Model for dense granular flows down bumpy inclines

We consider dense flows of spherical grains down an inclined plane on which spherical bumps have been affixed. We propose a theory that models stresses as the superposition of a rate-dependent contribution arising from collisional interactions and a rate-independent part related to enduring frictional contacts among the grains. We show that dense flows consist of three regions. The first is a thin basal layer where grains progressively gain fluctuation energy with increasing distance from the bottom boundary. The second is a core region where the solid volume fraction is constant and the production and dissipation of fluctuation energy are nearly balanced. The last is a thin collisional surface layer where the volume fraction abruptly vanishes as the free surface is approached. We also distinguish basal flows with the smallest possible height, in which the core and surface layers have disappeared. We derive simple closures of the governing equations for the three regions with insight from the numerical simulations of Silbert et al. [Phys. Rev. E64, 051302 (2001)] and the physical experiments of Pouliquen [Phys. Fluids 11, 542 (1999)]. The theory captures the range of inclination angles at which steady, fully developed flows are observed, the corresponding shape of the mean and fluctuation velocity profiles, the dependence of the flow rate on inclination, flow height, interparticle friction, and normal restitution coefficient, and the dependence of the height of basal flows on inclination.