Stability analysis of unbounded uniform dense granular shear flow based on a viscoplastic constitutive law

Abstract

Asymptotic and transient stability analyses of unbounded uniform granular shear flow at high solids volume fractions were carried out in the paper, based on a model composed of the viscoplastic constitutive law [P. Jop, Y. Forterre, and O. Pouliquen, Nature (London) 441, 727 (2006)] and the dilatancy law [O. Pouliquen et al., J. Stat. Mech.: Theory Exp. (2006) P07020]. We refer to this model as the VPDL (meaning of the “viscoplastic and dilatancy laws”) thereinafter. In this model, dense granular flows were treated as a viscoplastic fluid with a Drucker–Prager-like yielding criterion. We compared our results to those obtained using the frictional-kinetic model (FKM) [M. Alam and P. R. Nott, J. Fluid Mech. 343, 267 (1997)]. Our main result is that unbounded uniform dense granular shear flows are always asymptotically stable at large time based on the VPDL model, at least for two-dimensional perturbations. This is valid for disturbances of layering modes (i.e., the perturbations whose wavenumber vectors are aligned along the transverse coordinate) as well as for nonlayering modes (the streamwise component of the wavenumber vector is nonzero). By contrast, layering modes can be unstable based on the FKM constitutive laws. Interestingly, in the framework of the VPDL, the analysis shows that significant transient growth may occur owing to the non-normality of the linear system, although disturbances eventually decay at large time.