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On the analogy between thermal and rotational hydrodynamic stability

Published online by Cambridge University Press:  28 March 2006

Walter R. Debler
Walter R. Debler
Affiliation:
Department of Engineering Mechanics, The University of Michigan
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Abstract

The correspondence between the eigenvalues for the problem of the onset of convection in a fluid confined between two horizontal plates and for the stability of viscous flow between two cylinders rotating at almost the same angular velocity has been known for some time. The recent work of Chandrasekhar (1961) has prompted the extension of the analogy to a larger group of rotating cylinder problems and their associated convection cases in which the primary temperature distribution is parabolic. This paper shows the analogy between these two problems and presents data which give the corresponding temperature distribution for a given ratio of angular velocities between the two cylinders. The equivalent Rayleigh numbers are listed for the Taylor numbers given by Chandrasekhar (1954). The eigenfunctions for several of the parabolic temperature profiles are determined. These results show that the single vortex convection pattern becomes a double vortex for certain initial temperature distributions. The critical Rayleigh numbers for the stability of a layer of water which is near 4 °C is also found by analogy.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 24 , Issue 1 , January 1966 , pp. 165 - 176
Copyright
© 1966 Cambridge University Press

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References

Chandrasekhar, S. 1954 Mathematika, 1, 5.
Chandrasekhar, S. 1961 J. Math. Mech. 10, 68.
Davey, A. 1962 J. Fluid Mech. 14, 33.
Debler, W. 1959 Ph.D. Thesis, University of Michigan.
Pellew, A. & Southwell, R. V. 1940 Proc. Roy. Soc., A 176, 312.
Sparrow, E., Goldstein, R. J. & Jonsson, V. K. 1964 J. Fluid Mech. 18, 51.
Taylor, G. I. 1923 Phil. Trans., A 223, 289.