Pressure-driven flow in a thin pipe with rough boundary

Stationary incompressible Newtonian fluid flow governed by external force and external pressure is considered in a thin rough pipe. The transversal size of the pipe is assumed to be of the order epsilon, i.e., cross-sectional area is about epsilon(2), and the wavelength in longitudinal direction is modeled by a small parameter mu. Under general assumption epsilon, mu -> 0, the Poiseuille law is obtained. Depending on epsilon, mu-relation (epsilon << mu, epsilon/mu similar to constant, epsilon >> mu), different cell problems describing the local behavior of the fluid are deduced and analyzed. Error estimates are presented.