Transport coefficients for granular suspensions at moderate densities

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The Enskog kinetic theory for moderately dense granular suspensions is considered as a model to determine the Navier–Stokes transport coefficients. The influence of the interstitial gas on solid particles is modeled by a viscous drag force term plus a stochastic Langevin-like term. The suspension model is solved by means of the Chapman–Enskog method conveniently adapted to dissipative dynamics. The momentum and heat fluxes as well as the cooling rate are obtained to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the cooling terms (arising from collisional dissipation and viscous friction) cannot be compensated for by the energy gained by grains due to collisions with the interstitial gas, the reference distribution (zeroth-order approximation of the Chapman–Enskog solution) depends on time through its dependence on temperature. On the other hand, to simplify the analysis and given that we are interested in computing transport properties in the first order of deviations from the reference state, the steady-state conditions are considered. This simplification allows us to get explicit expressions for the Navier–Stokes transport coefficients. The present work extends previous results (Garzó et al 2013 Phys Rev. E 87 032201) since it incorporates two extra ingredients (an additional density dependence of the zeroth-order solution and the density dependence of the reduced friction coefficient) not accounted for by the previous theoretical attempt. While these two new ingredients do not affect the shear viscosity coefficient, the transport coefficients associated with the heat flux as well as the first-order contribution to the cooling rate are different from those obtained in the previous study. In addition, as expected, the results show that the dependence of the transport coefficients on both inelasticity and density is clearly different from that found in its granular counterpart (no gas phase). Finally, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state is performed. In contrast to the granular case (no gas-phase), no instabilities are found and hence, the homogeneous steady state is (linearly) stable.