Heat flux of driven granular mixtures at low density. Stability analysis of the homogeneous steady state

TL;DR

This paper derives heat flux transport coefficients for driven low-density granular mixtures and analyzes the stability of their homogeneous steady state, finding it stable under long-wavelength perturbations, contrasting with undriven cases.

Contribution

It provides explicit expressions for heat flux transport coefficients in driven granular mixtures and demonstrates their stability properties, extending previous theoretical frameworks.

Findings

Transport coefficients depend on system parameters.

Homogeneous steady state is linearly stable for long wavelengths.

Contrasts with undriven granular mixture stability results.

Abstract

The Navier--Stokes order hydrodynamic equations for a low-density driven granular mixture obtained previously [Khalil and Garz\'o, Phys. Rev. E \textbf{88}, 052201 (2013)] from the Chapman--Enskog solution to the Boltzmann equation are considered further. The four transport coefficients associated with the heat flux are obtained in terms of the mass ratio, the size ratio, composition, coefficients of restitution, and the driven parameters of the model. Their quantitative variation on the control parameters of the system is demonstrated by considering the leading terms in a Sonine polynomial expansion to solve the exact integral equations. As an application of these results, the stability of the homogeneous steady state is studied. In contrast to the results obtained in undriven granular mixtures, the stability analysis of the linearized Navier--Stokes hydrodynamic equations shows that the transversal and longitudinal modes are (linearly) stable with respect to long enough wavelength excitations. This conclusion agrees with a previous analysis made for single granular gases.