Qualitative/quantitative homogenization of some non-Newtonian flows in perforated domains

Abstract

In this paper, we consider the homogenization of stationary and evolutionary incompressible viscous non-Newtonian flows of Carreau-Yasuda type in domains perforated with a large number of periodically distributed small holes in $\mathbb{R}^{3}$, where the mutual distance between the holes is measured by a small parameter $\varepsilon>0$ and the size of the holes is $\varepsilon^{\alpha}$ with $\alpha \in (1, 3)$. The Darcy's law is recovered in the limit, thus generalizing the results from https://doi.org/10.1016/0362-546X(94)00285-P and [https://doi.org/10.1016/j.jde.2024.08.021] for $\alpha=1$. Instead of using their restriction operator to derive the estimates of the pressure extension by duality, we use the Bogovski\u{\i} type operator in perforated domains (constructed in [https://doi.org/10.1051/cocv/2016016]) to deduce the uniform estimates of the pressure directly. Moreover, quantitative convergence rates are given.