Stability analysis of unbounded uniform shear flows of dense, slightly inelastic spheres based on a frictional–kinetic theory

Abstract

Asymptotic and transient stability analyses of unbounded uniform shear flows of dense, slightly inelastic, spherical particles were carried out using a frictional–kinetic theory. This model proposed for describing dense flows is based on a critical state plasticity theory and a simplified kinetic theory. In this model, the bulk and shear viscosities, the “ thermal” conductivity, and the energy dissipation rate are proportional to a “mean pressure” which is composed of a quasistatic-frictional-contribution pressure considered for slow, plasticity deformations and a granular-kinetic-theory collisional-contribution pressure. We studied two-dimensional stability analyses of layering disturbances (i.e., the perturbations whose wave number vectors are aligned only in the gradient) as well as nonlayering disturbances (the wave number vectors have nonzero streamwise components). Although this model has a simpler framework, it predicted similar results to those obtained using a more elaborate frictional–kinetic model. For instance, nonlayering disturbances are asymptotically stable at large time; the maximum transient growth of disturbances increases as the solids fraction or the friction coefficient is increased; and transient growths of disturbances can be significant due to the non-normality of the system. However, the prediction of the asymptotic stability of layering disturbances may be questionable because the collisional-contribution terms of the present model were over-simplified.