Taylor vortices and the Goldreich—Schubert instability

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Abstract
A linear stability analysis is applied to the flow of an incompressible viscous fluid contained between two infinite coaxial rotating cylinders with radial gravity and a stabilising radial temperature gradient. Only fluids of low and moderate Prandtl number, Pr. (Pr≦2), are considered. The Boussinesq and narrow-gap approximations are used to obtain the linearised perturbation equations. The latter are solved numerically both for axisymmetric and non-axisymmetric disturbances and stability boundaries obtained. It is found that when Pr≦0.972 the most critical disturbance is always axisymmetric; when Pr>0.972 mode crossing occurs in a manner which appears to indicate that the onset of instability is favoured by the mode with the smallest wave-number. Overstable modes have not been found to exist when Pr<1. Transformation equations are given that enable existing Couette flow results to be applied to the model discussed here. A dispersion relation is obtained describing the growth rates of instabilities; this is compared with the dispersion relation obtained by Goldreich and Schubert (1967).

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