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

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.