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 asymptoticmethod in the shallow-flow approximation. The variables are expanded up to the first order ofaccuracy both in the sheared and pseudo-plug layers. The divergence of the strain rate, whichis obtained in classical approaches, is here avoided by removing the assumption of alignmentbetween the yield-stress tensor and the strain-rate tensor, but keeping the traceless property andthe equality between the norm of the yield-stress tensor and the yield stress. The model is derivedby averaging the mass, momentum and energy balance equations over the depth. This yields ahyperbolic 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 sourceterms and admits an exact balance energy equation. The velocity field in the depth is consistentlyreconstructed using only the variables of the depth-averaged model without any derivative. Thephysical relevance of the enstrophy is related to the shape of the velocity profile. The linearstability of a uniform solution is investigated for this model, showing a stabilizing effect of theplasticity. Roll waves are simulated numerically using a classical Godunov’s scheme. The modelfor a Newtonian fluid is presented as a particular cas