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Article Dans Une Revue Journal of Differential Equations Année : 2013

Effective boundary condition for Stokes flow over a very rough surface

Résumé

The main purpose of this paper is to derive a wall law for a flow over a very rough surface. We consider a viscous incompressible fluid filling a 3-dimensional horizontal domain bounded at the bottom by a smooth wall and at the top by a very rough wall. The latter consists in a plane wall covered with periodically distributed asperities which size depends on a small parameter $\varepsilon>0$ and with a fixed height. We assume that the flow is governed by the stationary Stokes equations. Using asymptotic expansions and boundary layer correctors we construct and analyze an asymptotic approximation of order $\mathcal{O}(\varepsilon^{3/2-\gamma})$ ($\gamma>0$ being arbitrary small) in the $H^1$-norm for the velocity and in the $L^2$-norm for the pressure. We derive an effective boundary condition of Navier type, then expressing the boundary layer terms in terms of the homogenized solution and the solution of a cell problem we obtain an effective approximation in the whole domain of the flow.

Dates et versions

hal-00794978 , version 1 (26-02-2013)

Identifiants

Citer

Youcef Amirat, Olivier Bodart, Umberto de Maio, Antonio Gaudiello. Effective boundary condition for Stokes flow over a very rough surface. Journal of Differential Equations, 2013, 254 (8), pp.3395-3430. ⟨10.1016/j.jde.2013.01.024⟩. ⟨hal-00794978⟩
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