Abstract
In the view of viscous potential flow theory, the hydromagnetic stability of the interface between two infinitely conducting, incompressible plasmas, streaming parallel to the interface and subjected to a constant magnetic field parallel to the streaming direction will be considered. The plasmas are flowing through porous media between two rigid planes and surface tension is taken into account. A general dispersion relation is obtained analytically and solved numerically. For Kelvin-Helmholtz instability problem, the stability criterion is given by a critical value of the relative velocity. On the other hand, a comparison between inviscid and viscous potential flow solutions has been made and it has noticed that viscosity plays a dual role, destabilizing for Rayleigh-Taylor problem and stabilizing for Kelvin-Helmholtz. For Rayleigh-Taylor instability, a new dispersion relation has been obtained in terms of a critical wave number. It has been found that magnetic field, surface tension, and rigid planes have stabilizing effects, whereas critical wave number and porous media have destabilizing effects.
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References
A. Lobatchev, Hydrodynamics of lnertial Fusion Capsules: Feed out and Deceleration Phase Instability, PhD Thesis, University of Rochester, New York (2000).
B. K. Shivamoggi, A generalized Theory of Stability of Superposed Fluids in Hydromagnetics, Astrophysics and Space Science, 84 (1981), 477–484.
K. Adham-Khodaparast, M. Kawaji and B. N. Antar, The Rayleigh-Taylor and Kelvin-Helmholtz Stability of A viscous Liquid-Vapor Interface with Heat and Mass Transfer, Phys. Fluids, 7 (1995), 359–364.
G. M. Moatirnid and M. H. Obied Allah, Electrohydrodynamic Linear Stability of Finitely Conducting Flows Through Porous Fluids with Mass and Heat Transfer, Applied Mathematical Modelling, 34 (2010), 3118–3129.
M. H. Obied Allah and A. A. Yahia, Nonlinear Rayleigh Taylor Instability in The Presence of Magnetic Field, Mass and Heat Transfer, Astrophysics and Space Science, 181 (1991), 183–194.
M. H. Obied Allah and M. A. Ibraheem, Thermal Effects on Linear and Nonlinear Rayleigh-Taylor Stability in The Presence of Mass, Heat Transfer and Magnetic Field, Indian J. Pure and Appl. Math., 31 (2000), 1545.
M. H. Obied Allah, Kelvin-Helmholtz Instability in The Presence of Mass, Heat Transfer, Porous Media, Surface Tension and Magnetic Field, Mechanics and Mechanical Engineering 9 (2005), 179–185.
P. Kumar, H. Mohan and R. Lal, Effect of Magnetic Field on Thermal Instability of A rotating Rivlin Ericksen Viscoelastic Fluid, Int. J. Mathematics and Mathematical Sciences, 2006 (2006), 1–10.
Y. O. El-Dib and A. Ghaly, Destabilizing Effect of Time-Dependent Oblique Magnetic Field on Magnetic Fluids Streaming in Porous Media, J. Colloid and Interface Sci., 269 (2004), 224–239.
A. M. AL-Khateeb and N. M. Laham, Stability Criteria of Rayleigh-Taylor Modes in Magnetized Plasma. Contrib. Plasma Phys., 43 (2003), 25–32.
A. R. F. Elhefhawy, Nonlinear Rayleigh-Taylor Instability in Magnetic Fluids with Uniform Horizontal and Vertical Magnetic Fields, ZAMP 44 (1993), 495–509.
D. D. Joseph, Viscous Potential How, J. Fluid Mechanics, 479 (2003), 191–197.
T. Funada and D. D. Joseph, Viscous Potential Flow Analysis of Kelvin-Helmholtz Instability in A channel, J. Fluid Mech., 445 (2001), 263–283.
D. D. Joseph, J. Belanger and G. S. Beavers, Breakup of A liquid Drop Suddenly Exposed to A high Speed Airstream, International Journal of Multiphase Flows, 25 (1999), 1263–1303.
R. Asthana and G. S. Agrawal, Viscous Potential Flow Analysis of Kelvin-Helmholtz Instability with Mass Transfer and Vaporization, Physica A, 382 (2007), 389–404.
M. A. Sirwah, Nonlinear Kelvin Helmholtz Instability of Magnetized Surface Waves on A subsonic Gas Viscous Potential Liquid Interface, Physica A, 375 (2007), 381400.
A. K. Elcoot, Electroviscous Potential Flow of Nonlinear Analysis of Capillary Instability, European J. Mechanics B/ Fluids, 26 (2007), 431–443.
P. K. Bhatia, Rayleigh Taylor Instability of Two Viscous Suspended Conducting Fluids in The Presence of A uniform Horizontal Magnetic Field, Nuov. Cim., 19B (1974), 161–168.
P. Kumar, Rayleigh-Taylor Instability of Rivlin-Ericksen Elastico-Viscous Fluids in The Presence of Suspended Particles Through Porous Media, Indian J. Pure and Applied Mathematics, 31 (2000), 533–539.
R. C. Sunil and R. S. Chandel, On Superposed Couple-Stress Fluids in Porous Medium in Hydromagnetics, Z. Naturforsch, 57a (2002), 955–960.
J. Peakall, Surface Tension in Small Hydraulic River Models-The Significance of The Weber Number, Journal of Hydrology (NZ), 35(2) (1996), 199–212.
A. Chadwiek and J. Marfett, Hydrulics in Civil Engineering, Allen and Unwin, London (1986).
J. J. Sharp, Hydraulic Modeling, Butterworth’s, London (1981).
B. S. Massey, Mechanics of Fluids, Sixth Edition, Chapman Hall, London (1989).
M. K. Awasthi and G. S. Agrawal, Viscous Potential Flow Analysis of Rayleigh-Taylor Instabihty With Mass and Heat Transfer, International. Journal of Applied Mathematics and Mechanics, 7(12) (2010), 73–84.
M. H. Obied Allah, Viscous Potential Flow Analysis of Interfacial Stability With Mass Transfer Through Porous Media, Applied Mathematics and Computation, 217 (2011), 7920–7931.
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Obied Allah, M.H. Viscous potential flow analysis of magnetohydrodynamic interfacial stability through porous media. Indian J Pure Appl Math 44, 419–441 (2013). https://doi.org/10.1007/s13226-013-0022-y
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DOI: https://doi.org/10.1007/s13226-013-0022-y