On the Asymptotic Limit of Flows Past a Ribbed Boundary

We consider a stationary Navier-Stokes flow in a bounded domain supplemented with the complete slip boundary conditions. Assuming the boundary of the domain is formed by a family of unidirectional asperities, whose amplitude as well as frequency is proportional to a small parameter E, we shall show that in the asymptotic limit the motion of the fluid is governed by the same system of the Navier-Stokes equations, however, the limit boundary conditions are different. Specifically, the resulting boundary conditions prevent the fluid from slipping in the direction of asperities, while the motion in the orthogonal direction is allowed without any constraint.