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

TL;DR

This paper studies the homogenization of non-Newtonian flows in perforated domains, deriving Darcy's law in the limit and providing quantitative convergence rates using advanced mathematical tools.

Contribution

It generalizes previous homogenization results to non-Newtonian flows with variable hole sizes and introduces a new approach using the Bogovski operator for pressure estimates.

Findings

Darcy's law is recovered in the homogenization limit.

Quantitative convergence rates are established.

The method improves pressure estimates 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.