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A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows. (English. Abridged French version) Zbl 1044.35056
Summary: We introduce a new model for shallow water flows with non-flat bottom. A prototype is the Saint Venant equation for rivers and coastal areas, which is valid for small slopes. An improved model, due to Savage–Hutter, is valid for small slope variations. We introduce a new model which relaxes all restrictions on the topography. Moreover it satisfies the properties (i) to provide an energy dissipation inequality, (ii) to be an exact hydrostatic solution of Euler equations. The difficulty we overcome here is the normal dependence of the velocity field, that we are able to establish exactly. Applications we have in mind concern, in particular, computational aspects of flows of granular material (for example in debris avalanches) where such models are especially relevant.

35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
74E20 Granularity
Full Text: DOI
[1] D. Bresch, B. Desjardins, On a viscous shallow water equations (Saint Venant model) and its quasi-geostrophic limit, Preprint · Zbl 1032.76012
[2] F. Bouchut, An introduction to finite volume methods for hyperbolic systems of conservation laws with source, École CEA-EDF-INRIA Écoulements peu profonds à surface libre, 7-10 octobre 2002, INRIA Rocquencourt, available at http://www.dma.ens.fr/ fbouchut/publications/fvcours.ps.gz
[3] Douady, S.; Andreotti, B.; Daerr, A., On granular surface flow equations, Eur. phys. J. B, 11, 131-142, (1999)
[4] Gerbeau, J.-F.; Perthame, B., Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Dcds(b), 1, 1, (2001) · Zbl 0997.76023
[5] Ghil, M.; Vautard, R., Nonlinear climate theory, (2000), Cambridge University Press Cambridge, London · Zbl 0709.62628
[6] Gray, J.M.N.T.; Wieland, M.; Hutter, K., Gravity driven free surface flow of granular avalanches over complex basal topography, Proc. roy soc. London ser. A, 455, 1841-1874, (1999) · Zbl 0951.76091
[7] Gwiazda, P., An existence result for a model of granular material with non-constant density, Asymptotic anal., 30, 43-60, (2002) · Zbl 1054.76082
[8] P. Hild, I.R. Ionescu, T. Lachand-Robert, I. Rosca, The blocking of an inhomogeneous Bingham fluid. Applications to landslides, Math. Modelling Numer. Anal., to appear · Zbl 1057.76004
[9] Lions, P.-L., Mathematical topics in fluid mechanics, incompressible models, Oxford lecture series in math. appl., 1, (1996), Oxford University Press
[10] Lions, P.-L.; Perthame, B.; Souganidis, P.E., Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and Lagrangian coordinates, Comm. pure appl. math., 49, 599-638, (1996) · Zbl 0853.76077
[11] A. Mangeney, J.-P. Vilotte, M.-O. Bristeau, B. Perthame, C. Simeoni, S. Yerneni, Numerical modeling of avalanches based on Saint Venant equations using a kinetic scheme, Preprint, 2002
[12] de Saint-Venant, A.J.C., Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit, C. R. acad. sci. Paris, 73, 147-154, (1871) · JFM 03.0482.04
[13] Savage, S.B.; Hutter, K., The dynamics of avalanches of granular materials from initiation to run-out, Acta mech., 86, 201-223, (1991) · Zbl 0732.73053
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