[Paper Review] Impermeability through a perforated domain for the incompressible 2D Euler equations

Christophe Lacave, Nader Masmoudi|arXiv (Cornell University)|Jul 10, 2014

Advanced Mathematical Modeling in Engineering31 references3 citations

TL;DR

This paper analyzes the asymptotic behavior of the 2D incompressible Euler equations in a domain perforated by small inclusions of size $\varepsilon$, with spacing $d_\varepsilon$. The key result shows that the fluid's limit behavior depends critically on the ratio $d_\varepsilon / \varepsilon$ (or $d_\varepsilon / \varepsilon^{2+1/\gamma}$ in 1D distributions): if this ratio tends to infinity, the fluid behaves as in the full plane; if it tends to zero, the porous medium acts as an impermeable wall, and the limit solution satisfies the Euler equations outside a solid obstacle.

ABSTRACT

We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $\varepsilon$ separated by distances $d_\varepsilon$ and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of $\frac{d_{\varepsilon}}\varepsilon$ when $\varepsilon$ goes to zero. If $\frac{d_{\varepsilon}}\varepsilon o \infty$, then the limit motion is not perturbed by the porous medium, namely we recover the Euler solution in the whole space. On the contrary, if $\frac{d_{\varepsilon}}\varepsilon o 0$, then the fluid cannot penetrate the porous region, namely the limit velocity verifies the Euler equations in the exterior of an impermeable square. If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1γ}}$ where $γ\in (0,\infty]$ is related to the geometry of the lateral boundaries of the obstacles. If $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1γ}} o \infty$, then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions. In particular it is equal to $\varepsilon^3$ for balls.

Motivation & Objective

• To understand the asymptotic behavior of the 2D incompressible Euler equations in a domain perforated by small inclusions.
• To determine under what conditions the porous medium becomes impermeable to fluid flow in the limit as the inclusion size $\varepsilon \to 0$.
• To characterize the transition between the fluid behaving as if the domain were unperforated and the fluid being blocked by an effective solid boundary.
• To identify the critical scaling of the distance between inclusions $d_\varepsilon$ relative to $\varepsilon$ that determines whether the porous medium influences the limit solution.
• To analyze how the geometry of the inclusions (especially lateral boundaries) affects the critical scaling in one-dimensional distributions.

Proposed method

• Model the fluid motion using the incompressible 2D Euler equations in a domain exterior to small inclusions of size $\varepsilon$.
• Consider two configurations: inclusions distributed on the unit square (2D) and on the unit segment (1D), with spacing $d_\varepsilon$.
• Analyze the limit as $\varepsilon \to 0$ by studying the behavior of the ratio $d_\varepsilon / \varepsilon$ in 2D and $d_\varepsilon / \varepsilon^{2+1/\gamma}$ in 1D, where $\gamma$ encodes the geometry of the obstacle boundaries.
• Use compactness arguments and weak convergence techniques to pass to the limit in the Euler equations on the perforated domain.
• Establish that the limit velocity field satisfies the Euler equations in the exterior of a solid obstacle when the critical ratio tends to zero.
• Prove that when the critical ratio tends to infinity, the limit solution coincides with the Euler solution in the full space, indicating no influence from the porous medium.

Experimental results

Research questions

• RQ1 Under what conditions does the presence of a porous medium in a 2D domain become impermeable to the fluid in the limit as $\varepsilon \to 0$?
• RQ2 How does the scaling of the distance between inclusions $d_\varepsilon$ relative to $\varepsilon$ determine whether the fluid can penetrate the porous region?
• RQ3 What role does the geometry of the inclusions—specifically the lateral boundary shape—play in determining the critical scaling for impermeability in one-dimensional distributions?
• RQ4 Can the limit solution be identified as the Euler solution in the full space, or does it satisfy a modified equation with a solid boundary?
• RQ5 What is the precise threshold scaling for $d_\varepsilon$ that separates the regime where the porous medium is effectively invisible from the regime where it acts as an impermeable wall?

Key findings

• When $d_\varepsilon / \varepsilon \to \infty$ in the 2D square configuration, the limit solution coincides with the Euler solution in the full plane, indicating no influence from the porous medium.
• When $d_\varepsilon / \varepsilon \to 0$ in the 2D square configuration, the fluid cannot penetrate the porous region, and the limit velocity satisfies the Euler equations in the exterior of an impermeable square.
• In the 1D segment configuration, the critical scaling is $d_\varepsilon / \varepsilon^{2+1\gamma} \to \infty$ for the porous medium to be invisible; if this ratio tends to zero, an impermeable wall emerges.
• For spherical inclusions in 1D, the critical scaling is $\varepsilon^3$, meaning impermeability occurs when $d_\varepsilon \ll \varepsilon^3$.
• The transition between regimes depends sensitively on the geometry of the obstacle boundaries, encoded by the parameter $\gamma$.
• The limit behavior is characterized by weak convergence of the velocity field and compactness arguments, leading to the identification of the limit as a solution to the Euler equations with a solid boundary in the impermeable regime.