Effective boundary condition for Stokes flow over a very rough surface

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 ε > 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.