The stability of unsteady cylinder flows

Abstract

First, the linear stability of the flow between two concentric cylinders when the outer one is a t rest and the inner has angular velocity Ω{1+ εcosωt} is considered. In the limit in which ε and ω tend to zero it is found that the critical Taylor number a t which instability first occurs is decreased by an amount of order ε2from its unmodulated value, the stabilizing effect a t order ε2ω2being slight. The limit in which ω tends to infinity with ε arbitrary is then studied. In this case it is found that the critical Taylor number is decreased by an amount of order ε2ω−3from its unmodulated value.Second, the effect of taking nonlinear terms into account is investigated. It is found that equilibrium perturbations of small but finite amplitude can exist under slightly supercritical conditions in both the high and low frequency limits. Some comparisons with experimental results are made, but these indicate that further theoretical work is needed for a broad band of values of ω. In appendix B it is shown how this can be done by an alternative formulation of the problem.