On the roughness-induced effective boundary conditions for a viscous flow

We consider the laminar viscous channel flow, with the lateral surface of the channel containing surface irregularities. It is supposed that a uniform pressure gradient is maintained in the longitudinal direction in the channel. After studying the corresponding boundary layers, we obtain rigorously the Navier friction condition. It is valid when the size and the amplitude of the imperfections tend to zero. Furthermore, the coefficient in the law is determined through an auxiliary boundary-layer type problem and the tangential drag force and the effective mass flow are determined at order corresponding to the characteristic size of the impurities at the power 3/2.