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The stability of a column of rotating liquid

Published online by Cambridge University Press:  26 February 2010

D. H. Michael
University College, London.
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The main result concerning the stability of an inviscid liquid column, which is at rest, is due to Rayleigh [1], who showed that, taking account of capillarity at the surface, the column will be unstable to small axisymmetric disturbances whose wavelengths in the axial direction are greater than the circumference of the cross-section. The reason for such instabilities is simply that disturbances of these wavelengths decrease the surface area of the column and hence make available excess surface energy which goes into building up the disturbance. The effects of rotation on the stability of a column having a free surface do not appear to have been studied and this note establishes some simple results concerning the effect of plane two-dimensional disturbances on a rotating column. With regard to the stability of rotating fluids in general there is another result due to Rayleigh [2] stating that the fluid will be unstable if the numerical value of the circulation decreases with the radius at any point. But this result is again for axisymmetric disturbances, and hence does not necessarily bear any relationship to the results for plane disturbances which, in the case of an incompressible liquid as is assumed here, cannot be axisymmetric. A result more closely related to this work is that of Kelvin [3] who considered the stability of a column of uniform vorticity in a fluid otherwise free of vorticity. Such a column is stable to two-dimensional and to three-dimensional disturbances.

Research Article
Copyright © University College London 1959

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1.Rayleigh, Lord, Proc. London Math. Soc., 10 (1878), 413.Google Scholar
2.Rayleigh, Lord, Proc. Royal Soc. (A), 93 (1916), 148154.Google Scholar
3.SirThomson, W., Phil. Mag. (5), 1 (1880), 155168.Google Scholar