Derivation of a consistent shallow-flow model for viscoplastic fluids and application to the dam-break problem and flow stoppage

A consistent three-equation shallow-flow model is derived for Herschel–Bulkley fluids propagating down an inclined plane. A rigorous asymptotic method is used to incorporate the complex rheological properties of the material into the model. Difficulties arise from the coexistence inside the flow of a sheared layer, in which the material is largely above the yielding threshold, and a pseudoplug layer, in which the material is just on the verge of yielding. The derivation of the model is based on two steps. First, the flow variables are expanded up to the first order of accuracy in terms of flow aspect ratio, in both the sheared and the pseudoplug layers. A specific generalization of the tensorial constitutive law is proposed, allowing us to implement a regular perturbation method in the entire domain. Second, the mass, momentum and energy balance equations are formally averaged over the flow depth. This results in three coupled equations for the fluid depth, the average velocity and a third variable, the enstrophy, related to the internal shearing of the flow. The final model has the structure of a fully hyperbolic system with relaxation source terms, and is adapted to represent dry fronts and material stoppage. A linear stability analysis is performed, showing a stabilizing effect of plasticity in good agreement with experimental results. A simple well-balanced numerical scheme is proposed, and various flow configurations are simulated, including roll waves and dam-break problems. The relative influences of plasticity, shear-thinning and shearing in the pseudoplug are investigated.