Flow of shear-thinning fluids through porous media

Pseudo-plastic fluids exhibit a non-linear stress-strain relationship which can provoke large, localized viscosity gradients. For the flow of such fluids in porous media the consequence is a strong variability of the effective permeability with porosity, angle of the macroscopic pressure gradient, and rheological parameters of the fluid. Such a variability is investigated on the basis of adjoint homogenization theory for a Carreau fluid in an idealized porous medium geometry, highlighting differences with respect to the Newtonian case. It is shown in particular that the more we depart from Newtonian conditions, the more the (often used) hypothesis of an effective viscosity in Darcy's law is a poor approximation, for the effective permeability tensor becomes strongly anisotropic.