Effective boundary condition at a rough surface starting from a slip condition |
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Authors: | Anne-Laure Dalibard David Gérard-Varet |
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Institution: | a DMA/CNRS, Ecole Normale Supérieure, 45 rue d?Ulm, 75005 Paris, France b IMJ, 175 rue du Chevaleret, 75013 Paris, France |
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Abstract: | We consider the homogenization of the Navier-Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength ?. It was shown recently that, for any non-degenerate roughness pattern, and for any reasonable condition imposed at the rough boundary, the homogenized boundary condition in the limit ε=0 is always no-slip. We give in this paper error estimates for this homogenized no-slip condition, and provide a more accurate effective boundary condition, of Navier type. Our result extends those obtained in Basson and Gérard-Varet (2008) 6] and Gerard-Varet and Masmoudi (2010) 13], in which the special case of a Dirichlet condition at the rough boundary was examined. |
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Keywords: | Wall laws Rough boundaries Homogenization Ergodicity Korn inequality |
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