Note on a heterogeneous shear flow

Author:

Miles John W.,Howard Louis N.

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.

Publisher

Cambridge University Press (CUP)

Subject

Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics

Reference7 articles.

1. Rayleigh, Lord 1887 Proc. Lond. Math. Soc. 19,67

2. see also Lord Rayleigh, The Theory of Sound, 368, Dover Publications, New York (1945).

3. Miles, J. W. 1963 J. Fluid Mech. 16,209.

4. Howard, L. N. 1961 J. Fluid Mech. 10,509.

5. Goldstein, S. 1931 Proc. Roy. Soc. A,132,524.

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