Enskog kinetic theory in a model of confined granular mixtures. Heat flux and some applications

Abstract: This work considers the Enskog kinetic theory for moderately dense confined granular mixtures within the framework of the $Δ$-model, an effective collisional model that accounts for the transfer of energy from the vertical to the horizontal degrees of freedom of grains through an additional velocity increment during collisions. The main objective is to complete the Navier--Stokes hydrodynamic description of dense binary granular mixtures by explicitly determining the heat-flux transport coefficients, namely the Dufour and thermal conductivity coefficients, together with the first-order contributions to the partial temperatures and the cooling rate. These quantities are determined in the steady state from the Enskog kinetic equation by means of the Chapman--Enskog method and are expressed in terms of the masses and diameters of the particles, mixture composition, density, and coefficients of restitution. The results show that finite-density effects have a significant influence on transport, leading to behaviors that differ markedly from those observed in the dilute regime. The complete set of transport coefficients is considered further to study two different problems. First, we perform a linear stability analysis of the homogeneous steady state (HSS). The analysis demonstrates that no unstable transverse or longitudinal hydrodynamic modes are found over the parameter space explored, indicating that the HSS is linearly stable even at moderate densities. As a second problem, the violation of the Onsager reciprocal relations for a confined dense granular mixture is quantified in terms of the parameter space of the problem.