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On the principle of exchange of stabilities for the magnetohydrodynamic thermal stability problem in completely confined fluids

Published online by Cambridge University Press:  28 March 2006

Michael Sherman  and
Simon Ostrach
Michael Sherman
Affiliation:
Engineering Division, Case Institute of Technology, Cleveland, Ohio
Simon Ostrach
Affiliation:
Engineering Division, Case Institute of Technology, Cleveland, Ohio
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Abstract

The principle of exchange of stabilities (exchange principle) for the thermal stability problem has been proved by Pellew & Southwell (1940) for fluids bounded by two infinite, horizontal parallel planes. Chandrasekhar (1952) discussed the establishment of the exchange principle for the same geometry when the fluid is an electrical conductor and when an arbitrary oriented, uniform, external magnetic field is applied in the vertical direction.

In this paper, the exchange principle is examined for fluids completely confined in an arbitrary region with rigid bounding surfaces that are good electrical conductors with respect to the fluid. The uniform magnetic field is applied in an arbitrary direction. A generalized thermal boundary condition is imposed which includes the fixed temperature and prescribed heat-flux conditions as special cases.

If no magnetic field is applied to the fluid, the present work reduces to a generalization (for completely confined fluids) of the Pellew & Southwell proof of the exchange principle. In the magnetohydrodynamic (MHD) thermal stability problem, the exchange principle is found to be valid if the total kinetic energy associated with an arbitrary disturbance is greater than or equal to its total magnetic energy. In a special case it is demonstrated that a sufficient condition which will establish the exchange principle is k ≤ η, where k is the fluid thermal diffusivity and η is the fluid electrical resistivity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 24 , Issue 4 , April 1966 , pp. 661 - 671
Copyright
© 1966 Cambridge University Press

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References

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