Dense, layered, inclined flows of spheres

We consider dense, inclined flows of spheres in which the particles translate in layers, whose existence may be promoted by the presence of a rigid base and/or sidewalls. We imagine that in such flows a sphere of a layer is forced up the back of a sphere of the layer below, lifting a column of spheres above it, and then falls down the front of the lower sphere, until it bumps against the preceding sphere of the lower layer. We calculate the forces and rate of momentum transfer associated with this process of rub, lift, fall, and bump and determine a relation between the ratio of shear stress to normal stress and the rate of strain that may be integrated to obtain the velocity profile. The fall of a sphere and that of the column above it results in a linear increase in the magnitude of the velocity fluctuations with distance from the base of the flow. We compare the predictions of the model with measured profiles of velocity and granular temperature in several different dense, inclined flows.