The Onset of Ferromagnetic Convection in a Micropolar Ferromagnetic Fluid Layer Heated from Below

Affiliation(s)

Department of Mathematics, Ambedkar Institute of Technology, Bangalore, India;.

UGC-CAS in Fluid Mechanics, Department of Mathematics, Bangalore University, Bangalore, India.

Department of Mathematics, East Point College of Engineering for Women, Bangalore, India..

Department of Mathematics, Ambedkar Institute of Technology, Bangalore, India;.

UGC-CAS in Fluid Mechanics, Department of Mathematics, Bangalore University, Bangalore, India.

Department of Mathematics, East Point College of Engineering for Women, Bangalore, India..

Abstract

The onset of ferromagnetic convection
in a micropolar ferromagnetic fluid layer heated from below in the presence of
a uniform applied vertical magnetic field has been investigated. The
rigid-isothermal boundaries of the fluid layer are considered to be either
paramagnetic or ferromagnetic and the eigenvalue problem is solved numerically
using the Galerkin method. It is noted that the paramagnetic boundaries with
large magnetic susceptibility *χ* delays the onset
of ferromagnetic convection the most when compared to very low magnetic
susceptibility as well as ferromagnetic boundaries. Increase in the value of
magnetic parameter *M*_{1} and spin diffusion (couple stress) parameter *N*_{3} is to hasten, while increase in the value of
coupling parameter *N*_{1} and micropolar heat conduction parameter *N*_{5} is to delay the onset of ferromagnetic
convection. Further, increase in the value of *M*_{1}, *N*_{1}, *N*_{5} and *χ** *as well as decrease
in *N*_{3} is to diminish the size of convection cells.

Cite this paper

C. Nanjundappa, I. Shivakumara and K. Srikumar, "The Onset of Ferromagnetic Convection in a Micropolar Ferromagnetic Fluid Layer Heated from Below,"*Journal of Electromagnetic Analysis and Applications*, Vol. 5 No. 3, 2013, pp. 120-133. doi: 10.4236/jemaa.2013.53020.

C. Nanjundappa, I. Shivakumara and K. Srikumar, "The Onset of Ferromagnetic Convection in a Micropolar Ferromagnetic Fluid Layer Heated from Below,"

References

[1] R. E. Rosensweig, “Ferrohydrodynamics,” Cambridge University Press, London, 1985.

[2] B. M. Berkovsky, V. F. Medvedev and M. S. Krakov, “Magnetic Fluids, Engineering Applications,” Oxford University Press, Oxford, 1993.

[3] R. Hergt, W. Andr?, C. G. Ambly, I. Hilger, U. Richter and H. G. Schmidt, “Physical Limitations of Hypothermia Using Magnetite Fine Particles,” IEEE Transictions of Magnetics, Vol. 34, No. 5, 1998, pp. 3745-3754.
doi:10.1109/20.718537

[4] J. L. Neuringer and R. E. Rosensweig, “Magnetic Fluids,” Physics of Fluids, Vol. 7, No. 12, 1964, pp. 1927-1937.
doi:10.1063/1.1711103

[5] B. A. Finlayson, “Convective Instability of Ferromagnetic Fluids,” Journal of Fluid Mechanics, Vol. 40, No. 4, 1970, pp. 753-767. doi:10.1017/S0022112070000423

[6] K. Gotoh and M. Yamada, “Thermal Convection in a Horizontal Layer of Magnetic Fluids,” Journal of Physics, Society of Japan, Vol. 51, 1982, pp. 3042-3048.
doi:10.1143/JPSJ.51.3042

[7] P. J. Stiles, F. Lin and P. J. Blennerhassett, “Heat Transfer through Weakly Magnetized Ferrofluids,” Journal of Colloidal and Interface Science, Vol. 151, No. 1, 1992, pp. 95-101. doi:10.1016/0021-9797(92)90240-M

[8] P. J. Blennerhassett, F. Lin and P. J. Stiles, “Heat Transfer through Strongly Magnetized Ferrofluids,” Proceeding of Royal Society A: A Mathematical, Physical and Engineering Sciences, Vol. 433, 1991, pp. 165-177.

[9] Sunil and A. Mahjan, “A Nonlinear Stability Analysis for Magnetized Ferrofluid Heated from Below,” Proceeding of Royal Society of London. A Mathematical, Physical and Engineering Sciences, Vol. 464, No. 2089, 2008, pp. 83-98. doi:10.1098/rspa.2007.1906

[10] C. E. Nanjundappa and I. S. Shivakumara, “Effect of Velocity and Temperature Boundary Conditions on Convective Instability in a Ferrofluid Layer,” ASME Journal of Heat Transfer, Vol. 130, 2008, Article ID: 104502.

[11] M. I. Shliomis, “Convective Instability of Magnetized Ferrofluids: Influence of Magneto-Phoresis and Soret Effect,” Thermal Non-Equilibrium Phenomena: Fluid Mixtures, Vol. 584, 2002, pp. 355-371.

[12] M. I. Shliomis and B. L. Smorodin, “Convective Instability of Magnetized Ferrofluids,” Journal of Magnetism and Magnetic Materials, Vol. 252, 2002, pp. 197-202.
doi:10.1016/S0304-8853(02)00712-6

[13] S. Odenbach, “Recent Progress in Magnetic Fluid Research,” Journal of Physics: Condensed Matter, Vol. 16, 2004, pp. 1135-1150. doi:10.1088/0953-8984/16/32/R02

[14] P. J. Stiles and M. Kagan, “Thermoconvective Instability of a Ferrofluid in a Strong Magnetic Field,” Journal Colloidal and Interface Science, Vol. 134, 1990, pp. 435- 448.

[15] P. N. Kaloni and J. X. Lou, “Convective Instability of Magnetic Fluids under Alternating Magnetic Fields,” Physical Review E, Vol. 71, 2004, Article ID: 066311.

[16] P. Ram and K. Sharma, “Revolving Ferrofluid Flow under the Influence of MFD Viscosity and Porosity with Rotating Disk,” Journal of Electromagnetic Analysis and Applications, Vol. 3, 2011, pp. 378-386.
doi:10.4236/jemaa.2011.39060

[17] A. C. Eringen, “Simple Microfluids,” International Journal of Engineering Sciences, Vol. 2, No. 2, 1964, pp. 205- 217. doi:10.1016/0020-7225(64)90005-9

[18] G. Lebon and C. Perez-Garcia, “Convective Instability of a Micropolar Fluid Layer by the Method of Energy,” International Journal of Engineering Sciences, Vol. 19, 1981, pp. 1321-1329.

[19] L. E. Payne and B. Straughan, “Critical Rayleigh Numbers for Oscillatory and Non-Linear Convection in an Isotropic Thermomicropolar Fluid,” International Journal of Engineering Sciences, Vol. 27, No. 7, 1989, pp. 827-836.
doi:10.1016/0020-7225(89)90048-7

[20] P. G. Siddheshwar and S. Pranesh, “Effect of a Non-Uniform Basic Temperature Gradient on Rayleigh-Benard Convection in a micropolar Fluid,” International Journal of Engineering Sciences, Vol. 36, No. 11, 1998, pp. 1183- 1196. doi:10.1016/S0020-7225(98)00015-9

[21] R. Idris, H. Othman and I. Hashim, “On Effect of NonUniform Basic Temperature Gradient on Bénard-Marangoni Convection in Micropolar Fluid,” International Communications in Heat and Mass Transfer, Vol. 36, No. 3, 2009, pp. 255-258.
doi:10.1016/j.icheatmasstransfer.2008.11.009

[22] M. N. Mahmud, Z. Mustafa and I. Hashim, “Effects of Control on the Onset of Bénard-Marangoni Convection in a Micropolar Fluid,” International Communications in Heat and Mass Transfer, Vol. 37, No. 9, 2010, pp. 1335- 1339. doi:10.1016/j.icheatmasstransfer.2010.08.013

[23] S. Pranesh and R. V. Kiran, “Study of Rayleigh-Bénard Magneto Convection in a Micropolar Fluid with Maxwell-Cattaneo Law,” Applied Mathematics, Vol. 1, 2010, pp. 470-480. doi:10.4236/am.2010.16062

[24] R. C. Sharma and P. Kumar, “On Micropolar Fluids Heated from Below in Hydromagnetics,” Journal of Non-Equilibrium Thermodynamics, Vol. 20, No. 2, 1995, pp. 150-159. doi:10.1515/jnet.1995.20.2.150

[25] R. C. Sharma and U. Gupta, “Thermal Convection in Micropolar Fluids in Porous Medium,” International Journal of Engineering Sciences, Vol. 33, No. 11, 1995, pp. 1887-1892. doi:10.1016/0020-7225(95)00047-2

[26] M. Zahn and D. R. Greer, “Ferrohydrodynamics Pumping in Spatially Uniform Sinusoidally Time Varying Magnetic Fields,” Journal of Magnetism and Magnetic Materials, Vol. 149, No. 1-2. 1995, pp. 165-173.
doi:10.1016/0304-8853(95)00363-0

[27] A. Abraham, “Rayleigh-Benard Convection in a Micropolar Magnetic Fluids,” International Journal of Engineering Sciences, Vol. 40, 2002, pp. 449-460.

[28] Reena and U. S. Rana, “Thermal Convection of Rotating Micropolar Fluid in Hydromagnetics Saturating a Porous Media,” International Journal of Engineering. Transactions A: Basics, Vol. 21, No. 4, 2008, pp. 375-396.

[29] Reena and U. S. Rana, “Effect of Dust Particles on a Layer of Micropolar Ferromagnetic Fluid Heated from Below Saturating a Porous Medium,” Applied Mathematics and Computation, Vol. 215, No. 7, 2009, pp. 2591-2607.
doi:10.1016/j.amc.2009.08.063

[30] Y. Qin and P. N. Kaloni, “A Thermal Instability Problem in a Rotating Micropolar Fluid,” International Journal of Engineering Sciences, Vol. 30, No. 9, 1992, pp. 1117-1126. doi:10.1016/0020-7225(92)90061-K

[31] Sunil, P. Chand, P. K. Bharti and A. Mahajan, “Thermal Convection a Micropolar Ferrofluid in the Presence of Rotation,” Journal of Magnetism and Magnetic Materials, Vol. 320, No. 3-4, 2008, pp. 316-324.
doi:10.1016/j.jmmm.2007.06.006