Global Weak Solutions to a Time-Periodic Body-Liquid Interaction Problem

TL;DR

This paper proves the existence of global, time-periodic weak solutions for a coupled fluid-structure interaction problem involving an incompressible Navier-Stokes fluid and a rigid body, with arbitrary frequency and amplitude of the external velocity field.

Contribution

It introduces a novel approach to establish global weak solutions without restrictions on the velocity field's magnitude or frequency, overcoming dissipation limitations.

Findings

Existence of time-periodic weak solutions for the coupled problem.

No restrictions on the amplitude or frequency of the external velocity field.

The method applies even at resonance frequencies, avoiding classical limitations.

Abstract

We prove existence of time-periodic weak solutions to the coupled liquid-structure problem constituted by an incompressible Navier-Stokes fluid interacting with a rigid body of finite size, subject to an {\em undamped} linear restoring force. The fluid flow is generated by a uniform, time-periodic velocity field $\bfV$ far from the body. {We emphasize that our result is global, in the sense that no restriction is imposed on the magnitude of $\bfV$ and, rather remarkably, the frequency of $\bfV$ is entirely arbitrary. Thus, in particular, it can coincide with any multiple of a natural frequency of vibration of the body so that, with this model, resonance cannot occur. Although based on the classical "invading domains" technique, our approach requires several new ideas.} Indeed, due to lack of sufficient dissipation, it appears quite unfeasible to show the existence of a fixed point of the Poincar\'e map at the finite-dimensional level along the Galerkin approximant. Therefore, unlike the usual strategy, such a result must be proven directly in a class of weak solutions, and therefore in the infinite-dimensional framework.