Qualitative/quantitative homogenization of some non-Newtonian flows in perforated domains (2406.17406v2)

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 $\alpha \in (1, \frac 32)$. The Darcy's law is recovered in the limit, thus generalizing the results from [\url{https://doi.org/10.1016/0362-546X(94)00285-P}] and [\url{https://doi.org/10.48550/arXiv.2310.05121}] 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 [\url{https://doi.org/10.1051/cocv/2016016}]) to deduce the uniform estimates of the pressure directly. Moreover, quantitative convergence rates are given.