Numerical approximation of laminar flows over rough walls with sharp asperities

We consider a viscous incompressible fluid filling an infinite horizontal domain bounded at the bottom by a plane wall and at the top by a rough wall. The latter is assumed to consist of a plane wall covered with periodically distributed asperities whose size depends on a small parameter ε>0. The assumption of sharp asperities is made, that is the height of the asperities does not vanish as ε→0. The rough wall is at rest and the plane wall is moving at a constant horizontal velocity. We assume that the flow is governed by the stationary Stokes equations. We give an asymptotic approximation of the Stokes flow, under the rugose region, by a Couette flow, depending on the size ε of the asperities. We then build a numerical approximation of the Couette flow based on a partial domain decomposition method.