A Linearized Viscous, Compressible Flow-Plate Interaction with Non-dissipative Coupling

TL;DR

This paper develops a mathematical model for viscous, compressible fluid flow interacting with an elastic plate, focusing on boundary conditions and semigroup well-posedness, especially considering non-dissipative coupling effects.

Contribution

It introduces a novel linearized model incorporating ambient flow effects at the interface and employs a Lumer-Phillips approach to establish well-posedness via semigroup theory.

Findings

Established semigroup well-posedness for the coupled system.

Derived a velocity matching condition involving the structure's material derivative.

Addressed the challenge of non-dissipative coupling in fluid-structure interaction.

Abstract

We address semigroup well-posedness for a linear, compressible viscous fluid interacting at its boundary with an elastic plate. We derive the model by linearizing the compressible Navier-Stokes equations about an arbitrary flow state, so the fluid PDE includes an ambient flow profile $\mathbf{U}$. In contrast to model in [Avalos, Geredeli, Webster, 2017], we track the effect of this term at the flow-structure interface, yielding a velocity matching condition involving the material derivative of the structure; this destroys the dissipative nature of the coupling of the dynamics. We adopt here a Lumer-Phillips approach, with a view of associating fluid-structure solutions with a $C_{0}$-semigroup $\left\{e^{\mathcal{A}t}\right\} _{t\geq 0}$ on a chosen finite energy space of data. Given this approach, the challenge becomes establishing the maximal dissipativity of an operator $\mathcal{A}$, yielding the flow-structure dynamics.