×

Onset of electroconvective instability of Oldroydian viscoelastic liquid layer in Brinkman porous medium. (English) Zbl 1168.76322

Summary: The problem of the onset of electrohydrodynamic instability in a horizontal layer of Oldroydian viscoelastic dielectric liquid through Brinkman porous medium under the simultaneous action of a certical ac electric field and a vertical temperature gradient is analyzed. Applying linear stability theory, we derive an equation of eight order. Under somewhat suitable boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Both the cases of stationary and oscillatory instabilities are discussed if the liquid layer is heated from below or above. The effects of the porosity of porous medium, the medium permeability, the Prandtl number, the ratio of retardation time to relaxation time, the elastic number, in the presence or absence of Rayleigh number are shown graphically for both cases. Some of the known results are derived as special cases. The electrical force has been shown to be the sole agency causing instability of the considered system since it is much more important than the buoyancy force even if the medium is porous.

MSC:

76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76E06 Convection in hydrodynamic stability
76A10 Viscoelastic fluids
76S05 Flows in porous media; filtration; seepage
80A20 Heat and mass transfer, heat flow (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Nield D.A. and Bejan A. (1999). Convection in Porous Media, 2nd edn. Springer, New York · Zbl 0924.76001
[2] Pop I. and Ingham D.B. (2001). Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon Press, Oxford
[3] Kumar P. and Singh M. (2006). On a viscoelastic fluid heated from below in a porous medium. J. NonEquilib. Thermodyn. 31: 189–203 · Zbl 1116.80010 · doi:10.1515/JNETDY.2006.009
[4] Lapwood E.R. (1948). Convection of a fluid in a porous medium. Proc. Cambridge Philos. Soc. 44: 508–521 · Zbl 0032.09203 · doi:10.1017/S030500410002452X
[5] Weber J.E. (1974). Convection in porous medium with horizontal and vertical temperature gradients. Int. J. Heat Mass Transf. 17: 241–248 · Zbl 0325.76125 · doi:10.1016/0017-9310(74)90085-4
[6] Cebeci T. and Bradshow P. (1984). Physical and Computational Aspects of Convective Heat Transfer. Springer, New York · Zbl 0545.76090
[7] Othman M.I.A. (2001). Electrohydrodynamic stability in a horizontal viscoelastic fluid layer in the presence of a vertical temperature gradient. Int. J. Eng. Sci. 39: 1217–1232 · doi:10.1016/S0020-7225(00)00092-6
[8] Othman M.I.A. and Zaki S.A. (2003). The effect of thermal relaxation time on a electrohydrodynamic viscoelastic fluid layer heated from below. Can. J. Phys. 81: 779–787 · doi:10.1139/p03-013
[9] Nepomnyashchy A.A., Simaanovskii I.B. and Legros J.-C. (2006). Interfacial Convection in Multilayer Systems. Springer, New York · Zbl 1100.76002
[10] Hennenberg H., Saghir M.Z., Rednikov A. and Legros J.C. (1997). Porous media and the Bénard Marangoni problen. Transp. Porous Media 27: 327–355 · doi:10.1023/A:1006564129233
[11] Vafai K. (eds) (2000). Handbook of Porous Media. Marcel Dekker, New York · Zbl 0954.00016
[12] Hsu C.T. and Cheng P. (1985). The Brinkman model for the natural convection about a semi-infinite vertical flat plate in a porous medium. Int. J. Heat Mass Transf. 28: 683–697 · Zbl 0576.76079 · doi:10.1016/0017-9310(85)90190-5
[13] Nazzar R., Amin N., Filip D. and Pop I. (2003). The Brinkman model for the mixed convection boundary layer flow past a horizontal circular cylinder in a porous medium. Int. J. Heat Mass Transf. 46: 3167–3178 · Zbl 1121.76391 · doi:10.1016/S0017-9310(03)00122-4
[14] El-Sayed, M.F.: The Brinkman model for the electrothermal instability of viscous rotating dielectric fluid in porous medium. J. NonEquilib. Thermodyn. (2007) (in press)
[15] Payne L.E. and Straughan B. (1998). Structural stability for the Darcy equations of flow in porous media. Proc. R. Soc. Ldn A 454: 1691–1698 · Zbl 0912.76088 · doi:10.1098/rspa.1998.0227
[16] Landau L.D. and Lifshitz E.M. (1960). Electrodynamics of Continuous Media. Pergamon Press, New York · Zbl 0122.45002
[17] Melcher J.R. (1981). Continuum Electromechanics. MIT Press, Cambridge
[18] Turnbull R.J. (1968). Electrohydrodynamic instability with a stabilizing temperature gradient. I. Theory. Phys. Fluids 11: 2588–2596 · Zbl 0191.26302 · doi:10.1063/1.1691864
[19] Turnbull R.J. (1968). Electrohydrodynamic instability with a stabilizing temperature gradient. II. Experimental results. Phys. Fluids 11: 2597–2603 · Zbl 0191.26302 · doi:10.1063/1.1691865
[20] Turnbull R.J. and Melcher J.R. (1969). Electrohydrodynamic Rayleigh–Taylor bulk instability. Phys. Fluids 12: 1160–1166 · Zbl 0177.56303 · doi:10.1063/1.1692646
[21] Turnbull R.J. (1969). Effect of dielectrophoretic forces on the B énard instability. Phys. Fluids 12: 1809–1815 · doi:10.1063/1.1692745
[22] Turnbull R.J. (1970). Thermal diffusion effects on the electrohydrodynamic Rayleigh–Taylor bulk instability. Phys. Fluids 13: 2615–2616 · doi:10.1063/1.1692836
[23] Takashima M. (1976). The effect of rotation on electrohydrodynamic instability. Can. J. Phys. 54: 342–347
[24] Ingham D.B., Pop I. (eds) (1998). Transport Phenomena in Porous Media. Pergamon Press, Oxford · Zbl 0918.76002
[25] El-Sayed M.F. (1999). Electrohydrodynamic interfacial stability conditions in the presence of heat and mass transfer and oblique electric fields. Z. Naturforsch. A 54: 470–476
[26] El-Sayed M.F., Mohamed A.A. and Metwaly T.M.N. (2005). Thermohydrodynamic instabilities of conducting liquid jets in the presence of time-dependent transverse electric fields. Phys. A 345: 367–394
[27] El-Sayed M.F., Mohamed A.A. and Metwaly T.M.N. (2007). Stability of cylindrical conducting fluids with heat and mass transfer in longitudinal periodic electric field. Phys. A 379: 59–80 · doi:10.1016/j.physa.2006.12.053
[28] El-Sayed M.F., Mohamed A.A. and Metwaly T.M.N. (2006). Effect of general applied electric field on comducting liquid jets instabilities in the presence of heat and mass transfer. Appl. Math. Comput. 172: 1078–1102 · Zbl 1093.76022 · doi:10.1016/j.amc.2005.03.009
[29] El-Sayed M.F. and Syam M.I. (2007). Numerical study for the electrified instability of viscoelastic cylindrical dielectric fluid film surrounded by a conducting gas. Phys. A 377: 381–400 · doi:10.1016/j.physa.2006.11.059
[30] El-Sayed M.F. (2006). Electrohydrodynamic intability of dielectric fluid layer between two semi-infinite identical conducting fluids in porous medium. Phys. A 367: 25–41 · doi:10.1016/j.physa.2005.10.057
[31] Oldroyd J.G. (1950). On the formulation of rheological equations of state. Proc. R. Soc. Ldn A 200: 523–541 · Zbl 1157.76305 · doi:10.1098/rspa.1950.0035
[32] Oldroyd J.G. (1958). Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. R. Soc. Ldn A 245: 278–297 · Zbl 0080.38805 · doi:10.1098/rspa.1958.0083
[33] Takashima M. and Ghosh A.K. (1979). Electrohydrodynamic instability in a viscoelastic fluid layer. J. Phys. Soc. Jpn 47: 1717–1722 · doi:10.1143/JPSJ.47.1717
[34] Nield D.A. (1991). Convection in porous medium with inclined temperature gradient. Int. J. Heat Mass Transf. 34: 87–92 · Zbl 0728.76105 · doi:10.1016/0017-9310(91)90176-F
[35] Nield D.A. (1994). Convection in porous medium with inclined temperature gradient: additional results. Int. J. Heat Mass Transf. 37: 3021–3025 · Zbl 0900.76600 · doi:10.1016/0017-9310(94)90356-5
[36] Nield D.A. (1998). Convection in porous medium with inclined temperature gradient and vertical throughflow. Int. J. Heat Mass Transf. 41: 241–243 · Zbl 0939.76590 · doi:10.1016/S0017-9310(97)00103-8
[37] Kaloni P.N. and Qiao Z.C. (1997). Non-linear stability of convection in a porous medium with inclined temperature gradient. Int. J. Heat Mass Transf. 40: 1611–1615 · Zbl 0922.76190 · doi:10.1016/S0017-9310(96)00204-9
[38] Siddheshwar P.G. and Srikrishna C.V. (2001). Rayleigh–Bénard convection in a viscoelastic fluid filled high porosity medium with non-uniform basic temperature gradient. Int. J. Math. Math Sci. 25: 609–619 · Zbl 1029.76019 · doi:10.1155/S0161171201001028
[39] Chandrasekhar S. (1981). Hydrodynamic and Hydromagnetic Stability. Dover Publications, New York · Zbl 0142.44103
[40] Yoon D.Y., Kim M.C. and Choi C.K. (2004). Onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid. Transp. Porous Media 55: 275–284 · doi:10.1023/B:TIPM.0000013328.69773.a1
[41] Takashima M. (1972). Thermal instability in a viscoelastic fluid layer I. J. Phys. Soc. Jpn 33: 511–518 · doi:10.1143/JPSJ.33.511
[42] Vest C.M. and Aspaci V.S. (1969). Overstability of a viscoelastic fluid layer heated from below. J. Fluid Mech. 36: 613–623 · Zbl 0175.24201 · doi:10.1017/S0022112069001881
[43] Malashetty M.S., Siddheshwar P.G. and Swamy M. (2006). Effect of thermal modulation on the onset of convection in a viscoelastic fluid saturated porous layer. Transp. Porous Media 62: 55–79 · doi:10.1007/s11242-005-4507-y
[44] Malashetty M.S. and Gaikwad S.M. (2003). Onset of convective instabilities in a binary liquid mixtures with fast chemical reactions in a porous medium. Heat Mass Transf. 39: 415–420
[45] Toms B.A. and Strawbridge G.J. (1953). Elastic and viscous properties of dilute solutions of polymethyl methaylate in organic liquids. Trans. Faraday Soc. 49: 1225–1232 · doi:10.1039/tf9534901225
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.