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Transport coefficients of solid particles immersed in a viscous gas

Transport properties of a suspension of solid particles in a viscous gas are studied. The dissipation in such systems arises from two sources: inelasticity in particle collisions and viscous dissipation due to the effect of the gas phase on the particles. Here, we consider a simplified case in which the mean relative velocity between the gas and solid phases is taken to be zero, such that "thermal drag" is the only remaining gas-solid interaction. Unlike the previous more general treatment of the drag force [Garzó \emph{et al.}, J. Fluid Mech. \textbf{712}, 129 (2012)], here we take into account contributions to the (scaled) transport coefficients $η^*$ (shear viscosity), $κ^*$ (thermal conductivity) and $μ^*$ (Dufour-like coefficient) coming from the temperature-dependence of the (dimensionless) friction coefficient $γ^*$ characterizing the amplitude of the drag force. At moderate densities, the thermal drag model (which is based on the Enskog kinetic equation) is solved by means of the Chapman-Enskog method and the Navier-Stokes transport coefficients are determined in terms of the coefficient of restitution, the solid volume fraction and the friction coefficient. The results indicate that the effect of the gas phase on $η^*$ and $μ^*$ is non-negligible (especially in the case of relatively dilute systems) while the form of $κ^*$ is the same as the one obtained in the dry granular limit. Finally, as an application of these results, a linear stability analysis of the hydrodynamic equations is carried out to analyze the conditions for stability of the homogeneous cooling state. A comparison with direct numerical simulations shows a good agreement for conditions of practical interest.