Homogenization of some evolutionary non-Newtonian flows in porous media

In this paper, we consider the homogenization of evolutionary incompressible purely viscous non-Newtonian flows of Carreau-Yasuda type in porous media with small perforation parameter $0< \varepsilon \ll 1$, where the small holes are periodically distributed. Darcy's law is recovered in the homogenization limit. Applying Poincar\'e type inequality in porous media allows us to derive the uniform estimates on velocity field, of which the gradient is small of size $\varepsilon$ in $L^{2}$ space. This indicates the nonlinear part in the viscosity coefficient does not contribute in the limit and a linear model (Darcy's law) is obtained. The estimates of the pressure rely on a proper extension from the perforated domain to the homogeneous non-perforated domain. By integrating the equations in time variable such that each term in the resulting equations has certain continuity in time, we can establish the extension of the pressure by applying the dual formula with the restriction operator.