On interaction of an elastic wall with a Poiseuille type flow

TL;DR

This paper analyzes the interaction between a fluid flow modeled by Navier--Stokes equations and an elastic plate boundary, demonstrating stability and the existence of a finite-dimensional attractor under certain conditions.

Contribution

It establishes the existence of a global attractor and exponential stability for a coupled fluid-structure system without mechanical damping in the elastic component.

Findings

Existence of a compact finite-dimensional global attractor.

Exponential stability of the linearized coupled system.

Fluid viscosity alone suffices for system stabilization.

Abstract

We study dynamics of a coupled system consisting of the 3D Navier--Stokes equations which is linearized near a certain Poiseuille type flow in an (unbounded) domain and a classical (possibly nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We first show that this problem generates an evolution semigroup $S_t$ on an appropriate phase space. Then under some conditions concerning the underlying (Poiseuille type) flow we prove the existence of a compact finite-dimensional global attractor for this semigroup and also show that $S_t $ is an exponentially stable $C_0$-semigroup of linear operators in the fully linear case. Since we do not assume any kind of mechanical damping in the plate component, this means that dissipation of the energy in the fluid flow due to viscosity is sufficient to stabilize the system.