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On the steady motion of a Navier-Stokes flow across a sieve with prescribed pressure drop in a finite pipe

Gianmarco Sperone

Abstract

The steady motion of a viscous incompressible fluid through a sieve (that is, a wall perforated with a large number of small holes), in a pipe of finite length, is modeled through the Navier-Stokes equations under mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while the pressure drop is prescribed along the pipe. Applying the classical energy method in homogenization theory, we study the asymptotic behavior of the solutions to this system, without any restriction on the magnitude of the data, as the diameters of the perforations vanish. Regardless of the initial scaling and distribution of the holes, we show that the sieve asymptotically becomes a wall, meaning that the effective equations are two, independent, stationary Navier-Stokes systems with a no-slip boundary condition on the wall. In the absence of external forces we prove, furthermore, that the fluid motion becomes quiescent in the homogenization limit.

On the steady motion of a Navier-Stokes flow across a sieve with prescribed pressure drop in a finite pipe

Abstract

The steady motion of a viscous incompressible fluid through a sieve (that is, a wall perforated with a large number of small holes), in a pipe of finite length, is modeled through the Navier-Stokes equations under mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while the pressure drop is prescribed along the pipe. Applying the classical energy method in homogenization theory, we study the asymptotic behavior of the solutions to this system, without any restriction on the magnitude of the data, as the diameters of the perforations vanish. Regardless of the initial scaling and distribution of the holes, we show that the sieve asymptotically becomes a wall, meaning that the effective equations are two, independent, stationary Navier-Stokes systems with a no-slip boundary condition on the wall. In the absence of external forces we prove, furthermore, that the fluid motion becomes quiescent in the homogenization limit.

Paper Structure

This paper contains 4 sections, 10 theorems, 141 equations, 3 figures.

Table of Contents

  1. Introduction and presentation of the problem
  2. Solvability of the boundary-value problem at the $\varepsilon$-level
  3. Boundary-value problem at the $\varepsilon$-level: uniform bounds
  4. Asymptotic behavior as $\varepsilon \to 0^{+}$: homogenized equations

Key Result

Theorem 1.1

Let $(\Omega_{\varepsilon})_{\varepsilon \in I_{*}}$ be the family of domains perfordomain and $\alpha >0$. Given any $p^{\pm} \in \mathbb{R}$ and $f \in L^{2}(\Omega)$, let $(u_{\varepsilon},\Phi_{\varepsilon}) \in H^{1}(\Omega_{\varepsilon}) \times L^{2}(\Omega_{\varepsilon})$ be a weak solution o and

Figures (3)

  • Figure 1.1: Representation of an aperture domain in the sense of Heywood.
  • Figure 1.2: Representation of the fluid domain $\Omega_{\varepsilon}$ and the sieve $\Gamma_{\varepsilon}$.
  • Figure 4.1: Representation of an admissible, smooth and distorted pipe of finite length.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 16 more