Viscoplastic shallow flow equations with topography

The shallow flow of a viscoplastic fluid over a general basal topography is investigated. The plasticity (flow/no flow) criteria of the constitutive law may include Von-Mises (Bingham fluid) or Drucker–Prager (Mohr–Coulomb) models. Coulomb frictional conditions on the bottom are considered. Assuming that the shear stresses are small with respect to the stresses associated to the tangent plane, an asymptotic analysis is developed for small thickness aspect ratio. A Saint-Venant model and a new depth integrated theory is presented. The resulting shallow flow equations have the same structure as the three dimensional ones. The 2-D (tangent) momentum balance law and the thickness evolution equation are closed with a “shallow constitutive equation” which links the averaged stresses to the rate of deformations in the tangent plane. The shallow flow/no flow (yield) condition and the shallow viscosity are not the same as in the three dimensional case but the constitutive law has the same structure. The curvature of the bottom surface is included in the model in the expression of the differential operators as well as in the frictional terms.To illustrate the capabilities of the shallow model to reproduce the flow, the sheet flow is analyzed. Two comparisons between the (2-D) in-plane channel flow and the asymptotic (1-D) flow for the Drucker–Prager fluids are considered. One comparison involves the experimental data and the other one includes ALE (Arbitrary Lagrangian–Eulerian) computations.A couple of boundary value problems, modeling shallow dense avalanches, for different viscoplastic laws are selected to illustrate the predictive capabilities of the model: spreading a Drucker–Prager dome on a talweg and the role of barriers in stopping a viscoplastic avalanche.

► A new shallow flow model for visco-plastic fluids with cohesion was deduced.
► The deduced model works for a general basal topography.
► The model points out a “shallow constitutive equation” of the visco-plastic fluid.