Effective slip boundary conditions for arbitrary one-dimensional surfaces

In many applications it is advantageous to construct effective slip boundary conditions, which could fully characterize flow over patterned surfaces. Here we focus on laminar shear flows over smooth anisotropic surfaces with arbitrary scalar slip $b(y)$, varying in only one direction. We derive general expressions for eigenvalues of the effective slip-length tensor, and show that the transverse component is equal to a half of the longitudinal one with twice larger local slip, $2b(y)$. A remarkable corollary of this relation is that the flow along any direction of the 1D surface can be easily determined, once the longitudinal component of the effective slip tensor is found from the known spatially nonuniform scalar slip.