January  2009, 8(1): 123-142. doi: 10.3934/cpaa.2009.8.123

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

1. 

Department of Mathematics,Institute for Physical Science and Technology, and Institute for Systems Research, University of Maryland, College Park, MD 20742, United States

2. 

Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany

Received  June 2008 Revised  September 2008 Published  October 2008

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 ω, 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 ω by using a Fourier-finite element method. Computational results show that the steady solution loses its stability by a process suggestive of a Takens-Bogdanov bifurcation. In Part II we prove convergence of the numerical method.
Citation: Stuart S. Antman, David Bourne. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part I: Formulation, Analysis, and Computations. Communications on Pure & Applied Analysis, 2009, 8 (1) : 123-142. doi: 10.3934/cpaa.2009.8.123
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