The Taylor-Couette problem is a fundamental example in bifurcation theory and hydrodynamic stability, and has been the subject of over 1500 papers. This thesis treats a generalization of this problem in which the rigid outer cylinder is replaced by a deformable (nonlinearly viscoelastic) cylinder whose motion is not prescribed, but responds to the forces exerted on it by the moving liquid. The inner cylinder is rigid and rotates at a prescribed angular velocity, driving the liquid, which in turn drives the deformable 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, where the domain occupied by the liquid depends on the deformation of the outer cylinder.

This thesis treats the stability of Couette flow, a steady solution of the nonlinear fluid-solid system that can be found analytically, first with respect to perturbations that are independent of z, then with respect to axisymmetic perturbations. The linearized stability problems are governed by quadratic eigenvalue problems. For each problem, this thesis gives a detailed characterization of how the spectrum of the linearized operator depends on the control parameter, which is the angular velocity of the rigid inner cylinder. In particular, there are theorems detailing how the eigenvalues cross the imaginary axis. The spectrum is computed by a mixed Fourier-finite element method. The spectral properties determine the conditions under which the system loses its linearized stability. The same conditions support theorems on nonlinear stability. New physical phenomena are discovered that are not observed in the classical Taylor-Couette problem. The fluid-solid interaction models that are developed have applications in structural engineering and human physiology.