The Fourier state of a dilute granular gas described by the inelastic Boltzmann equation

TL;DR

This paper investigates two stationary solutions of the inelastic Boltzmann equation for dilute granular gases, revealing a Fourier-like heat flux law and comparing theoretical predictions with molecular dynamics simulations.

Contribution

It introduces and analyzes two new stationary solutions of the inelastic Boltzmann equation with a scaling property, extending understanding beyond weak dissipation or small gradients.

Findings

One solution is singular in the elastic limit.

The other solution's predictions agree well with simulations under moderate inelasticity.

A Fourier-like law for heat flux is established in the granular gas context.

Abstract

The existence of two stationary solutions of the nonlinear Boltzmann equation for inelastic hard spheres or disks is investigated. They are restricted neither to weak dissipation nor to small gradients. The one-particle distribution functions are assumed to have an scaling property, namely that all the position dependence occurs through the density and the temperature. At the macroscopic level, the state corresponding to both is characterized by uniform pressure, no mass flow, and a linear temperature profile. Moreover, the state exhibits two peculiar features. First, there is a relationship between the inelasticity of collisions, the pressure, and the temperature gradient. Second, the heat flux can be expressed as being linear in the temperature gradient, i.e. a Fourier-like law is obeyed. One of the solutions is singular in the elastic limit. The theoretical predictions following from the other one are compared with molecular dynamics simulation results and a good agreement is obtained in the parameter region in which the Fourier state can be actually observed in the simulations, namely not too strong inelasticity.