A consistent three-equation shallow-flow model for Bingham fluids
Résumé
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 removing the assumption of alignment
between the yield-stress tensor and the strain-rate tensor, but keeping the traceless property and
the equality between the norm of the yield-stress tensor and the yield stress. 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
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