Steady inclined flows of granular-fluid mixtures

We extend a recent theory for steady, uniform, gravity-driven flow of a highly concentrated granular--fluid mixture over an erodible bed between frictional sidewalls. We first include angles of inclination greater than the angle of repose of the particles; then, we introduce a boundary condition for flow over a rigid bumpy bed. We compare the predictions of the resulting theory with the volume flow rates, depths, and angles of inclination measured in experiments on dry and variously saturated flows over rigid and erodible boundaries. Finally, we employ the resulting theory, with the assumption that the flow is shallow, to solve, in an approximate way, for the variation of height and average velocities along a steady, non-uniform, inclined flow of a granular--fluid mixture that moves over a rigid bumpy bed. The solutions exhibit features of the flow seen in the experiments - for example, a dry, bulbous snout in advance of the fluid, whose length increases with increasing number of the particles and that disappears with increasing velocity - for which satisfactory explanations were lacking.