Transient and asymptotic stability of granular shear flow

Abstract

The linear stability of granular material in an unbounded uniform shear flow is considered. Linearized equations of motion derived from kinetic theories are used to arrive at a linear initial-value problem for the perturbation quantities. Two cases are investigated: (a) wavelike disturbances with time constant wavenumber vector, and (b) disturbances that will change their wave structure in time owing to a shear-induced tilting of the wavenumber vector. In both cases, the stability analysis is based on the solution operator whose norm represents the maximum possible amplification of initial perturbations. Significant transient growth is observed which has its origin in the non-normality of the involved linear operator. For case (a), regions of asymptotic instability are found in the two-dimensional wavenumber plane, whereas case (b) is found to be asymptotically stable for all physically meaningful parameter combinations. Transient linear stability phenomena may provide a viable and fast mechanism to trigger finite-amplitude effects, and therefore constitute an important part of pattern formation in rapid particulate flows.