An analytic approach for the evolution of the static-flowing interface in viscoplastic granular flows
Résumé
Observed avalanche flows of dense granular material have the property to present two possible behaviours: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive analytically a set of equations that is able to manage the dynamics of such interface, in the so-called shallow regime where the flow is supposed to be thin compared to its downslope extension. It is obtained via an asymptotics starting from an incompressible viscoplastic model with Drucker-Prager yield stress, in which we have to make several assumptions. Additionally to the classical ones that are that the curvature of the topography, the width of the layer, and the viscosity are small, we assume that the internal friction angle is close to the slope angle, the velocity is small, and the pressure is convex with respect to the normal variable. This last assumption is for the stability of the double layer static-flowing configuration. The resulting model takes the form of a formally overdetermined initial-boundary problem in the variable normal to the topography. It handles arbitrary velocity profiles, and is therefore more general than depth-averaged models. It includes a new non-hydrostatic nonlinear coupling term. It has the property to be numerically solvable, at least in the uncoupled case, with or without viscosity.
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