Effect of rugosity on a flow governed by stationary Navier-Stokes equations

We study the stationary flow of a fluid occupying a 3D infinite horizontal domain bounded by a rough wall that is at rest and by a plane that moves with a constant velocity. The rough wall is a plane covered with periodically distributed asperities of size ε ε . We prove that, outside a neighbourhood of the rough region, the flow behaves asymptotically as a Couette flow, as ε → 0 ε → 0 , up to an exponentially small error.