THE LINEAR STRABILITY OF THE FLOW IN THE NARROW GAP BETWEEN TWO CONCENTRIC ROTATING SPHERES

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Abstract
The flow of a viscous fluid contained in the narrow gap between two concentric spheres rotating with different angular velocities about a common rotation axis is considered. The onset of instability of this flow is investigated analytically, and two special cases are given particular attention. In the first case the outer sphere is at rest, while in the second case the fluid is in almost rigid rotation with the inner sphere rotating slightly faster than the outer. Instability first sets in near the equator, but the critical Taylor number is greater than that for the corresponding cylinder problem. The WKBJ method is used. Difficulties which arose in previous treatments are resolved by identifying the turning points. They are located not on the real latitude axis but in its extension to the complex plane. The implementation of the procedure leads to an ordinary-differential-equation eigenvalue problem which can be solved by standard numerical techniques.

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