Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. II. Stability analysis

TL;DR

This paper performs a linear stability analysis of dilute granular gases of inelastic and rough hard disks or spheres, revealing conditions for stability and comparing theoretical predictions with simulations.

Contribution

It extends stability analysis to inelastic and rough particles, providing novel results for hard disks and comparing with simulations.

Findings

Identifies a high-inelasticity region with divergent wave number for hard disks.

Shows good agreement between theory and simulations at moderate inelasticity.

Suggests the high-inelasticity instability may be an artifact of approximations.

Abstract

Conditions for the stability under linear perturbations around the homogeneous cooling state are studied for dilute granular gases of inelastic and rough hard disks or spheres with constant coefficients of normal ($\alpha$) and tangential ($\beta$) restitution. After a formally exact linear stability analysis of the Navier--Stokes--Fourier hydrodynamic equations in terms of the translational ($d_t$) and rotational ($d_r$) degrees of freedom, the transport coefficients derived in the companion paper [A. Meg\'ias and A. Santos, "Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. I. Transport coefficients," Phys. Rev. E 104, 034901 (2021)] are employed. Known results for hard spheres [V. Garz\'o, A. Santos, and G. M. Kremer, Phys. Rev. E 97, 052901 (2018)] are recovered by setting $d_t=d_r=3$, while novel results for hard disks ($d_t=2$, $d_r=1$) are obtained. In the latter case, a high-inelasticity peculiar region in the $(\alpha,\beta)$ parameter space is found, inside which the critical wave number associated with the longitudinal modes diverges. Comparison with event-driven molecular dynamics simulations for dilute systems of hard disks at $\alpha=0.2$ shows that this theoretical region of absolute instability may be an artifact of the extrapolation to high inelasticity of the approximations made in the derivation of the transport coefficients, although it signals a shrinking of the conditions for stability. In the case of moderate inelasticity ($\alpha=0.7$), however, a good agreement between the theoretical predictions and the simulation results is found.