysis of the homogeneous steady state is performed showing this state is always linearly stable.