The evolution equations for Taylor vortices in the small gap limit

The centrifugal instability of the viscous fluid flow between concentric circular cylinders in the small gap limit is considered. The amplitude of the Taylor vortex is allowed to depend on a slow time variable, a slow axial variable, and the polar angle. It is shown that the amplitude of the vortex cannot in general be described by a single amplitude equation. However, if the axial variations are periodic a single amplitude equation can be derived. In the absence of any slow axial variations it is shown that a Taylor vortex remains stable to wavy vortex perturbations. Furthermore, in this situation, stable nonaxisymmetric modes can occur but do not bifurcate from the Taylor vortex state. The stability of these modes is shown to be governed by a modified form of the Eckhaus criterion.