Low-volume-fraction limit for polymer fluids

We study a stationary, purely viscous polymer flow through a porous medium modelled as a periodic array of cells consisted of a fluid part and a solid one. Solid parts of the domain present impermeable obstacles, whose impact on fluid flow may be seen as a slowing factor through averaged quantities such as the permeability function, obtained by the homogenization process. In that way, the influence of the microstructure is implemented in the homogenized equations through a kind of nonlinear Darcyʹs law. Our goal is to find more explicitly the dependence of the permeability function on the size η of the obstacle in the unit cell and the so-called low-volume-fraction limit. Main difficulties arise from the nonlinear character of the power-law viscosity and the apparent weak convergence of the solutions involved.