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Instability of a viscous fluid of variable density in a magnetic field

Published online by Cambridge University Press:  28 March 2006

N. T. Dunwoody
N. T. Dunwoody
Affiliation:
Department of Engineering Mathematics, The Queen's University, Belfast
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Abstract

The instability to small two-dimensional disturbances of an electrically conducting fluid of variable density is investigated. The viscous fluid is bounded between two vertical parallel planes normal to which a magnetic field of constant intensity is applied. Significant parameters upon which the behaviour of the Rayleigh number at neutral stability depends are the Hartmann number M and the wave-number α which is associated with a periodic disturbance with periodicity in the unbounded horizontal direction.

The solution may be sought by considering basic disturbances which are either symmetric or antisymmetric about the median plane parallel to the boundary planes. It is found that for a given magnetic field strength the critical Rayleigh number governing stability is associated with an antisymmetric disturbance of zero wave-number. The least stable symmetric disturbance which arises when the wave-number is zero is less easily excited. This trend is seen again in the purely hydrodynamic case (M = 0) where, corresponding to a finite wave-numbe value, the more unstable mode at neutral stability is found to be an antisymmetric one.

The most unstable situation occurs when both the Hartmann number and the wave number zero. In this case the result of Wooding (1960) that the minimum critical Rayleigh number is zero and is associated with a symmetric disturbance is reobtained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 20 , Issue 1 , September 1964 , pp. 103 - 113
Copyright
© 1964 Cambridge University Press

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References

Ostrach, S. 1955 50 Jahre Grenzschichtforschung. Verlag Friedr. Vieweg und Sohn.
Taylor, G. I. 1954 Proc. Phys. Soc. B, 67, 868.
Wooding, R. A. 1960 J. Fluid Mech. 7, 501.
Yih, C.-S. 1959 Quart. Appl. Math. 17, 25.
Yih, C.-S. 1960 Quart. Appl. Math. 18, 300.