Multi-Regime Shallow Free-Surface Flow Models for Quasi-Newtonian Fluids

International audience ; The mathematical modeling of thin free-surface laminar flows for quasi-Newtonian fluids (power-law rheology) is addressed with a particular attention to geophysical flows (e.g. ice or lava flows). Asymptotic thin-layer flow models (one-equation and two-equation models) consistent with various viscous regimes, corresponding to different basal boundary conditions (from adherence to pure slip), are derived. The challenge being to derive models consistent from slip to no-slip basal boundary condition, though at the price of balancing small friction by small mean slope. Starting from reference flows (the steady-state uniform ones) corresponding to different shear regimes, the exact expressions of all fields $(\bsigma, \bu, p)$ are calculated formally by a perturbation expansion method.The calculations are such that all field expressions remain valid for any laminar viscous regimes. The calculations are presented either in a mean slope coordinatesystem with local variations of the topography or in the Prandtl coordinate system, hence valid in presence of any non flat basal topography.Formal error estimates proving the consistency of the derivations are stated. An unified one-equation model (lubrication type in the depth variable $h$) is derived at order $1$. Next, few unified two-equation models in variable $(q,h)$ (shallow water type) are stated and discussed.The classical first order models from the literature are recovered if considering the corresponding particular cases (generally, flat bottom with a particular regime and/or specific basal boundary condition). Two one-dimensional numerical examples illustrate the robustness of these new multi-regime formulations (the change of flow regimes being due either to a sharp change of the mean-slope topography or to a sharp change of basal boundary condition).