Sharp pressure estimates for the Navier-Stokes system in thin porous media

TL;DR

The paper analyzes Newtonian fluid flow in thin, porous media with periodic perforations, introducing a Reynolds-number scaling $Re^\varepsilon=\varepsilon^{-\gamma}$ to determine when Darcy laws remain valid across three geometric regimes. Using vertical dilation, pressure extension, and an unfolded-function framework, the authors derive sharp pressure estimates that control inertial effects and establish regime-specific 2D Darcy limits with permeability tensors $\mathcal{K}_H$, $\mathcal{K}_P$, and $\mathcal{K}_V$ obtained from local Stokes or Hele‑Shaw problems. The results identify critical Reynolds numbers $Re_c=\varepsilon^{-\gamma_c}$ for each TPM type, below which Darcy laws hold and above which inertia matters, offering rigorous justification for reduced models in thin heterogeneous domains. These findings provide a theoretical basis for accurate, computationally efficient modeling of Newtonian flows in thin porous structures and guidance for developing numerical methods that respect regime-dependent inertia effects.

Abstract

A relevant problem for applications is to model the behavior of Newtonian fluids through thin porous media, which is a domain with small thickness $ε$ and perforated by periodically distributed cylinders of size and period $ε^δ$, with $δ>0$. Depending on the relation between thickness and the size of the cylinders, it was introduced in (Fabricius et al., Transp. Porous Media, 115, 473-493, 2016), (Anguiano and Suárez-Grau, Z. Angew. Math. Phys., 68:45, 2017) and (Anguiano and Suárez-Grau, Mediterr. J. Math., 15:45, 2018) that there exist three regimes depending on the value of $δ$: $δ\in (0,1)$, $δ=1$ and $δ>1$. In each regime, the asymptotic behavior of the fluid is governed by a lower-dimensional Darcy's law. In previous studies, the Reynolds number is considered to be of order one and so, the question that arises is for what range of values of the Reynolds number the lower-dimensional Darcy laws are still valid in each regime, which represents the main the goal of this paper. In this sense, considering a fluid governed by the Navier-Stokes system and assuming the Reynolds number written in terms of the thickness $ε$, we prove that, for each regime, there exists a critical Reynolds number $Re_c$ such that for every Reynolds number $Re$ with order smaller or equal than $Re_c$, the lower-dimensional Darcy law is still valid. On the contrary, for Reynolds numbers $Re$ greater than $Re_c$, the inertial term of the Navier-Stokes system has to be taken into account in the asymptotic behavior and so, the Darcy law is not valid.

Figures (2)

• Figure 1: View of HTPM ($\delta>1$) (left) and PTPM ($\delta=1$) (right)
• Figure 2: View of VTPM ($0<\delta<1$)

Theorems & Definitions (26)

• Theorem 3.1: Sharp pressure estimates
• Theorem 4.1: Darcy's law for HTPM
• Theorem 4.2: Darcy's law for PTPM
• Theorem 4.3: Darcy's law for VTPM
• Lemma 5.1: Poincaré's inequality
• Lemma 5.2: Velocity estimates
• Remark 5.3: Extension of velocity to $\Omega$
• Lemma 5.4: Lemma 4.5-(i) in Anguiano_SuarezGrau
• Remark 5.5
• Lemma 5.6