Stability of stratified shear flows. (English) Zbl 0575.76106

Jain, R. K.; Kochar, G. T.

J. Math. Anal. Appl. 96, 269-282 (1983).

For a steady plane parallel flow of an inviscid, incompressible fluid of variable density under gravity, it is shown that the complex wave velocity for any unstable mode lies in a semiellipse-type region whose major axis coincides with the diameter of Howard’s semicircle, while its minor axis depends on the stratification. If \(kc_ i\) denotes the complex part of wave frequency and \(J_ 0\) the minimum of the local Richardson number over the flow domain, it is further established that \(kc_ i\to 0+\) as \(J_ 0\to 1/4-\). The case of free upper surface and conditional reduction dependent on the curvature of the basic velocity of the unstable region is also studied.

MSC:

76V05	Reaction effects in flows
76E05	Parallel shear flows in hydrodynamic stability
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76F10	Shear flows and turbulence

Keywords:

semiellipse-type region of complex wave velocity; steady plane parallel flow; fluid of variable density under gravity; unstable mode; Howard’s semicircle

References:

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