Sonine approximation for collisional moments of granular gases of inelastic rough spheres

TL;DR

This paper uses Sonine approximation to evaluate collisional moments in a granular gas of inelastic rough spheres, analyzing temperature ratios and cumulants in different states, revealing non-Maxwellian effects especially in the cooling state.

Contribution

It introduces a Sonine approximation method to compute collisional rates of moments in inelastic rough spheres, improving understanding of non-Maxwellian properties in granular gases.

Findings

Maxwellian approximation closely predicts temperature ratio.

Cumulants reveal significant non-Maxwellian behavior in cooling state.

Singular behavior of cumulants near quasi-smooth limit in cooling state.

Abstract

We consider a dilute granular gas of hard spheres colliding inelastically with coefficients of normal and tangential restitution $\alpha$ and $\beta$, respectively. The basic quantities characterizing the distribution function $f(\mathbf{v},\bm{\omega})$ of linear ($\mathbf{v}$) and angular ($\bm{\omega}$) velocities are the second-degree moments defining the translational ($T^\text{tr}$) and rotational ($T^\text{rot}$) temperatures. The deviation of $f$ from the Maxwellian distribution parameterized by $T^\text{tr}$ and $T^\text{rot}$ can be measured by the cumulants associated with the fourth-degree velocity moments. The main objective of this paper is the evaluation of the collisional rates of change of these second- and fourth-degree moments by means of a Sonine approximation. The results are subsequently applied to the computation of the temperature ratio $T^\text{rot}/T^\text{tr}$ and the cumulants of two paradigmatic states: the homogeneous cooling state and the homogeneous steady state driven by a white-noise stochastic thermostat. It is found in both cases that the Maxwellian approximation for the temperature ratio does not deviate much from the Sonine prediction. On the other hand, non-Maxwellian properties measured by the cumulants cannot be ignored, especially in the homogeneous cooling state for medium and small roughness. In that state, moreover, the cumulant directly related to the translational velocity differs in the quasi-smooth limit $\beta\to -1$ from that of pure smooth spheres ($\beta=-1$). This singular behavior is directly related to the unsteady character of the homogeneous cooling state and thus it is absent in the stochastic thermostat case.