Effective slip law for general viscous flows over an oscillating surface

摘要

We consider the non-stationary three-dimensional viscous flow in a bounded domain, with the lateral surface containing microscopic surface irregularities. Under the assumption of a smooth flow in the domain without roughness, we prove that there is a smooth solution to a problem with the rough boundary. In the papers by J?ger and Mikelic?, the friction law was obtained as a perturbation of the Poiseuille flows. Here, the situation is more complicated. Nevertheless, after studying the corresponding boundary layers and using the results on solenoidal vector fields in domains with rough boundaries, we obtain rigorously the Navier friction condition. It is valid when the size and amplitude of the imperfections tend to zero. Furthermore, the friction matrix in the law is determined through a family of auxiliary boundary-layer type problems. Effective equations approximate velocity at order O (ε) in the H1-norm, uniformly in time, and O(ε~(3/2)) in the L ~2-norm, also uniformly in time. Approximation for the pressure is O(ε~(3/2)) in the Lloc2-norm.