The influence of friction on the stability of unbounded granular shear flow

Abstract

Some recent studies have considered the stability of unbounded rapid granular shear flow, with the sole mechanism for stress generation being instantaneous inelastic collisions between grains. This paper extends these studies by presenting a linear stability analysis in which stress generation due to grain friction is also accounted for. This is accomplished by using the ‘frictional–kinetic’ model, which integrates in a simple manner the stress arising from the two mechanisms. Solution of the linearized equations of motion is obtained by allowing the wavenumber vector of the disturbances to rotate as a function of time. As in the case of a purely kinetic stress, it is found that the flow is stable to non-layering disturbances. Disturbances in the form of layering modes may lead to instability, depending on the solids fraction and material parameters. Instability is absent altogether if the balance of fluctuational energy is not considered or if the material is assumed to be incompressible. Friction may stabilize or destabilize the flow, depending on the inelasticity of grain collisions and the effective roughness of the medium. When a purely frictional stress is considered, it is found that the system is always neutrally stable. Even if the flow is asymptotically stable, there may be significant transient growth of disturbances due to the non-normality of the associated linear operator. The initial transient growth rate, as well as the temporal maximum of transient growth is enhanced by friction.