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We consider the Navier-Stokes equation in a domain with a rough boundary. The roughness is modeled by a small amplitude and small wavelength oscillation, with typical scale $${\varepsilon \ll 1}$$. For periodic oscillation, it is well-known that the best homogenized (that is regular in $${\varepsilon}$$) boundary condition is of Navier type. Such result still holds for random stationary irregularities, as shown recently by the first author [5, 15]. We study here arbitrary irregularity patterns.

Relevance of the Slip Condition for Fluid Flows Near an Irregular Boundary