Approximation of time-periodic flow past a translating body by flows in bounded domains

TL;DR

This paper studies time-periodic flow around a moving body, proving existence and uniqueness of solutions in unbounded and bounded domains, and analyzing how well truncated domain solutions approximate the exterior flow as the boundary expands.

Contribution

It establishes the existence and uniqueness of solutions for time-periodic Navier-Stokes flow in exterior and truncated domains, and quantifies the approximation error as the boundary radius increases.

Findings

Existence and uniqueness of strong solutions in exterior domains.

Existence and uniqueness of weak solutions in truncated domains.

Error estimates showing convergence of truncated solutions to exterior flow as R increases.

Abstract

We consider a time-periodic incompressible three-dimensional Navier-Stokes flow past a translating rigid body. In the first part of the paper, we establish the existence and uniqueness of strong solutions in the exterior domain $\Omega \subset {\mathbb R}^3$ that satisfy pointwise estimates for both the velocity and pressure. The fundamental solution of the time-periodic Oseen equations plays a central role in obtaining these estimates. The second part focuses on approximating this exterior flow within truncated domains $\Omega \cap B_R$, incorporating appropriate artificial boundary conditions on $\partial B_R$. For these bounded domain problems, we prove the existence and uniqueness of weak solutions. Finally, we estimate the error in the velocity component as a function of the truncation radius $R$, showing that, as $R \to \infty$, the velocities of the truncated problems converge, in an appropriate norm, to the velocity of the exterior flow.