Instabilities in granular gas–solid flows

Abstract

A linear stability analysis of the hydrodynamic equations with respect to the homogeneous cooling state is performed to study the conditions for stability of a suspension of solid particles immersed in a viscous gas. The dissipation in such systems arises from two different sources: inelasticity in particle collisions and viscous friction dissipation due to the influence of the gas phase on the solid particles. The starting point is a suspension model based on the (inelastic) Enskog kinetic equation. The effect of the interstitial gas phase on the dynamics of grains is modeled though a viscous drag force. The study is carried out in two different steps. First, the transport coefficients of the system are obtained by solving the Enskog equation by means of the Chapman–Enskog method up to first order in spatial gradients. Explicit expressions for the Navier–Stokes transport coefficients are obtained in terms of the volume fraction, the coefficient of restitution and the friction coefficient characterizing the amplitude of the external force. Once the transport properties are known, then the corresponding linearized hydrodynamic equations are solved to get the dispersion relations. In contrast to previous studies (Garzó et al 2016 Phys. Rev. E 93 012905), the hydrodynamic modes are analytically obtained as functions of the parameter space of the system. For a d-dimensional system, as expected linear stability shows d − 1 transversal (shear) modes and a longitudinal ‘heat’ mode to be unstable with respect to long enough wavelength excitations. The results also show that the main effect of the gas phase is to decrease the value of the critical length Lc (beyond which the system becomes unstable) with respect to its value for a dry granular fluid. Comparison with direct numerical simulations for Lc shows a qualitative good agreement for conditions of practical interest.