Effective boundary condition at a rough surface starting from a slip condition

Anne-Laure Dalibard (1) , David Gérard-Varet (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IMJ - Institut de Mathématiques de Jussieu

We consider the homogenization of the Navier-Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength $\epsilon$. 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 $\epsilon = 0$ 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 previous works, in which the special case of a Dirichlet condition at the rough boundary was examined.

Mots clés: Wall laws; Rough boundaries; Homogenization; Ergodicity; Korn inequality

Domaines: Equations aux dérivées partielles [math.AP]

Anne-Laure Dalibard, David Gérard-Varet. Effective boundary condition at a rough surface starting from a slip condition. Journal of Differential Equations, 2011, 251 (12), pp.3450-3487. ⟨10.1016/j.jde.2011.07.017⟩. ⟨hal-00498555⟩