Long-time behavior of an anisotropic rigid body interacting with a Poiseuille flow in an unbounded 2D channel

TL;DR

This paper investigates the long-term dynamics of an elliptic rigid body in a 2D channel under Poiseuille flow, proving global existence of solutions and return to equilibrium for small flow amplitudes.

Contribution

It provides the first long-time analysis of fluid-solid interaction with a specified non-trivial final state, including existence and stability results.

Findings

Global-in-time existence of weak solutions.

Return to equilibrium for small flow amplitudes.

Detailed analysis of the body's motion near channel boundaries.

Abstract

We study the long-time behavior of an elliptic rigid body which is allowed to vertically translate and rotate in a 2D unbounded channel under the action of a Poiseuille flow at large distances. The motion of the fluid is modelled by the incompressible Navier-Stokes equations, while the motion of the solid is described through Newton's laws. In addition to the solid inertia and the hydrodynamic forces, we assume the dynamics of the solid is driven by internal elastic restoring forces but without any structural damping. Through a precise description of the motion of the elliptic body whenever it comes close to the channel boundaries, we prove global-in-time existence of weak solutions. Our second main contribution is a proof of return to equilibrium in case the amplitude of the Poiseuille flow is small. \black To our knowledge, this represents the first long-time analysis of fluid-solid interaction problems with a given non-trivial final state.