Instability of unbounded uniform granular shear flow

Abstract

The paper presents a linear stability analysis of cohesionless granular materials during rapid shear flow. The analysis is based on the governing equations developed in the kinetic theory of Lun et al. (1984) for granular flows of smooth, nearly elastic, uniform spherical particles. The primary flow is taken to be a uniform, simple shear flow and the effects of small perturbations in velocity components, granular temperature and solids fraction are considered. The inelasticity of the particles is characterized by a constant coefficient of restitution which is assumed to be close to unity. Some permissible solutions are sinusoidal plane waves in which the wavenumber vector is continuously turned by the mean shear flow and its magnitude varied as time proceeds. The initial growth (or decay) rates for these perturbations are sought. The resulting linearized equations for the flow perturbations turn out to be exceedingly long and complex; they are determined by the use of computer algebra. It is found that, in general, long wavelengths are the most unstable and that short wavelengths are dampened by ‘viscous action’. ‘Instability’ increases with decreasing coefficient of restitution. Numerical results for initial growth rates were obtained for several values of mean solids fraction and particle coefficient of restitution. Flows tend to be more stable at both high and very low concentrations than at moderate concentrations. These results appear to be consistent in the main with recent computer simulations of granular flows of disk-like particles by Hopkins & Louge (1991).