Viscous flow around a rigid body performing a time-periodic motion

TL;DR

This paper proves the existence of time-periodic solutions for viscous incompressible flow around a rigid body performing a prescribed periodic motion, using advanced Fourier and Sobolev space techniques.

Contribution

It develops a novel linear theory in homogeneous Sobolev spaces to handle the ill-posed resolvent problem in classical spaces, establishing existence results.

Findings

Existence of time-periodic solutions under specific motion conditions.

Development of a linear theory in homogeneous Sobolev spaces.

Application of Fourier series analysis to fluid-structure interaction.

Abstract

The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.