On the Roughness-Induced Effective Boundary Conditions for an Incompressible 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 of the channel. After studying the corresponding boundary layers, we obtain rigorously the Navier friction condition. It is valid when the size and 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 up to order O( 3/2). The value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of O( ).