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Bounds on the phase velocity in the linear instability of viscous shear flow problem in the \(\beta\)-plane. (English) Zbl 1049.76026

Summary: Results obtained by D. D. Joseph [J. Fluid Mech. 33, 617–621 (1968; Zbl 0164.28703)] for the viscous parallel shear flow problem are extended to the problem of viscous parallel shear flow problem in the beta plane, and a sufficient condition for stability has also been derived.

MSC:

76E20 Stability and instability of geophysical and astrophysical flows
76E07 Rotation in hydrodynamic stability
86A05 Hydrology, hydrography, oceanography

Citations:

Zbl 0164.28703
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References:

[1] Banerjee, M. B.; Shandil, R. G.; Gourala, M. G.; Chauhan, S. S., Eigenvalue bounds for the Orr-Sommerfeld’s equation and their relevance to the existence of backward wave motion, Stud. Appl. Math., 103, 43-50 (1999) · Zbl 1136.76358 · doi:10.1111/1467-9590.00119
[2] Banerjee, M. B.; Shandil, R. G.; Chauhan, S. S.; Sharma, D., Eigenvalue bounds for the Orr-Sommerfeld’s equation and their relevance to the existence of backward wave motion-II, Stud. Appl. Math., 105, 31-34 (2000) · Zbl 1136.76357 · doi:10.1111/1467-9590.00140
[3] Drazin, P. G.; Howard, L. N., Hydrodynamic stability of parallel flow of inviscid fluid, Advances in Applied Mechanics (1966), New Delhi: Acdemic Press, New Delhi
[4] Fjortoft, R., Application of integral theorems in deriving criteria of stability of laminar flow and for the baroclinic circular vortex, Geofys. Publ., 17, 1-52 (1950)
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[6] Hickernel, F. J., An upper bound on the growth rate of a linear instability in a homogeneous shear flow, Stud, Appl. Math., 72, 87-93 (1985) · Zbl 0588.76068
[7] Joseph, D. D., Eigenvalue bounds for the Orr-Sommerfeld equation, J. Fluid Mech., 33, 617-621 (1968) · Zbl 0164.28703 · doi:10.1017/S0022112068001552
[8] Kuo, H. L., Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere, J. Meteorol., 6, 105-105 (1949)
[9] Rayleigh, J. W S., On the instability of certain fluid motions, Proc. London Math. Soc., 9, 57-70 (1880) · JFM 12.0711.02
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