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

TL;DR

The paper develops a rigorous homogenization theory for stationary and evolutionary incompressible non-Newtonian flows with Carreau–Yasuda viscosity in 3D perforated domains, where hole size scales as $\varepsilon^\alpha$ and holes are periodically distributed. Under the regime $1<\alpha<3/2$ and $\lambda>\alpha$ (with $g_r$ nonzero allowing a broader class of viscosities), the velocity converges to Darcy flow with a positive definite permeability $M_0$, and the limit system is $\frac{1}{2}\eta_0 M_0 {\bf u} = {\bf f}-\nabla p$, ${\rm div}\,{\bf u}=0$, with appropriate boundary conditions. The authors combine Allaire-type oscillating test functions with a Bogovski\u{}ı operator in perforated domains to obtain uniform pressure estimates and to derive both qualitative convergence and quantitative rates via a relative-energy framework; the torus case yields explicit rates, while bounded domains require boundary-layer corrections. They also provide a detailed analysis of pressure convergence, decomposing the pressure into a convergent part and a residual that vanishes in $L^q$ spaces, and extend the results to the evolutionary NSE, ensuring the convergence to Darcy's law in time-dependent settings. The work broadens the scope of homogenization for non-Newtonian fluids beyond the traditional Carreau model, delivering precise convergence rates and highlighting the role of hole size exponents and inertia in the limiting behavior.

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 $\varepsilon^α$ with $α\in (1, 3)$. 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 $α=1$. Instead of using their restriction operator to derive the estimates of the pressure extension by duality, we use the Bogovskiĭ 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.

Theorems & Definitions (30)

• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.5
• Remark 2.6
• Proposition 2.7
• Lemma 2.8
• Theorem 3.1
• Remark 3.2