Substantiation of the Darcy law for a porous medium with condition of partial adhesion

A study is made of a stationary Stokes's system in a periodically perforated domain with boundary conditions of mixed type, which describes the motion of a viscous incompressible fluid in a porous medium in the presence of friction between the fluid and the walls of the pores. The relation between the leading terms of the asymptotic expansions with respect to $\varepsilon$ for the fluid velocity and the pressure is obtained, where $\varepsilon$ is the parameter characterizing the fineness of the porous structure.