Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-29T13:39:51.894Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear Kelvin—Helmholtz instability in magnetic fluids of finite thickness: effects of a tangential magnetic field

Published online by Cambridge University Press:  13 March 2009

Abdel Raouf F. Elhefnawy
Abdel Raouf F. Elhefnawy
Affiliation:
Department of Mathematics, Faculty of Science, Banha University, Banha 13518, Egypt
Get access Rights & Permissions [Opens in a new window]

Abstract

The nonlinear stability of a horizontal interface separating two streaming magnetic fluids of finite thickenss is investigated in two dimensions. The fluids are considered to be inviscid and incompressible. The magnetic field is applied along the direction of streaming. The method of multiple scales, in both space and time, is used to examine the stability properties of the system arising from second-harmonic resonance. A pair of partial differential equations for the amplitude of the wave and its second harmonic are derived. These describe the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrödinger equation that occurs in the non-resonant case. The stability condition of this equation is discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. The nonlinear cut-off wavenumber separating the regions of stability from those of instability is obtained. The equation governing the evolution of the amplitude at the critical point is also obtained, which leads to a nonlinear Klein—Gordon equation.

Type
Research Article
Information
Journal of Plasma Physics , Volume 51 , Issue 3 , June 1994 , pp. 451 - 465
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chandrasekhar, S. 1961 Hydrodynamic and Hydrornagnetic Stability. Oxford University Press.Google Scholar
Drazin, P. G. 1970 J. Fluid Mech. 42, 321.CrossRefGoogle Scholar
Elhefnawy, A. R. F. 1992 a Physica A 182, 419.Google Scholar
Elhefnawy, A. R. F. 1992 b Int. J. Theor. Phys. 31, 1505.Google Scholar
Grimshaw, R. 1990 Nonlinear Ordinary Differential Equations. Blackwell.Google Scholar
Hasimoto, H., & Ono, H. 1972 J. Phys. Soc. Japan 33, 805.CrossRefGoogle Scholar
Infeld, E., & Rowlands, G. 1990 Nonlinear Waves, Solitons and Chaos. Cambridge University Press.Google Scholar
Lang, C. G., & Newell, A. C. 1971 SIAM J. Appl. Maths 21, 605.Google Scholar
Malik, S. K., & Singh, M. 1986 Phys. Fluids 29, 2853.CrossRefGoogle Scholar
Mohamed, A. A., & El Shehawey, E. F. 1989 Fluid Dyn. Res. 5, 117.Google Scholar
Murakami, V. 1986 J. Phys. Soc. Japan 55, 3851.CrossRefGoogle Scholar
Nayfeh, A. H. 1976 J. Appl. Mech. 98, 584.Google Scholar
Nayfeh, A. H., & Saric, W. S. 1972 J. Fluid Mech. 55, 311.Google Scholar
Parkes, E. J. 1991 Wave Motion 13, 261.Google Scholar
Rosensweig, R. E. 1985 Ferrohydrodynarnics. Cambridge University Press.Google Scholar
Sánchez, A., & Vázquez, L. 1991 Int. J. Mod. Phys. 5B, 2825.CrossRefGoogle Scholar
Weissman, M. A. 1979 Phil. Trans. R. Soc. Lond. A 290, 639.Google Scholar