Kinetic theory of granular particles immersed in a molecular gas

Abstract

The transport coefficients of a dilute gas of inelastic hard spheres immersed in a gas of elastic hard spheres (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 the granular particles. In this situation, the molecular gas may be treated as a thermostat (or bath) of elastic hard spheres at a fixed temperature. The Boltzmann kinetic equation is the starting point of the present work. The 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. In dimensionless form, the Navier–Stokes–Fourier transport coefficients of the granular gas are given in terms of the mass ratio $m/m_g$ ( $m$ and $m_g$ being the masses of a granular and a gas particle, respectively), the (reduced) bath temperature and the coefficient of restitution. Interestingly, previous results derived from a suspension model based on an effective fluid–solid interaction force are recovered in the Brownian limit ( $m/m_g \to \infty$ ). Finally, as an application of the theory, a linear stability analysis of the homogeneous steady state is performed showing that this state is always linearly stable.