TY - JOUR

T1 - A non-self-adjoint quadratic eigenvalue problem describing a fluid-solid interaction Part I : formulation, analysis, and computations

AU - Bourne, D.P.

AU - Antman, S.S.

PY - 2009

Y1 - 2009

N2 - 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 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.

AB - 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 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.

U2 - 10.3934/cpaa.2009.8.123

DO - 10.3934/cpaa.2009.8.123

M3 - Article

VL - 8

SP - 123

EP - 142

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

SN - 1534-0392

IS - 1

ER -