Laminar shallow viscoplastic fluid flowing through an array of vertical obstacles

A new Bingham-Darcy shallow depth approximation flow model is proposed in this paper. This model is suitable for a shallow viscoplastic fluid flowing on a general topography and crossing an array of vertical obstacles. An analogous porous medium is first introduced to reduce the array of obstacles. The reduction model is based on a continuum model similar to the Brinkman equations, where the usual Darcy model is extended for viscoplastic Bingham fluids. A specific asymptotic analysis of this Bingham-Darcy porous medium for the case of shallow depth flows allows us to produce a new reduced model. Some assumptions are needed for the reduction: laminar flow, small degrees of slope variation of the underlying topography and a yield stress that is small when compared with gravity effects. The resulting solution is a highly nonlinear parabolic equation in terms of the flow height only, and is efficiently solved by a Newton method, without any regularization. However, our numerical predictions compares well, both qualitatively and quantitatively with both experimental measurements and full tridimensional simulations. Finally, a new experiment for a viscoplastic flow over an inclined plane through a network of obstacles is proposed and numerical simulations are provided for future comparison with experiments.