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

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. Megías 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. Garzó, 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.

arXiv comment13 pages, 11 figures; v2: Supplemental material (videos) added

References in corpus (20)

• Size distribution of particles in Saturn's rings from aggregation and fragmentation
• Cluster-Growth in Freely Cooling Granular Media
• Translations and Rotations are correlated in Granular Gases
• Information Entropy in Cosmology
• Modified Sonine approximation for the Navier-Stokes transport coefficients of a granular gas
• A model for the atomic-scale structure of a dense, nonequilibrium fluid: the homogeneous cooling state of granular fluids
• Breakdown of the Sonine expansion for the velocity distribution of Granular Gases
• Transport coefficients of a granular gas of inelastic rough hard spheres
• Role of roughness on the hydrodynamic homogeneous base state of inelastic spheres
• Sonine approximation for collisional moments of granular gases of inelastic rough spheres
• Steady state in a gas of inelastic rough spheres heated by a uniform stochastic force
• Numerical Simulations of Saturn's B Ring: Granular Friction as a Mediator between Self-Gravity and Viscous Overstability
• Impact of roughness on the instability of a free-cooling granular gas
• Inhomogeneous Cooling of the Rough Granular Gas in Two Dimensions
• Driven and undriven states of multicomponent granular gases of inelastic and rough hard disks or spheres
• Energy Production Rates of Multicomponent Granular Gases of Rough Particles. A Unified View of Hard-Disk and Hard-Sphere Systems
• Experimental study of clusters in dense granular gas and implications for the particle stopping time in protoplanetary disks
• Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. I. Transport coefficients
• Relative entropy as a measure of inhomogeneity in general relativity
• Properties of the Homogeneous Cooling State of a Gas of Inelastic Rough Particles