Revisiting the Jones eigenproblem in fluid-structure interaction

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Department of Mathematical Sciences


The Jones eigenvalue problem first described in [D. S. Jones, Quart. J. Mech. Appl. Math., 36 (1983), pp. 111--138] 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 any nontrivial modes unless the domain possesses significant geometric symmetries. Indeed, Jones modes were shown to not be possible in most $C^\infty$ domains in [T. Hargé, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), pp. 857--859]. However, in this paper we will see that while the existence of Jones modes sensitively depends on the domain geometry, such modes 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.

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SIAM Journal on Applied Mathematics