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Note on a heterogeneous shear flow

Published online by Cambridge University Press:  28 March 2006

John W. Miles  and
Louis N. Howard
John W. Miles
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University
Louis N. Howard
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology
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Abstract

Goldstein (1931) has considered the stability of a shear layer within which the velocity and the density vary linearly and outside which they are constant. Rayleigh (1880, 1887) had found that the corresponding, homogeneous shear flow is unstable in and only in a finite band of wave-numbers. Goldstein concluded that a small density gradient renders the flow unstable for all wave-numbers. This conclusion appears to depend on the acceptance of all possible branches of a multi-valued eigenvalue equation, and it is shown that the principal branch of this eigenvalue equation yields one and only one unstable mode if and only if the wave-number lies in a band that decreases from Rayleigh's band to zero as the Richardson number increases from 0 to ¼.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 20 , Issue 2 , October 1964 , pp. 331 - 336
Copyright
© 1964 Cambridge University Press

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References

Goldstein, S. 1931 Proc. Roy. Soc. A, 132, 524.
Howard, L. N. 1961 J. Fluid Mech. 10, 509.
Miles, J. W. 1961 J. Fluid Mech. 10, 496.
Miles, J. W. 1963 J. Fluid Mech. 16, 209.
Rayleigh, Lord 1880 Proc. Lond. Math. Soc. 11, 57.
Rayleigh, Lord 1887 Proc. Lond. Math. Soc. 19, 67; see also Lord Rayleigh, The Theory of Sound, 368, Dover Publications, New York (1945).