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

TL;DR

This paper analyzes the steady flow of viscous incompressible fluids through a perforated sieve in a finite pipe, demonstrating that as perforations vanish, the flow behaves as if the sieve is a solid wall, with the fluid eventually becoming still.

Contribution

It establishes the asymptotic behavior of Navier-Stokes solutions in perforated domains without restrictions on data magnitude, showing the sieve acts as a wall in the homogenization limit.

Findings

The sieve asymptotically becomes a wall with no-slip boundary conditions.

Flow becomes quiescent in the absence of external forces in the limit.

Effective equations are two independent stationary Navier-Stokes systems.

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.