Nonlinear stability of rotating two superposed magnetized fluids with the technique of the homotopy perturbation

Document Type: Research Paper

Authors

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

Abstract

In the present work, the Rayleigh-Taylor instability of two rotating superposed magnetized fluids within the presence of a vertical or a horizontal magnetic flux has been investigated. The nonlinear theory is applied, by solving the equation of motion and uses the acceptable nonlinear boundary conditions. However, the nonlinear characteristic equation within the elevation parameter is obtained. This equation features a transcendental integro-Duffing kind. The homotopy perturbation technique has been applied by exploitation the parameter growth technique that results in constructing the nonlinear frequency. Stability conditions are derived from the frequency equation. It's illustrated that the rotation parameter plays a helpful result. It's shown that the stability behavior within the extremely uniform rotating fluids equivalents to the system while not rotation. A periodic solution for the elevation function has been performed. Numerical calculations area unit created for linear analysis furthermore the nonlinear scope. Moreover, the elevation function has been premeditated versus the time parameter. The strategy adopted here is vital and powerful for solving nonlinear generator systems with a really high nonlinearity arising in nonlinear science and engineering.

Keywords

Main Subjects


[1]           A. S. Ramsey, W. H. Besant, 1954, A Treatise on Hydromechanics: Hydrodynamics, G. Bell,

[2]           CC Lin:" The Theory of Hydrodynamic Stability", Cambridge University Press, 1955, 155 頁, 15× 23cm, 22s 6d, Vol. 11, No. 5, pp. 217, 1956.

[3]           S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability Oxford University Press Oxford Google Scholar, 1961.

[4]           H. F. Bauer, Coupled oscillations of a solidly rotating liquid bridge, Acta Astronautica, Vol. 9, No. 9, pp. 547-563, 1982.

[5]           Y. O. El-Dib, Capillary instability of an oscillating liquid column subjected to a periodic rigid-body rotation, Fluid dynamics research, Vol. 18, No. 1, pp. 17, 1996.

[6]           G. M. Moatimid, Y. O. El-Dib, Effects of an unsteady rotation on the electrohydrodynamic stability of a cylindrical interface, International journal of engineering science, Vol. 32, No. 7, pp. 1183-1193, 1994.

[7]           F. F. Hatay, S. Biringen, G. Erlebacher, W. Zorumski, Stability of high‐speed compressible rotating Couette flow, Physics of Fluids A: Fluid Dynamics, Vol. 5, No. 2, pp. 393-404, 1993.

[8]           G. Sarma, P. C. Lu, S. Ostrach, Film Stability in a Vertical Rotating Tube with a Core‐Gas Flow, The Physics of Fluids, Vol. 14, No. 11, pp. 2265-2277, 1971.

[9]           A. E.-M. A. Mohammed, A. G. El-Sakka, G. M. Sultan, Electrohydrodynamic stability of m= 0 mode of a rotating jet under a periodic field, Physica Scripta, Vol. 31, No. 3, pp. 193, 1985.

[10]         S. Leblanc, C. Cambon, Effects of the Coriolis force on the stability of Stuart vortices, Journal of Fluid Mechanics, Vol. 356, pp. 353-379, 1998.

[11]         Y. O. El-Dib, The stability of a rigidly rotating magnetic fluid column effect of a periodic azimuthal magnetic field, Journal of Physics A: Mathematical and General, Vol. 30, No. 10, pp. 3585, 1997.

[12]         K. Schwarz, Effect of rotation on the stability of advective flow in a horizontal fluid layer at a small Prandtl number, Fluid Dynamics, Vol. 40, No. 2, pp. 193-201, 2005.

[13]         R. HIDE, THE CHARACTER OF THE EQUILIBRIUM OF A HEAVY, VISCOUS, INCOMPRESSIBLE, ROTATING FLUID OF VARIABLE DENSITY: II. TWO SPECIAL CASES, The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 9, No. 1, pp. 35-50, 1956.

[14]         L. Debnath, Exact solutions of the unsteady hydrodynamic and hydromagnetic boundary layer equations in a rotating fluid system, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 55, No. 7‐8, pp. 431-438, 1975.

[15]         L. Dávalos‐Orozco, J. Aguilar‐Rosas, Rayleigh–Taylor instability of a continuously stratified fluid under a general rotation field, Physics of Fluids A: Fluid Dynamics, Vol. 1, No. 7, pp. 1192-1199, 1989.

[16]         L. Dávalos-Orozco, Rayleigh-Taylor instability of two superposed fluids under imposed horizontal and parallel rotation and magnetic fields, Fluid dynamics research, Vol. 12, No. 5, pp. 243, 1993.

[17]         B. Chakraborty, J. Chandra, Rayleigh–Taylor instability in the presence of rotation, The Physics of Fluids, Vol. 19, No. 12, pp. 1851-1852, 1976.

[18]         B. Chakraborty, Hydromagnetic Rayleigh–Taylor instability of a rotating stratified fluid, The Physics of Fluids, Vol. 25, No. 5, pp. 743-747, 1982.

[19]         L. Davalos-Orozco, Rayleigh-Taylor stability of a two-fluid system under a general rotation field, Dynamics of atmospheres and oceans, Vol. 23, No. 1-4, pp. 247-255, 1996.

[20]         P. Sharma, R. Chhajlani, Effect of finite Larmor radius on the Rayleigh-Taylor instability of two component magnetized rotating plasma, Zeitschrift für Naturforschung A, Vol. 53, No. 12, pp. 937-944, 1998.

[21]         P. Hemamalini, S. A. Devi, Rayleigh-Taylor Instability of a Two-fluid Layer Subjected to Rotation and a Periodic Tangential Magnetic Field, FDMP: Fluid Dynamics & Materials Processing, Vol. 10, No. 4, pp. 491-501, 2014.

[22]         G. M. Moatimid, A. F. El-Bassiouny, Nonlinear interfacial Rayleigh–Taylor instability of two-layers flow with an ac electric field, Physica Scripta, Vol. 76, No. 2, pp. 105, 2007.

[23]         S. A. Devi, P. Hemamalini, Nonlinear Rayleigh–Taylor instability of two superposed magnetic fluids under parallel rotation and a normal magnetic field, Journal of magnetism and magnetic materials, Vol. 314, No. 2, pp. 135-139, 2007.

[24]         J. M. Stone, T. Gardiner, The magnetic Rayleigh-Taylor instability in three dimensions, The Astrophysical Journal, Vol. 671, No. 2, pp. 1726, 2007.

[25]         H. Yu, D. Livescu, Rayleigh–Taylor instability in cylindrical geometry with compressible fluids, Physics of Fluids, Vol. 20, No. 10, pp. 104103, 2008.

[26]         A. Joshi, M. C. Radhakrishna, N. Rudraiah, Rayleigh–Taylor instability in dielectric fluids, Physics of Fluids, Vol. 22, No. 6, pp. 064102, 2010.

[27]         H. G. Lee, K. Kim, J. Kim, On the long time simulation of the Rayleigh–Taylor instability, International Journal for Numerical Methods in Engineering, Vol. 85, No. 13, pp. 1633-1647, 2011.

[28]         L. Wang, W. Ye, X. He, Density gradient effects in weakly nonlinear ablative Rayleigh-Taylor instability, Physics of Plasmas, Vol. 19, No. 1, pp. 012706, 2012.

[29]         J. Tao, X. He, W. Ye, F. Busse, Nonlinear Rayleigh-Taylor instability of rotating inviscid fluids, Physical Review E, Vol. 87, No. 1, pp. 013001, 2013.

[30]         A. Piriz, Y. Sun, N. Tahir, Rayleigh-Taylor stability boundary at solid-liquid interfaces, Physical Review E, Vol. 88, No. 2, pp. 023026, 2013.

[31]         Y. Murakami, Second harmonic resonance on the marginally neutral curve in the Kelvin-Helmholtz flow, Physics Letters A, Vol. 131, No. 6, pp. 368-372, 1988.

[32]         Y. O. El-Dib, Nonlinear stability of surface waves in magnetic fluids: effect of a periodic tangential magnetic field, Journal of plasma physics, Vol. 49, No. 2, pp. 317-330, 1993.

[33]         Y. O. El-Dib, Nonlinear hydromagnetic Rayleigh–Taylor instability for strong viscous fluids in porous media, Journal of magnetism and magnetic materials, Vol. 260, No. 1-2, pp. 1-18, 2003.

[34]         J.-H. He, Homotopy perturbation technique, Computer methods in applied mechanics and engineering, Vol. 178, No. 3-4, pp. 257-262, 1999.

[35]         J.-H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons & Fractals, Vol. 26, No. 3, pp. 695-700, 2005.

[36]         J.-H. He, Homotopy perturbation method with two expanding parameters, Indian journal of Physics, Vol. 88, No. 2, pp. 193-196, 2014.

[37]         Y. O. El-Dib, Multiple scales homotopy perturbation method for nonlinear oscillators, Nonlinear Sci. Lett. A, Vol. 8, No. 4, pp. 352-364, 2017.

[38]         J.-H. He, Homotopy perturbation method with an auxiliary term, in Proceeding of, Hindawi, pp.

[39]         Y. El-Dib, Stability Analysis of a Strongly Displacement Time-Delayed Duffing Oscillator Using Multiple Scales Homotopy Perturbation Method, Journal of Applied and Computational Mechanics, Vol. 4, No. 4, pp. 260-274, 2018.

[40]         R. Rosensweig, Ferrohydrodynamics Cambridge University Press Cambridge, New York, Melbourne, 1985.

[41]         J. R. Melcher, 1963, Field-coupled surface waves, MIT,

[42]         P. Weidman, M. Goto, A. Fridberg, On the instability of inviscid, rigidly rotating immiscible fluids in zero gravity, Zeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 48, No. 6, pp. 921-950, 1997.

[43]         Y. O. El-Dib, Viscous interface instability supporting free-surface currents in a hydromagnetic rotating fluid column, Journal of plasma physics, Vol. 65, No. 1, pp. 1-28, 2001.

[44]         Y. O. El-Dib, A. Y. Ghaly, Nonlinear interfacial stability for magnetic fluids in porous media, Chaos, Solitons & Fractals, Vol. 18, No. 1, pp. 55-68, 2003.