Multi-Regime Shallow Free-Surface Flow Models for Quasi-Newtonian Fluids
Résumé
The mathematical modeling of thin free-surface flows for quasi-Newtonian fluids (power-law rheology: shear-thickening, shear-thinning and newtonian) is addressed. Asymptotic shallow flow models, one-equation and two-equations, consistent with any laminar viscous regimes, are derived formally. To do so, first reference flows, corresponding to different regimes, are considered. Second, the exact expressions of all fields $(\bsigma, \bu, p)$ are written as perturbations of the reference flows; all expressions are unified in the sense they are available for any viscous regimes. The asymptotic derivations are presented either in a mean slope coordinate system with local variations of the topography or in the Prandtl's coordinate system since it is shown to be equivalent. Then, all expressions and equations remain valid in presence of large topography variations. Formal error estimates proving the consistency of the derivations are stated. Next, one equation model (lubrication-type) on the flow thickness is derived at order $0$ and at order $1$. Then, two-equations models (shallow water type) are stated; also a gradually varied flow version is presented. All these models are unified in the sense that they remain valid for any viscous regimes, with any boundary conditions at bottom (from adherence to pure slip). The classical models from the literature are retrieved if considering the corresponding assumptions (generally a particular regime or a specific boundary condition at bottom). Preliminary 1D numerical examples illustrate the robustness of the unified one-equation model, in presence of two-regime flows (either due to a sharp change of the mean-slope topography or due to a sharp change of boundary condition at bottom).
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