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For the extended Rayleigh problem of hydrodynamics stability dealing with homogeneous shear flows with variable cross section, we have obtained a parabolic instability region. This improved parabolic instability region intersects with the semicircular instability region under certain condition. The validity of the result is illustrated with an example of basic flows. Furthermore, we have obtained a bound for the complex part of the phase velocity.

In this paper exponential stability of nonlinear fractional order stochastic system with Poisson jumps is studied in finite dimensional space. Existence and uniqueness of solution, stability and exponential stability results are established by using boundedness properties of Mittag-Leffler matrix function, fixed point route and local assumptions on nonlinear terms. A numerical example is given to illustrate the efficiency of the obtained results. Finally, conclusion is drawn.

We consider extended Taylor-Goldstein problem of hydrodynamic stability dealing with incompressible, inviscid, stratified shear flows of arbitrary cross section. For this problem we obtained an instability region for an unstable mode which depends on breadth function, basic velocity profile, vorticity variation, wave number, Richardson number and stratification parameter. Furthermore, long wave stability result, namely, if k ≤ k c > 0 (for some critical wave number k c) implies stability of the mode.

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