On the Oseen-type resolvent problem associated with time-periodic flow past a rotating body

TL;DR

This paper studies the mathematical properties of time-periodic flows of viscous fluids past rotating bodies, establishing conditions for the existence of solutions based on the ratio of rotational velocities.

Contribution

It introduces a Sobolev space framework for the resolvent problem and analyzes how the ratio of rotational velocities affects solution existence.

Findings

Existence of solutions when the velocity ratio is rational.

Counterexample showing non-existence when the ratio is irrational.

Uniform resolvent estimates depend on additional restrictions.

Abstract

Consider the time-periodic flow of an incompressible viscous fluid past a body performing a rigid motion with non-zero translational and rotational velocity. We introduce a framework of homogeneous Sobolev spaces that renders the resolvent problem of the associated linear problem well posed on the whole imaginary axis. In contrast to the cases without translation or rotation, the resolvent estimates are merely uniform under additional restrictions, and the existence of time-periodic solutions depends on the ratio of the rotational velocity of the body motion to the angular velocity associated with the time period. Provided that this ratio is a rational number, time-periodic solutions to both the linear and, under suitable smallness conditions, the nonlinear problem can be established. If this ratio is irrational, a counterexample shows that in a special case there is no uniform resolvent estimate and solutions to the time-periodic linear problem do not exist.