Transport properties for driven granular fluids in situations close to homogeneous steady states

TL;DR

This paper derives transport coefficients for driven granular fluids near steady states using kinetic theory, compares results with simulations, and analyzes stability, revealing good viscosity agreement and stability due to external driving.

Contribution

It provides explicit expressions for transport coefficients in driven granular fluids considering time-dependent reference distributions, a novel approach in this context.

Findings

Excellent agreement with simulations for kinematic viscosity.

Discrepancies observed in longitudinal viscosity and thermal diffusivity at high densities.

No instabilities found in the stability analysis due to external driving.

Abstract

The transport coefficients of a granular fluid driven by a stochastic bath with friction are obtained by solving the inelastic Enskog kinetic equation from the Chapman-Enskog method. The heat and momentum fluxes as well as the cooling rate are determined to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the collisional cooling cannot be compensated locally for the heat produced by the external driving force, the reference distribution $f^{(0)}$ (zeroth-order approximation) depends on time through its dependence on temperature. This fact gives rise to conceptual and practical difficulties not present in the undriven case. On the other hand, to simplify the analysis and given that we are interested in computing transport in the first order of deviations from the reference state, the steady-state conditions are considered to get explicit forms for the transport coefficients and the cooling rate. A comparison with recent molecular dynamics simulations for driven granular fluids shows an excellent agreement for the kinematic viscosity although some discrepancies are observed for the longitudinal viscosity and the thermal diffusivity at large densities. Finally, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state is performed. As expected, no instabilities are found thanks to the presence of the external bath.