A consistent three-equation shallow-flow model for Bingham fluids

We derive a model for Bingham fluid flows down an inclined plane with a consistent asymptotic method in the shallow-flow approximation. The variables are expanded up to the first order of accuracy both in the sheared and pseudo-plug layers. The divergence of the strain rate, which is obtained in classical approaches, is here avoided by a specific regularization of the rheology allowing to implement a regular perturbation method in the whole fluid domain. Unlike classical regularization methods, the material is here characterized by a true yield stress. Below the yield point, the behavior is perfectly rigid. An alternative tensor expression of the constitutive law is proposed. In particular, the assumption of an alignment between the yield-stress tensor and the strain-rate tensor is removed. The model is derived by averaging the mass, momentum and energy balance equations over the depth. This yields a hyperbolic model of three equations for the fluid depth, the average velocity and a third variable, called enstrophy, related to the variance of the velocity. The model features new relaxation source terms and admits an exact balance energy equation. The velocity field in the depth is consistently reconstructed using only the variables of the depth-averaged model without any derivative. The physical relevance of the enstrophy is related to the shape of the velocity profile. The linear stability of a uniform solution is investigated for this model, showing a stabilizing effect of the plasticity. Roll waves are simulated numerically using a classical Godunov's scheme. The model for a Newtonian fluid is presented as a particular case.