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Nonlinear stability of Kelvin–Helmholtz waves in magnetic fluids stressed by a time-dependent acceleration and a tangential magnetic field

Published online by Cambridge University Press:  13 March 2009

Yusry O. El-Dib
Yusry O. El-Dib
Affiliation:
Department of Mathematics, Faculty of Education, Am Shams University, Heliopolis, Cairo, Egypt
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Abstract

The nonlinear stability of surface waves propagating between two superposed streaming magnetic fluids is investigated. The fluids are stressed by a constant tangential magnetic field and a vertical periodic acceleration. The solution employs the method of multiple scales. Owing to the periodicity, resonant cases appear. Two parametrically nonlinear Schrödinger equations are derived for the resonant cases to describe the elevation of weakly nonlinear capillary waves. The standard nonlinear Schrödinger equation is satisfied for the non resonant cases. Necessary and sufficient conditions for stability are obtained. A formula for the surface elevation is obtained in each case. It is found that the magnetic field, the velocities and the frequency of the applied periodic force play dual roles in the resonant region. Investigation of the stability criterion by nonlinear perturbation shows that an increase in the acceleration frequency has a stabilizing effect. The stabilizing role of the frequency is due to the destabilizing effect of the amplitude of the periodic acceleration.

Type
Research Article
Information
Journal of Plasma Physics , Volume 55 , Issue 2 , April 1996 , pp. 219 - 234
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Benjamin, T. B. & Ursell, F. 1954 Proc. R. Soc. Lond. A225, 505–517.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydrodynamic Stability. Oxford University Press.Google Scholar
Cowley, M. D., & Rosensweig, R. E., 1967 J. Fluid Mech. 30, 671–688.CrossRefGoogle Scholar
Deysikder, S., & Das, K. P., 1990 Contrib. Plasma Phys. 30, 469.CrossRefGoogle Scholar
Drazin, P. G., 1970 J. Fluid Mech. 42, 321–335.CrossRefGoogle Scholar
El-dib, Y. O. 1993 J. Plasma Phys. 49, 317–330.CrossRefGoogle Scholar
El-dib, Y. O., 1994 a Can J. Phys. 72, 578–590.CrossRefGoogle Scholar
El-dib, Y. O., 1994 b Appl. Maths Lett. 7, 89–92.CrossRefGoogle Scholar
El-dib, Y. O., 1995 Fluid Dyn. Res. 15, 385–404.CrossRefGoogle Scholar
Elhefnawy, A. R. F. 1992 Int. J. Theor. Phys. 31, 1505–1520.CrossRefGoogle Scholar
Faraday, M. 1831 Phil. Trans. R. Soc. Lond. 121, 319–340.Google Scholar
Kambe, T. & Umeki, M. 1990 J. Fluid Mech. 212, 373–393.CrossRefGoogle Scholar
Kant, R. & Malik, S. K., 1985 Phys. Fluids. 28, 3534–3537.CrossRefGoogle Scholar
Kelly, R. E., 1965 J. Fluid Mech. 22, 547–560.CrossRefGoogle Scholar
Lyell, M. J., & Michael, R. 1991 AIAA J. 29, 1894–1900.CrossRefGoogle Scholar
Malik, S. K., & Singh, M., 1988 Phys. Fluids. 31, 1069–1073.CrossRefGoogle Scholar
Mohamed, A. A. & Elshehawey, E. F., 1989 Fluid Dyn. Res. 5, 117–133.CrossRefGoogle Scholar
Nayfeh, A. H. 1973 Perturbation Methods. Wiley, New York.Google Scholar
Nayfeh, A. H. 1976 Trans. ASME E: J. Appl. Mech. 43, 584–588.Google Scholar
Nayfeh, A. H., & Moox, D. T. 1979 Nonlinear Oscillations. Wiley, New York.Google Scholar
Nayfeh, A. H. & Saric, W. S. 1971 J. Fluid Mech. 46, 209–231.CrossRefGoogle Scholar
Rosensweig, R. E. 1985 Ferrohydrodynamics. Cambridge University Press.Google Scholar
Thorpe, S. A. 1968 J. Fluid Mech. 32, 693–704.CrossRefGoogle Scholar
Umeki, M & Kambe, T. 1989 J. Phys. Soc. Jpn. 48, 140–154.CrossRefGoogle Scholar
Weissman, M.A. 1979 Phil. Trans. R. Soc. Lond. A290, 639–685.Google Scholar