Steady Bifurcating Solutions of the Couette–Taylor Problem for Flow in a Deformable Cylinder
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The classical Couette–Taylor problem is to describe the motion of a viscous incompressible fluid in the region between two rigid coaxial cylinders, which rotate at constant angular velocities. This paper treats a generalization of this problem in which the rigid outer cylinder is replaced by a deformable (nonlinearly elastic) cylinder. The inner cylinder is rigid and rotates at a prescribed angular velocity. We study steady rotationally symmetric motions of the fluid coupled with steady axisymmetric motions of the deformable outer cylinder in which it rotates at a prescribed constant angular velocity, typically different from that of the inner cylinder. The motion of the outer cylinder is governed by a geometrically exact theory of shells and the motion of the liquid by the Navier–Stokes equations, with the domain occupied by the liquid depending on the deformation of the outer cylinder. The nonlinear fluid-solid system admits a (trivial) steady solution, termed the Couette solution, which can be found explicitly. This paper treats the global (multiparameter) bifurcation of steady-state solutions from the Couette solution. This problem exhibits technical mathematical difficulties directly due to the fluid-solid interaction: The smoothness of the shell’s configuration restricts the smoothness of the fluid variables, and their boundary values on the shell determine the smoothness of the shell’s configuration. It is essential to ensure that this cycle of implications is consistent.
KeywordsFluid-solid interaction Navier–Stokes equations Bifurcation Nonlinear elasticity Couette–Taylor problem
The research reported here was supported in part by grants from the NSF. This work represents a considerable extension of Chapter 5 of Bourne . Some of the work of Bourne was carried out at the Max Planck Institute for Mathematics in the Sciences in Leipzig, the University of Bonn, the Eindhoven University of Technology, and the University of Glasgow. We are grateful to James C. Alexander for helpful comments.
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