He’s Multiple-Scale Solution for the Three-dimensional Nonlinear KH Instability of Rotating Magnetic Fluids

  • Yusry Osman El-Dib Department of Mathematics, Faculty of Education, Ain Shams University
  • Amal A Mady Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

Abstract

This paper elucidates a trend in solving nonlinear oscillators of the rotating Kelvin-Helmholtz instability. The system is constituted by the vertical flux or the horizontal flux. He’s multiple-scales perturbation methodology has been applied and therefore the system is represented by a generalized homotopy equation. This approach ends up in a periodic answer to a nonlinear oscillator with high nonlinearity. The cubic-quintic nonlinear Duffing equation is obligatory as a condition to uniformly answer. This equation is employed to derive the stability criteria. The transition curves are plotted to investigate the stability image. It's shown that the angular velocity suppresses the instability. The tangential flux plays a helpful role, whereas the vertical field encompasses a destabilizing influence. Within the existence of the rotation, the velocity ratio reduces stability configuration.

Keywords: He’s-Multiple-Scale Method, Kelvin-Helmholtz Instability, Nonlinear Stability, Rotating Fluids, Magnetic Fluids.

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References

[1]        Roberts, P. H., and Soward, A. M., “Rotating Fluid in Geophysics”, Academic Press INC. London LTD., (1978).


[2]        Pearlstein, A. J., “Effect of rotation on the stability of a doubly diffusive fluid layer”, J. Fluid Mech., 103 , (1981), 389.


[3]        Patil, P. R., Parvath, C. P., and Venkatakrishnan, K. S., “Effect of rotation on the stability of a doubly diffusive fluid layer in a porous medium”, Int. J. Heat & Mass Trans., 33 (6), (1990), 1073.


[4]        Pedlosky, J., “Geophysical Fluid Dynamics”, Springer Verlag, New York (1979).


[5]        Davalos-Orozco, L. A., “Rayleigh-Taylor instability of a two fluid layers under a general rotation field and a horizontal magnetic field”, Astrophys. Space Sci., 243, (1996), 291.


[6]       Leblanc, S., and Cambon, C., “On the three-dimensional instabilities of plane flows subjected to Coriolis force”,  Phys. Fluids, 9 (5), (1997), 1307.


[7]       Ramsey, A. S., “A Treatise on Hydromechanics ”, London G. Bell and Sons. Landan, (1954).


[8]       Landau, L. D., and Lifshitz, E. M., “Fluid Mechanics”, Pergamon, New York, (1959).


[9]       Melcher, J. R., “Field Coupled Surface Waves ”, MIT Press, Cambridge, (1963).


[10]     Chandrasekhar, S., “Hydrodynamic and Hydromagnetic Stability ”, Oxford University Press, Oxford, (1961).


[11]     Alterman, Z., “Effect of magnetic field and rotation on Kelvin-Helmholtz instability”, Phys. Fluids, 4, (1961), 1207.


[12]     El-Sayed, M. F., “Electrohydrodynamic Kelvin-Helmholtz instability of two rotating dielectric fluids ”, Z. Naturforsch, 53a, (1998), 17.


[13]      El-Sayed, M. F., “Effect of rotation and an oblique electric field on the developments of Kelvin-Helmholtz instability”, Il: Nuovo Cimento, 114B (11),  (1999), 1305.


[14]      Drazin, P. G., “Kelvin-Helmholtz instability of finite amplitude ”, J. Fluid Mech., 42 (2), (1970), 321.


[15]      Nayfeh, A. H., and Saric, W. S., “Nonlinear waves in a Kelvin-Helmholtz  flow”, J. Fluid Mech., 55, (1972), 311.


[16]      Weissman, M. A., “Nonlinear wave packets in the Kelvin-Helmholtz instability ”, Phil. Trans. R. Soc., 290A, (1979), 639.


[17]      El-Dib, Y. O., “Nonlinear stability of Kelvin-Helmholtz waves in magnetic fluids stressed by a time-dependent acceleration and a tangential magnetic field”, J. Plasma Phys., 55 (2), (1996), 219.


[18]      Zakaria, K., “Kelvin-Helmholtz instability of a horizontal interface between a finite subsonic gas and a finite magnetic liquid”, Can. J. Phys., 76, (1998), 361.


[19]     Chavaraddi, K.,B., Katagi, N.N. and Awati,V.,B.,” Kelvin-Helmholtz Instability in a Fluid Layer Bounded Above by a Porous Layer and Below by a Rigid Surface in Presence of Magnetic Field”, Applied Mathematics, 3,(2012), 564-570.


[20]      Chavaraddi, K.,B., Katagi, N.N. and Pai, N. P, “Electro-hydrodynamic Kelvin-Helmholtz Instability in a Fluid Layer Bounded Above by a Porous Layer and Below by a Rigid Surface,” International Journal of Engineering and Technoscience, 2(4), 2011, 281-288.


[21]     Davalos-Orozco, L. A., “Kelvin-Helmholtz instability under horizontal rotation and magnetic fields”, J. Plasma Phys., 59, (1998), 193.


[22]      El-Dib, Y.O., Motimed, G.M. and Mady A.A.,” A novelty to the nonlinear rotating Rayleigh–Taylor instability”, Pramana – J. Phys, 93:82 (2019)


[23]      Whitham, G. B., “Linear and Nonlinear Waves”, Wiley, New York, (1974).


[24]      El-Dib, Y.O., 2017, “Multiple scales homotopy perturbation method for nonlinear oscillators”, Nonlinear Sciences Letters A, 8(4), 352.


[25]      El-Dib, Y.O., “Stability of a strongly displacement time-delayed Duffing oscillator by the multiple scales-homotopy perturbation method”, J. Applied and Comp.  Mech., 4(4), (2018), 260.


[26]      El-Dib, Y.O., “Periodic Solution and Stability Behavior for Nonlinear Oscillator Having a Cubic Nonlinearity Time-Delayed”, Int. Ann. Sci., 5(1), (2018),12-25.


[27]      Chakraborty, B. B., and Chandra, J., “Rayleigh-Taylor instability in the presence of rotation”, Phys. Fluids, 19, (1976), 1851.


[28]      Chakraborty, B. B.,”Hydromagnetic Rayleigh-Taylor instability of a rotating stratified fluid”, Phys. Fluids, 25, (1982), 743.


[29]      Mohamed, A. A., El-Dib, Y. O., and Mady, A. A., “Nonlinear gravitational stability of streaming in an electrified viscous flow through porous media”, Choas Solitons & Fractals, 14, (2002),1027.


 [30]     El-Dib, Y. O.,” Nonlinear stability of  surface waves in magnetic fluids Effect of a  periodic tangential magnetic field”. J. Plasma Phys., 49, (1993),317 - 330.


[31]      El-Dib, Y.O., 2003, ”Nonlinear hydromagnetic Rayleigh-Taylor instability for strong viscous fluids in porous media”, J.M.M., 260, 1.


[32]      He, J.H.,” Homotopy perturbation technique”, Computer Method in Applied Mechanics and Engineering, 178, (1999), 257.


[33]      He, J.H.,” Application of homotopy perturbation method to nonlinear wave equation”, Chaos Soliton and  Fractals, 26,(2005), 695.


[34]      He, J.H.,” Homotopy perturbation method with two expanding parameters”, Indian Journal of Physics, 88(2), (2014), 193.


[35]      He, J.H.,” Homotopy perturbation method with an auxiliary term”, Abstract and Applied Analysis, (2012), 857612.


[36]      El-Dib, Y. O., “Instability of parametrically second – and third subharmonic resonances governed by nonlinear Schrödinger equations with complex coefficients”, Chaos Solitons & Fractals, 11, (2000), 1773.


[37]      Nayfeh, A. H., and Mook, D. T., “Nonlinear Oscillations ”, Wiley, New York, (1979).


[38]     El-Dib, Y. O., “Periodic solution of the cubic nonlinear Klein–Gordon equation and the stability criteria via the He-multiple scales”, Pramana, 92(1), (2019),7.


[39]      El-Dib, Y. O., and Matoog, R. T., “Electrorheological Kelvin-Helmholtz instability of a fluid sheet”, J. Coll. Int. Sci., 289, (2005), 223.

Published
2019-12-06
How to Cite
[1]
Y. El-Dib and A. Mady, “He’s Multiple-Scale Solution for the Three-dimensional Nonlinear KH Instability of Rotating Magnetic Fluids”, Int. Ann. Sci., vol. 9, no. 1, pp. 52-69, Dec. 2019.
Section
Research Article