Effect of Non-Uniform Basic Temperature Gradient on the Onset of Rayleigh-Bénard-Marangoni Electro-Convectionin a Micropolar Fluid

Affiliation(s)

Department of Mathematics, Christ University, Bangalore, India.

Department of Mathematics, BNMIT, Bangalore, India.

Department of Mathematics, Christ University, Bangalore, India.

Department of Mathematics, BNMIT, Bangalore, India.

ABSTRACT

The effects of electric field and non-uniform basic temperature gradient on the onset of Rayleigh-Bénard-Marangoni convection in a micropolar fluid are studied using the Galerkin technique. The eigenvalues are obtained for an upper free/adiabatic and lower rigid/isothermal boundaries. The microrotation is assumed to vanish at the boundaries. A linear stability analysis is performed. The influence of various micropolar fluid parameters and electric Rayleigh number on the onset of convection has been analysed. Six different non-uniform temperature profiles are considered and their comparative influence on onset is discussed.

Cite this paper

T. Joseph, S. Manjunath and S. Pranesh, "Effect of Non-Uniform Basic Temperature Gradient on the Onset of Rayleigh-Bénard-Marangoni Electro-Convectionin a Micropolar Fluid,"*Applied Mathematics*, Vol. 4 No. 8, 2013, pp. 1180-1188. doi: 10.4236/am.2013.48158.

T. Joseph, S. Manjunath and S. Pranesh, "Effect of Non-Uniform Basic Temperature Gradient on the Onset of Rayleigh-Bénard-Marangoni Electro-Convectionin a Micropolar Fluid,"

References

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[2] D. A. Nield, “Surface Tension and Buoyancy Effects in Cellular Convection,” ZAMM, Vol. 17, No. 1, 1966, pp. 131-139. doi:10.1007/BF01594092

[3] N. Rudraiah, V. Ramachandramurthy and O. P. Chandana, “Surface Tension Driven Convection Subjected to Rota tion and Non-Uniform Temperature Gradient,” Interna tional Journal of Heat and Mass Transfer, Vol. 28, No. 8, 1985, pp. 1621-1624. doi:10.1016/0017-9310(85)90264-9

[4] T. Maekawa and I. Tanasawa, “Effect of Magnetic Field and Buoyancy on Onset of Marangoni Convection,” In ternational Journal of Heat and Mass Transfer, Vol. 31, No. 2, 1988, pp. 285-293. doi:10.1016/0017-9310(88)90011-7

[5] G. S. R. Sarma, “Marangoni Convection in a Fluid Layer under the Action of a Transverse Magnetic Field,” Space Research, Vol. 19, 1979, pp. 575-578.

[6] G. S. R. Sarma, “Marangoni Convection in a Liquid Lay er under the Simultaneous Action of a Transverse Mag netic Field and Rotation,” Advances in Space Research, Vol. 1, No. 5, 1981, pp. 55-58. doi:10.1016/0273-1177(81)90151-4

[7] M. Takashima, “Surface Tension Driven Instability in a Horizontal Liquid Layer with a Deformable Free Surface Part I Steady Convection,” Journal of the Physical Soci ety of Japan, Vol. 50, No. 8, 1981, pp. 2745-2750. doi:10.1143/JPSJ.50.2745

[8] S. K. Wilson, “The Effect of a Uniform Magnetic Field on the Onset of Steady Bénard-Marangoni Convection in a Layer of Conducting Fluid,” Journal of Engineering Mathematics, Vol. 27, No. 2, 1993, pp. 161-188. doi:10.1007/BF00127480

[9] S. K. Wilson, “The Effect of a Uniform Magnetic Field on the Onset of Steady Marangoni Convection in a Layer of Conducting Fluid with a Prescribed Heat Flux at Its Lower Boundary,” Physics of Fluids, Vol. 6, No. 11, 1994, pp. 3591-3560. doi:10.1063/1.868417

[10] I. Hashim and S. K. Wilson, “The Onset of Bénard-Ma rangoni Convection in a Horizontal Layer of Fluid,” In ternational Journal of Engineering Science, Vol. 37, No. 5, 1999, pp. 643-662. doi:10.1016/S0020-7225(98)00084-6

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[13] M. Takashima and K. D. Aldridge, “The Stability of a Horizontal Layer of Dielectric Fluid under the Simulta neous Action of a Vertical dc Electric Field and a Vertical Temperature Gradient,” The Quarterly Journal of Mecha nics and Applied Mathematics, Vol. 29, No. 1, 1976, pp. 71-87.

[14] P. J. Stiles, “Electro Thermal Convection in Dielectric Li quids,” Chemical Physics Letters, Vol. 179, No. 3, 1991, pp. 311-315. doi:10.1016/0009-2614(91)87043-B

[15] P. J. Stiles, F. Lin and P. J. Blennerhassett, “Convective Heat Transfer through Polarized Dielectric Liquids,” Phy sics Fluids, Vol. 5, No. 12, 1993, pp. 3273-3279. doi:10.1063/1.858684

[16] P. G. Siddeshwar, “Oscillatory Convection in Viscoelas tic, Ferromagnetic/Dielectric Liquids,” International Jour nal of Modern Physics B, Vol. 16, No. 17-18, 2002, pp. 2629-2635. doi:10.1142/S0217979202012761

[17] P. G. Siddeshwar and A. Abraham, “Effect of Time-Peri odic Boundary Temperatures/Body Force on Rayleigh Bénard Convection in a Ferromagnetic Fluid,” Acta Me chanica, Vol. 161, No. 3-4, 2003, pp. 131-150.

[18] P. G. Siddeshwar and A. Abraham, “Rayleigh-Bénard Con vection in a Dielectric Liquid: Imposed Time-Periodic Boundarytemperatures,” Chamchuri Journal of Mathema tics, Vol. 1, No. 2, 2009, pp. 105-121.

[19] P. G. Siddeshwar and A. T. Y. Chan, “Ferrohydrodyna mic and Electrohydrodynamics Instability in Viscoelastic Liquids: An Analogy,” Proceedings 4th International Conference on Fluid Mechanics, Dalian, 20-30 July 2004, pp. 167-172.

[20] I. S. Shivakumara, M. S. Nagashree and K. Hemalatha, “Electrothermoconvective Instability in a Heat Generat ing Dielectric Fluid Layer,” International Communica tions in Heat and Mass Transfer, Vol. 34, No. 9-10, 2007, pp. 1041-1047. doi:10.1016/j.icheatmasstransfer.2007.05.006

[21] N. Rudraiah, B. M. Shankar and C.-O. Ng, “Electrohy drodynamic Stability of Couple Stress Fluid Flow in a Channel Occupied by a Porous Medium,” Special Topics & Reviews in Porous Media—An International Journal, Vol. 2, No. 1, 2011, pp. 11-22.

[22] P. G. Siddeshwar and D. Radhakrishna, “Linear and Nonlinear Electroconvection under AC Electric Field,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 7, 2012, pp. 2883-2895. doi:10.1016/j.cnsns.2011.11.009

[23] A. C. Eringen, “Theory of Micropolar Fluids,” International Journal of Engineering Science, Vol. 16, No. 1, 1966, p. 1.

[24] B. Datta and V. U. K. Sastry, “Thermal Instability of a Horizontal Layer of Micropolar Fluid Heated from Be low,” International Journal of Engineering Science, Vol. 14, No. 7, 1976, pp. 631-637. doi:10.1016/0020-7225(76)90005-7

[25] P. G. Siddheshwar and S. Pranesh, “Magnetoconvection in a Micropolar Fluid,” International Journal of Engi neering Science, USA, Vol. 36, No. 10, 1998, pp. 1173 1181. doi:10.1016/S0020-7225(98)00013-5

[26] P. G. Siddheshwar and S. Pranesh, “Magnetoconvection in Fluids with Suspended Particles under 1 g and ug,” In ternational Journal of Aerospace Science and Technology, France, Vol. 6, No. 2, 2001, pp. 105-114. doi:10.1016/S1270-9638(01)01144-0

[27] P. G. Siddheshwar and S. Pranesh, “Suction-Injection Effects on the Onset of Rayleigh-Bénard-Marangoni Con vection in a Fluid with Suspended Particles,” Acta Me chanica, Germany, Vol. 152, No. 1-4, 2001, pp. 241-252. doi:10.1007/BF01176958

[28] S. Pranesh and R. Baby, “Effect of Non-Uniform Tem perature Gradient on the Onset of Rayleigh-Bénard Elec tro Convection in a Micropolar Fluid,” Applied Mathe matics, Vol. 3, No. 5, 2012, pp. 442-450. doi:10.4236/am.2012.35067

[29] S. Pranesh and A. Kumar, “Effect of Non-Uniform Basic Concentration Gradient on the Onset of Double-Diffusive Convection in Micropolar Fluid,” Applied Mathematics, Vol. 3, No. 5, 2012, pp. 417-424. doi:10.4236/am.2012.35064

[1] J. R. A. Pearson, “On Convection Cells Induced by Sur face Tension,” Journal of Fluid Mechanics, Vol. 4, No. 5, 1958, pp. 489-500. doi:10.1017/S0022112058000616

[2] D. A. Nield, “Surface Tension and Buoyancy Effects in Cellular Convection,” ZAMM, Vol. 17, No. 1, 1966, pp. 131-139. doi:10.1007/BF01594092

[3] N. Rudraiah, V. Ramachandramurthy and O. P. Chandana, “Surface Tension Driven Convection Subjected to Rota tion and Non-Uniform Temperature Gradient,” Interna tional Journal of Heat and Mass Transfer, Vol. 28, No. 8, 1985, pp. 1621-1624. doi:10.1016/0017-9310(85)90264-9

[4] T. Maekawa and I. Tanasawa, “Effect of Magnetic Field and Buoyancy on Onset of Marangoni Convection,” In ternational Journal of Heat and Mass Transfer, Vol. 31, No. 2, 1988, pp. 285-293. doi:10.1016/0017-9310(88)90011-7

[5] G. S. R. Sarma, “Marangoni Convection in a Fluid Layer under the Action of a Transverse Magnetic Field,” Space Research, Vol. 19, 1979, pp. 575-578.

[6] G. S. R. Sarma, “Marangoni Convection in a Liquid Lay er under the Simultaneous Action of a Transverse Mag netic Field and Rotation,” Advances in Space Research, Vol. 1, No. 5, 1981, pp. 55-58. doi:10.1016/0273-1177(81)90151-4

[7] M. Takashima, “Surface Tension Driven Instability in a Horizontal Liquid Layer with a Deformable Free Surface Part I Steady Convection,” Journal of the Physical Soci ety of Japan, Vol. 50, No. 8, 1981, pp. 2745-2750. doi:10.1143/JPSJ.50.2745

[8] S. K. Wilson, “The Effect of a Uniform Magnetic Field on the Onset of Steady Bénard-Marangoni Convection in a Layer of Conducting Fluid,” Journal of Engineering Mathematics, Vol. 27, No. 2, 1993, pp. 161-188. doi:10.1007/BF00127480

[9] S. K. Wilson, “The Effect of a Uniform Magnetic Field on the Onset of Steady Marangoni Convection in a Layer of Conducting Fluid with a Prescribed Heat Flux at Its Lower Boundary,” Physics of Fluids, Vol. 6, No. 11, 1994, pp. 3591-3560. doi:10.1063/1.868417

[10] I. Hashim and S. K. Wilson, “The Onset of Bénard-Ma rangoni Convection in a Horizontal Layer of Fluid,” In ternational Journal of Engineering Science, Vol. 37, No. 5, 1999, pp. 643-662. doi:10.1016/S0020-7225(98)00084-6

[11] R. J. Turnbull, “Electro Convective Instability with a Sta bilizing Temperature Gradient. I. Theory,” Physics of Fluids, Vol. 11, No. 12, 1968, pp. 2588-2596. doi:10.1063/1.1691864

[12] R. J. Turnbull and R. J. Melcher, “Electrohydrodynamic Rayleigh-Taylor Bulk Instability,” Physics of Fluids, Vol. 12, No. 6, 1969, pp. 1160-1166. doi:10.1063/1.1692646

[13] M. Takashima and K. D. Aldridge, “The Stability of a Horizontal Layer of Dielectric Fluid under the Simulta neous Action of a Vertical dc Electric Field and a Vertical Temperature Gradient,” The Quarterly Journal of Mecha nics and Applied Mathematics, Vol. 29, No. 1, 1976, pp. 71-87.

[14] P. J. Stiles, “Electro Thermal Convection in Dielectric Li quids,” Chemical Physics Letters, Vol. 179, No. 3, 1991, pp. 311-315. doi:10.1016/0009-2614(91)87043-B

[15] P. J. Stiles, F. Lin and P. J. Blennerhassett, “Convective Heat Transfer through Polarized Dielectric Liquids,” Phy sics Fluids, Vol. 5, No. 12, 1993, pp. 3273-3279. doi:10.1063/1.858684

[16] P. G. Siddeshwar, “Oscillatory Convection in Viscoelas tic, Ferromagnetic/Dielectric Liquids,” International Jour nal of Modern Physics B, Vol. 16, No. 17-18, 2002, pp. 2629-2635. doi:10.1142/S0217979202012761

[17] P. G. Siddeshwar and A. Abraham, “Effect of Time-Peri odic Boundary Temperatures/Body Force on Rayleigh Bénard Convection in a Ferromagnetic Fluid,” Acta Me chanica, Vol. 161, No. 3-4, 2003, pp. 131-150.

[18] P. G. Siddeshwar and A. Abraham, “Rayleigh-Bénard Con vection in a Dielectric Liquid: Imposed Time-Periodic Boundarytemperatures,” Chamchuri Journal of Mathema tics, Vol. 1, No. 2, 2009, pp. 105-121.

[19] P. G. Siddeshwar and A. T. Y. Chan, “Ferrohydrodyna mic and Electrohydrodynamics Instability in Viscoelastic Liquids: An Analogy,” Proceedings 4th International Conference on Fluid Mechanics, Dalian, 20-30 July 2004, pp. 167-172.

[20] I. S. Shivakumara, M. S. Nagashree and K. Hemalatha, “Electrothermoconvective Instability in a Heat Generat ing Dielectric Fluid Layer,” International Communica tions in Heat and Mass Transfer, Vol. 34, No. 9-10, 2007, pp. 1041-1047. doi:10.1016/j.icheatmasstransfer.2007.05.006

[21] N. Rudraiah, B. M. Shankar and C.-O. Ng, “Electrohy drodynamic Stability of Couple Stress Fluid Flow in a Channel Occupied by a Porous Medium,” Special Topics & Reviews in Porous Media—An International Journal, Vol. 2, No. 1, 2011, pp. 11-22.

[22] P. G. Siddeshwar and D. Radhakrishna, “Linear and Nonlinear Electroconvection under AC Electric Field,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 7, 2012, pp. 2883-2895. doi:10.1016/j.cnsns.2011.11.009

[23] A. C. Eringen, “Theory of Micropolar Fluids,” International Journal of Engineering Science, Vol. 16, No. 1, 1966, p. 1.

[24] B. Datta and V. U. K. Sastry, “Thermal Instability of a Horizontal Layer of Micropolar Fluid Heated from Be low,” International Journal of Engineering Science, Vol. 14, No. 7, 1976, pp. 631-637. doi:10.1016/0020-7225(76)90005-7

[25] P. G. Siddheshwar and S. Pranesh, “Magnetoconvection in a Micropolar Fluid,” International Journal of Engi neering Science, USA, Vol. 36, No. 10, 1998, pp. 1173 1181. doi:10.1016/S0020-7225(98)00013-5

[26] P. G. Siddheshwar and S. Pranesh, “Magnetoconvection in Fluids with Suspended Particles under 1 g and ug,” In ternational Journal of Aerospace Science and Technology, France, Vol. 6, No. 2, 2001, pp. 105-114. doi:10.1016/S1270-9638(01)01144-0

[27] P. G. Siddheshwar and S. Pranesh, “Suction-Injection Effects on the Onset of Rayleigh-Bénard-Marangoni Con vection in a Fluid with Suspended Particles,” Acta Me chanica, Germany, Vol. 152, No. 1-4, 2001, pp. 241-252. doi:10.1007/BF01176958

[28] S. Pranesh and R. Baby, “Effect of Non-Uniform Tem perature Gradient on the Onset of Rayleigh-Bénard Elec tro Convection in a Micropolar Fluid,” Applied Mathe matics, Vol. 3, No. 5, 2012, pp. 442-450. doi:10.4236/am.2012.35067

[29] S. Pranesh and A. Kumar, “Effect of Non-Uniform Basic Concentration Gradient on the Onset of Double-Diffusive Convection in Micropolar Fluid,” Applied Mathematics, Vol. 3, No. 5, 2012, pp. 417-424. doi:10.4236/am.2012.35064