Hydraulic Theory for a Frictional Debris Flow on a Collisional Shear Layer
We consider a heap of grains that is driven by gravity down an incline. We assume that the heap is deforming by frictional shearing while supported at its base by a relatively thin region of intense shear in which grains interact through collisions. We describe the frictional shearing using the Mohr-Coulomb yield condition and distinguish between active and passive states when relating the normal stress in the flow direction to the weight of the material. We use the balance laws, constitutive relations, and boundary conditions of the kinetic theory for dense granular flows to describe the region of colliding grains at the base. We determine the relationship between the shear stress, normal stress, and relative velocity of the boundaries in this shear layer using an analysis of a steady shearing flow between identical bumpy boundaries. This relationship permits us to close the hydraulic equations governing the evolution of the shape of the heap and the depth-averaged velocity. We integrate the resulting equations numerically for values of the parameters typical for glass or acrylic spheres.
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