Enskog kinetic theory of binary granular suspensions: heat flux and stability analysis of the homogeneous steady state

TL;DR

This paper extends the Enskog kinetic theory to calculate heat flux transport coefficients in binary granular suspensions, analyzing their dependence on mixture parameters and stability of the homogeneous steady state.

Contribution

It provides explicit forms of heat flux transport coefficients using the Chapman--Enskog method and second Sonine approximation, including their variation with mixture parameters.

Findings

Heat flux coefficients depend differently on inelasticity compared to dry mixtures.

Homogeneous steady state is linearly stable for long wavelengths.

Stability results align with previous suspensions but differ from dry granular mixtures.

Abstract

The Enskog kinetic theory of multicomponent granular suspensions employed previously [G\'omez Gonz\'alez, Khalil, and Garz\'o, Phys. Rev. E \textbf{101}, 012904 (2020)] is considered further to determine the four transport coefficients associated with the heat flux. These transport coefficients are obtained by solving the Enskog equation by means of the application of the Chapman--Enskog method around the local version of the homogeneous state. Explicit forms of the heat flux transport coefficients are provided in steady-state conditions by considering the so-called second Sonine approximation to the distribution function of each species. Their quantitative variation on the control parameters of the mixture (masses and diameters, coefficients of restitution, concentration, volume fraction, and the background temperature) is demonstrated and the results show that in general the dependence of the heat flux transport coefficients on inelasticity is clearly different from that found in the absence of the gas phase (\emph{dry} granular mixtures). As an application of the general results, the stability of the homogeneous steady state is analyzed by solving the linearized Navier--Stokes hydrodynamic equations. The linear stability analysis (which holds for wavelengths long compared with the mean free path) shows that the transversal and longitudinal modes are always stable with respect to long-enough wavelength excitations. This conclusion agrees with previous results derived for monocomponent and (dilute) bidisperse granular suspensions but contrasts with the instabilities found in previous works in dry (no gas phase) granular mixtures.