The linear stability of the flow in a narrow spherical annulus

Abstract

The flow of a fluid in a narrow spherical annulus is considered. When the outer sphere remains fixed and the angular velocity of the inner one is increased beyond a critical value an instability resembling Taylor vortices appears. This instability is investigated by expanding the solution in powers of the small parameter ε, the ratio of the gap thickness to the radius, and assuming that two length scales, O(1) and O(ε), are important in the latitudinal direction.The perturbation then takes the form of cells which are of roughly square cross-section, at least near the equator, but whose amplitude decays rapidly with latitude; it is also subject to a slow spatial modulation. The critical value of the Taylor number at which the instability first appears is shown to be that for infinite concentric cylinders plus a correction O(ε) due to secondary motions and a correction not greater than O(ε) due to the domain being bounded.