A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction. Part I: Formulation, Analysis, and Computations

David Bourne and Stuart S. Antman

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Submission date: 03. Sep. 2008
Pages: 26
published in: Communications on pure and applied analysis, 8 (2009) 1, p. 123-142 
DOI number (of the published article): 10.3934/cpaa.2009.8.123
Bibtex
MSC-Numbers: 65N25, 74F10, 76D05, 74D10
Keywords and phrases: Non-self-adjoint quadratic eigenvalue problem, fluid-solid interaction, viscous fluid, nonlinear viscoeastic shell
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Abstract:
This two-part paper treats the numerical approximation of a tricky quadratic eigenvalue problem arising from the following generalization of the classical Taylor-Couette problem: A viscous incompressible fluid occupies the region between a rigid inner cylinder and a deformable outer cylinder, which we take to be a nonlinearly viscoelastic membrane. The inner cylinder rotates at a prescribed angular velocity , driving the fluid, which in turn drives the deformable outer cylinder. The motion of the outer cylinder is not prescribed, but responds to the forces exerted on it by the moving fluid. A steady solution of this coupled fluid-solid system, analogous to the Couette solution of the classical problem, can be found analytically. Its linearized stability is governed by a non-self-adjoint quadratic eigenvalue problem.

In Part I, we give a careful formulation of the geometrically exact problem. We compute the eigenvalue trajectories in the complex plane as functions of by using a Fourier-finite element method. Computational results show that steady solution loses its stability by a process suggestive of a Takens-Bogdanov bifurcation. In Part II we prove convergence of the numerical method.