Collections of Fluid Loaded Plates: A Nonlocal Approach

TL;DR

This paper introduces a novel nonlocal boundary value problem approach to analyze the motion of fluid-loaded elastic plates in a channel, providing explicit solutions, stability conditions, and a unified framework.

Contribution

It develops a new unified boundary value problem approach for fluid-loaded plates, deriving explicit solutions and stability criteria using a matrix-based analysis.

Findings

Explicit solutions for the linear problem

Conditions for asymptotic stability identified

The problem is characterized by a matrix depending on physical parameters

Abstract

We consider the motion of a collection of fluid loaded elastic plates, situated horizontally in an infinitely long channel. We use a new, unified approach to boundary value problems, introduced by A.S. Fokas in the late 1990s, and show the problem is equivalent to a system of 1-parameter integral equations. We give a detailed study of the linear problem, providing explicit solutions and well-posedness results in terms of standard Sobolev spaces. We show that the associated Cauchy problem is completely determined by a matrix, which depends solely on the mean separation of the plates and the horizontal velocity of each of the driving fluids. This matrix corresponds to the infinitesimal generator of the semigroup for the evolution equations in Fourier space. By analysing the properties of this matrix, we classify necessary and sufficient conditions for which the problem is asymptotically stable.