Abstract: The Taylor-Couette problem is a fundamental example in hydrodynamic stability and bifurcation theory, and has been the subject of over 1500 papers. This lecture treats the following generalization: a viscous incompressible liquid occupies the region between two coaxial cylinders. The inner cylinder is rigid and rotates at a constant angular velocity while the outer cylinder is deformable and its motion is not prescribed, but responds to the forces exerted on it by the moving liquid.

We model the deformable cylinder using a geometrically exact theory for a nonlinearly viscoelastic shell. A steady solution of this liquid-solid interaction problem can be found analytically; its linear stability is governed by a tricky quadratic eigenvalue problem. The main purpose of this lecture is to describe the analysis and computation of the spectrum, and its consequences for nonlinear stability.