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Homogenization of Leray's flux problem for the steady-state Navier-Stokes equations in a multiply-connected planar domain

Clara Patriarca, Gianmarco Sperone

TL;DR

This work analyzes the homogenization of Leray's flux problem for steady 2D incompressible Navier–Stokes flow in a multiply-connected perforated domain as the hole size vanishes. Using an ε-uniform energy framework, relative capacity, and a uniform solenoidal extension of the boundary data, the authors establish ε-independent bounds for velocity and pressure under symmetry or small boundary-flux assumptions. The main result shows that the homogenized limit solution in the intact domain solves the same Navier–Stokes system with boundary data $v_{*}$, i.e., the effective equations remain unchanged in the limit. The paper also discusses the limitations of extending these results to fully general 2D configurations, highlighting the role of pressure extensions and boundary-layer effects near the holes.

Abstract

The steady motion of a viscous incompressible fluid in a multiply-connected, planar, bounded domain (perforated with a large number of small holes) is modeled through the Navier-Stokes equations with non-homogeneous Dirichlet boundary data satisfying the general outflow condition. Under either a symmetry assumption on the data or under a smallness condition on each of the boundary fluxes (therefore, no constraints on the magnitude of the boundary velocity are imposed), we apply the classical energy method in homogenization theory and study the asymptotic behavior of the solutions to this system as the size of the perforations goes to zero: it is shown that the effective equations remain unmodified in the limit. The main novelty of the present work lies in the obtainment of the required uniform bounds, which are achieved by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.

Homogenization of Leray's flux problem for the steady-state Navier-Stokes equations in a multiply-connected planar domain

TL;DR

This work analyzes the homogenization of Leray's flux problem for steady 2D incompressible Navier–Stokes flow in a multiply-connected perforated domain as the hole size vanishes. Using an ε-uniform energy framework, relative capacity, and a uniform solenoidal extension of the boundary data, the authors establish ε-independent bounds for velocity and pressure under symmetry or small boundary-flux assumptions. The main result shows that the homogenized limit solution in the intact domain solves the same Navier–Stokes system with boundary data , i.e., the effective equations remain unchanged in the limit. The paper also discusses the limitations of extending these results to fully general 2D configurations, highlighting the role of pressure extensions and boundary-layer effects near the holes.

Abstract

The steady motion of a viscous incompressible fluid in a multiply-connected, planar, bounded domain (perforated with a large number of small holes) is modeled through the Navier-Stokes equations with non-homogeneous Dirichlet boundary data satisfying the general outflow condition. Under either a symmetry assumption on the data or under a smallness condition on each of the boundary fluxes (therefore, no constraints on the magnitude of the boundary velocity are imposed), we apply the classical energy method in homogenization theory and study the asymptotic behavior of the solutions to this system as the size of the perforations goes to zero: it is shown that the effective equations remain unmodified in the limit. The main novelty of the present work lies in the obtainment of the required uniform bounds, which are achieved by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.
Paper Structure (6 sections, 7 theorems, 134 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 134 equations, 2 figures.

Table of Contents

  1. Introduction and presentation of the problem
  2. Symmetric domain with periodic perforations
  3. Boundary-value problem at the $\varepsilon$-level: uniform bounds
  4. $\varepsilon$-Uniform bounds in the symmetric case
  5. Asymptotic behavior as $\varepsilon \to 0^{+}$: homogenized equations
  6. Remarks on the fully general two-dimensional case

Key Result

Lemma 2.1

Let $\Omega_{\varepsilon}$ be as in perfordomain and $\phi_{\varepsilon} \in H_{0}^{1}(\Omega)$ be the function satisfying cap1. Then, $\phi_{\varepsilon} \in H^{2}(\Omega_{\varepsilon})$ and the following estimates hold for some constant $C_{*} > 0$ that depends on $\Omega$ and $\{ \delta_{0}, \delta_{1} \}$, but is independent of $\varepsilon \in I_{*}$.

Figures (2)

  • Figure 1.1: Representation of the domain $\Omega$ (left) and the perforated domain $\Omega_{\varepsilon}$ (right).
  • Figure 1.2: The symmetric domain $\Omega$ (left) and the symmetric perforated domain $\Omega_{\varepsilon}$ (right).

Theorems & Definitions (19)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.1
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • ...and 9 more