Revisiting the Jones eigenproblem in fluid-structure interaction

TL;DR

This paper investigates the Jones eigenproblem in fluid-structure interaction, revealing that Jones modes can exist in broad classes of Lipschitz domains, contrary to previous beliefs about their rarity.

Contribution

It provides the first detailed theoretical and computational analysis of Jones eigenmodes in Lipschitz domains, demonstrating their existence in various geometries.

Findings

Jones modes exist in a broad class of Lipschitz domains.

Existence of Jones modes depends on domain geometry.

Analytical demonstration of Jones modes on simple geometries.

Abstract

The Jones eigenvalue problem first described by D.S. Jones in 1983 concerns unusual modes in bounded elastic bodies: time-harmonic displacements whose tractions and normal components are both identically zero on the boundary. This problem is usually associated with a lack of unique solvability for certain models of fluid-structure interaction. The boundary conditions in this problem appear, at first glance, to rule out {\it any} non-trivial modes unless the domain possesses significant geometric symmetries. Indeed, Jones modes were shown to not be possible in most $C^\infty$ domains (see article by T. Harg\'e 1990). However, we should in this paper that while the existence of Jones modes sensitively depends on the domain geometry, such modes {\it do} exist in a broad class of domains. This paper presents the first detailed theoretical and computational investigation of this eigenvalue problem in Lipschitz domains. We also analytically demonstrate Jones modes on some simple geometries.