Stability of freely cooling granular mixtures at moderate densities

TL;DR

This paper analyzes the stability of freely cooling granular mixtures at moderate densities, deriving an explicit critical length scale for instability onset that accounts for nonlinear dissipation effects, extending previous dilute-limit results.

Contribution

It provides a comprehensive stability analysis of granular mixtures at moderate densities, including nonlinear dissipation effects, and derives an explicit expression for the critical system size for instability.

Findings

Transversal velocity modes become unstable when system size exceeds critical length

Explicit formula for critical length scale $L_c$ in terms of mixture parameters

Quantitative discrepancies with previous low-dissipation theoretical models

Abstract

The formation of velocity vortices and density clusters is an intriguing phenomenon of freely cooling granular flows. In this work, the critical length scale $L_c$ for the onset of instability is determined via stability analysis of the linearized Navier-Stokes hydrodynamic equations of $d$-dimensional granular binary mixtures at moderate densities. In contrast to previous attempts, the analysis is not restricted to nearly elastic systems since it takes into account the nonlinear dependence of the transport coefficients and the cooling rate on the collisional dissipation. As expected from previous results obtained in the very dilute regime, linear stability shows $d-1$ transversal (shear) modes and a longitudinal ("heat") mode to be unstable with respect to long enough wavelength excitations. The theoretical predictions also show that the origin of the instability is driven by the transversal component of the velocity field that becomes \emph{unstable} when the system length $L>L_c$. An explicit expression of $L_c$ is obtained in terms of the masses and diameters of the mixture, the composition, the volume fraction and the coefficients of restitution. Previous results derived in the limit of both mechanically equivalent particles and low-density mixtures are consistently recovered. Finally, a comparison with previous theoretical works which neglect the influence of dissipation on the transport coefficients shows quantitative discrepancies for strong dissipation.