Effective wall-laws for the Stokes equations over curved rough boundaries

We derive effective wall-laws for Stokes systems with inhomogeneous boundary conditions in three dimensional bounded domains with curved rough boundaries. No-slip boundary condition is given on the locally periodic rough boundary parts with characteristic roughness size $\epsilon$ and boundary data is assumed to be supported in the nonoscillatory smooth boundary. Based on the analysis of a boundary layer cell problem depending on geometry of the fictitious boundary and roughness shape, boundary layer approximations are constructed using orthogonal tangential vectors and normal vector on the fictitious boundary, which have $O(\epsilon^{3/2})$-order in $L^2$-norm and $O(\epsilon)$-order in energy-norm. Then, a Navier wall-law with error estimates of $O(\epsilon^{3/2})$-order in $L^2$-norm and $O(\epsilon)$-order in $W^{1,1}$-norm is obtained, which is proved to be irrespective of the choice of the orthogonal tangent vectors.