On equilibrium modes and their stability in the small-gap Taylor problem by a new method

We consider small disturbances to the steady viscous flow between concentric rotating circular cylinders. It is shown that as the dimensionless gap width δ approaches zero the terms from ∂/∂θ in the governing equations are O(δ1/2) and so do not appear in the first approximation. This is in contrast to the classical small-gap approximation, in which terms from ∂/∂θ do appear in the first approximation. Here θ is the usual azimuthal angle. Let ε be a measure of the size of the small perturbation. In Sec. III it is argued that the asymptotic relation ε∼δ2 is needed to obtain a correct approximation to weakly nonlinear axisymmetric (Taylor-vortex) equilibrium flow for small values of δ. We use this relation between ε and δ to obtain a unified amplitude equation to describe both axisymmetric modes and those with a general dependence on the azimuthal angle, taking into account weakly nonlinear effects. This allows a description of single-mode equilibrium states and of their stability. Some unexpected results are obtained about the stability of these modes. The mode with m=0 is shown to be stable for all relevant values of the Taylor number T. This is to be expected under the particular limiting process used. However the modes with m=1, 2, and 3 are shown to become stable for successively higher values of T. It is also shown that these nonaxisymmetric equilibrium states may be obtained by the solution of some initial value problems, thus leaving open the possibility of attaining wavy disturbances by a different method than the usual one of instability of Taylor vortices.