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Stability of parallel flow in a parallel magnetic field at small magnetic Reynolds numbers

Published online by Cambridge University Press:  28 March 2006

P. G. Drazin
P. G. Drazin
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology
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Abstract

The hydromagnetic stability of a basic two-dimensional parallel flow of an incompressible conducting fluid in a uniform magnetic field parallel to the flow is considered. By use of the generalization of the Orr–Sommerfeld equation for an electrically conducting fluid, it is shown that any given small wave disturbance can be stabilized by a sufficiently strong magnetic field if the Reynolds number is finite and the magnetic Reynolds number small.

Stability of velocity profiles with a point of inflexion at small magnetic Reynolds number and infinite Reynolds number is considered in detail. Perturbation methods are developed to find stability characteristics in two cases, when the magnetic field is weak, and when the disturbance is a long wave. These methods are applied to the jet and the half-jet, which are both found to be unstable to long-wave disturbances, however strong the magnetic field. Nonetheless, these two flows can be stabilized for any given harmonic disturbance of finite wavelength. The analysis of the jet reveals the surprising result that the magnetic field makes inviscid long-wave disturbances more unstable.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 8 , Issue 1 , May 1960 , pp. 130 - 142
Copyright
© 1960 Cambridge University Press

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References

Curle, N. 1956 Aero. Res. Council, Lond., Unpublished Rep. no. 18426.
Esch, R. E. 1957 J. Fluid Mech. 3, 289.
Lessen, M. 1952 Quart. Appl. Math. 10, 184.
Lessen, M. & Fox, J. A. 1955 50 Jahre Grenzschichtforschung, p. 122. Braunschweig: Friedr. Vieweg & Sohn.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Michael, D. H. 1953 Proc. Camb. Phil. Soc. 49, 166.
Michael, D. H. 1955 Proc. Camb. Phil. Soc. 51, 528.
Murgatroyd, W. 1953 Phil. Mag. (7), 44, 1348.
Nisbet, I. C. T. 1960 (to be published).
Rayleigh, J. W. S. 1945 The Theory of Sound, Vol. II, Chap. XXI. New York: Dover.
Savic, P. 1941 Phil. Mag. (7), 32, 245.
Stuart, J. T. 1954 Proc. Roy. Soc. A, 221, 189.
Synge, J. L. 1938 Hydrodynamic Stability. Semi -centennial publications of the Amer. Math. Soc. 2 (Addresses), 227.Google Scholar