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Numerical and analytical investigations of thermosolutal instability in rotating Rivlin-Ericksen fluid in porous medium with Hall current. (English) Zbl 1400.76080

Summary: Numerical and analytical investigations of the thermosolutal instability in a viscoelastic Rivlin-Ericksen fluid are carried out in the presence of a uniform vertical magnetic field to include the Hall current with a uniform angular velocity in a porous medium. For stationary convection, the stable solute gradient parameter and the rotation have stabilizing effects on the system, whereas the magnetic field and the medium permeability have stabilizing or destabilizing effects on the system under certain conditions. The Hall current in the presence of rotation has stabilizing effects for sufficiently large Taylor numbers, whereas in the absence of rotation, the Hall current always has destabilizing effects. These effects have also been shown graphically. The viscoelastic effects disappear for stationary convection. The stable solute parameter, the rotation, the medium permeability, the magnetic field parameter, the Hall current, and the vis-coelasticity introduce oscillatory modes into the system, which are non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.

MSC:

76S05 Flows in porous media; filtration; seepage
76E06 Convection in hydrodynamic stability
76U05 General theory of rotating fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York (1981)
[2] Brandt, H. and Fernando, H. J. S. Double-Diffusive Convection, American Geophysical Union Monograph, Washington, D.C. (1996)
[3] Veronis, G, On finite amplitude instability in thermohaline convection, Journal of Marine Research, 23, 1-17, (1965)
[4] Nield, D A, The thermohaline Rayleigh-Jeffreys problem, Journal of Fluid Mechanics, 29, 545-558, (1967) · doi:10.1017/S0022112067001028
[5] Nield, D A, Onset of thermohaline convection in porous medium, Water Resources Research, 4, 553-560, (1965) · doi:10.1029/WR004i003p00553
[6] Tabor, H; Matz, R, Solar pond project, Solar Energy, 9, 177-180, (1965) · doi:10.1016/0038-092X(65)90044-7
[7] Shirtcliffe, T G L, Lake bonney, antarctica: cause of the elevated temperatures, Journal of Geophysical Research, 69, 5257-5268, (1964) · doi:10.1029/JZ069i024p05257
[8] Sherman, A. and Sutton, G.W. Magnetohydrodynamics,Northwestern University Press, Evanston (1962)
[9] Gupta, A S, Hall effects on thermal instability, Romanian Journal of Pure and Applied Mathematics, 12, 665-677, (1967) · Zbl 0148.45807
[10] Sharma, R C; Sunil; Chand, S, Thermosolutal instability of Rivlin-ericksen rotating fluid in porous medium, Indian Journal of Pure and Applied Mathematics, 29, 433-440, (1998) · Zbl 0906.76024
[11] Reiner, M, A mathematical theory of dilatancy, American Journal of Mathematics, 67, 350-362, (1945) · Zbl 0063.06464 · doi:10.2307/2371950
[12] Rivlin, R S, The hydrodynamics of non-Newtonian fluids, I. Proceedings of the royal society of London, series A, Mathematical and Physical Sciences, A193, 260-281, (1948) · Zbl 0031.43001 · doi:10.1098/rspa.1948.0044
[13] Ericksen, J L, Characterstic surfaces of equations of motion of non-Newtonian fluids, Zeitschrift fjr Angewandte Mathematik und Physik, 4, 260-267, (1953) · Zbl 0051.18203 · doi:10.1007/BF02074635
[14] Truesdell, C. and Noll, W. The Noninear Field Theories of Mechanics, Handbunch der Physik III/3, Springer, Berlin/Heidelberg/New York (1965) · Zbl 0137.19501
[15] Fredricksen, A. G. Principles and Applications of Rheology, Prentice-Hall Inc., New Jersey (1964)
[16] Joseph, D. D. Stability of Fluid Motion II, Springer-Verlag, New York (1976) · Zbl 0345.76023
[17] Rivlin, R S; Ericksen, J L, Stress-deformation relations for isotropic materials, Journal of Rational Mechanics and Analysis, 4, 323-425, (1955) · Zbl 0064.42004
[18] Garg, A; Srivastava, R K; Singh, K K, Drag on sphere oscillating dusty Rivlin-ericksen elastico-viscous liquid, Proceedings of the Indian National Science Academy, 64A, 355-363, (1994) · Zbl 0928.76013
[19] Sharma, V; Kumar, S, Magnetogravitational instability of a thermally conducting rotating viscoelastic fluid with Hall current, Indian Journal of Pure and Applied Mathematics, 31, 1559-1578, (2000) · Zbl 1012.76029
[20] Kumar, S, The instability of streaming elastico-viscous fluids in porous medium in hydromagnetics, Journal of Applied Mathematics and Mechanics, 7, 33-45, (2011) · Zbl 1431.76126
[21] Sharma, R C; Bhardwaj, V K, Thermosolutal convection in a rotating fluid in hydromagnetics in porous medium, Acta Physica Hungarica, 73, 59-66, (1993)
[22] Sharma, R C; Kumari, V, Effect of magnetic field and rotation on thermosolutal convection in porous medium, Japan Journal of Industrial and Applied Mathematics, 9, 79-90, (1992) · Zbl 0743.76097 · doi:10.1007/BF03167195
[23] Sharma, R C; Kango, S K, Thermal convection in Rivlin-ericksen elastico-viscous fluid in porous medium in hydromagnetics, Czechoslovak Journal of Physics, 49, 197-203, (1999) · Zbl 1044.76518 · doi:10.1023/A:1022849927803
[24] Sharma, R C; Sunil, a; Chand, S, Hall effect on thermal instability of Rivlin-ericksen fluid, Indian Journal of Pure and Applied Mathematics, 31, 49-59, (2000) · Zbl 0960.76032
[25] Kumar, S; Sharma, V; Kishor, K, Stability of stratified Rivlin-ericksen fluid in the presence of horizontal magnetic field and uniform horizontal rotation, International Journal of Mathematical Sciences and Applications, 4, 263-274, (2010)
[26] Rana, G C; Kumar, S, Thermal instability of Rivlin-ericksen elastico-viscous rotating fluid permeating with suspended particles under variable gravity field in porous medium, Studia Geotechnica et Mechanica, 32, 39-54, (2010)
[27] Kumar, S; Sharma, V; Kishor, K, Effect of suspended particles on thermosolutal instability in Rivlin-ericksen elastico-viscous fluid, (2010), New Delhi
[28] Kumar, S; Gupta, D; Jaswal, R; Rana, G C, Thermal instability of elastico-viscous compressible fluid in the presence of Hall current, Studia Geotechnica et Mechanica, 33, 61-72, (2011)
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