Derivation and numerical resolution of 2D Shallow Water equations for multi-regime flows of Herschel-Bulkley fluids
Résumé
This paper presents mathematical modelling and simulation of thin free-surface flows of viscoplastic fluids with a Herschel–Bulkley rheology over complex topographies. Using the asymptotic expansion method, depth-averaged models (lubrication and shallow water type models) are derived for three-dimensional multi-regime flows on non-flat inclined topographies with varying basal conditions. Starting from the Navier–Stokes equations, two flow regimes corresponding to different balances between shear and pressure forces are presented. Flow models corresponding to these regimes are calculated as perturbations of the zeroth-order solutions. The classical reference models in the literature are recovered by considering their respective cases on a flat-inclined surface. In the second regime case, a pressure term is non-negligible. Mathematically, it leads to a corrective term to the classical regime equations. Flow solutions of the two regimes are compared; the difference appears in particular in the vicinity of sharp changes of slopes. Nonetheless, both regime models are compared with experiments and are found to be in good agreement.
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