A Mathematical Model for Magnetohydrodynamic Convection Flow in a Rotating Horizontal Channel with Inclined Magnetic Field, Magnetic Induction and Hall Current Effects
Abstract: Closed-form and asymptotic solutions are derived for the steady, fully-developed hydromagnetic free and forced convection flow in a rotating horizontal parallel-plate channel under the action of an inclined magnetic field and constant pressure gradient along the longitudinal axis of the channel. The magnetic field is strong enough to generate Hall current effects and the magnetic Reynolds number of sufficient magnitude that induced magnetic field effects are also present. Secondary flow is present owing to the Hall current effect. The channel plates are also taken to be electrically-conducting. The conservation equations are formulated in an (x, y, z) coordinate system and non-dimensionalized using appropriate transformations. The resulting non-dimensional coupled ordinary differential equations for primary and secondary velocity components and primary and secondary induced magnetic field components and transformed boundary conditions are shown to be controlled by the dimensionless pressure gradient parameter (px), Hartmann number (M2), Grashof number (G), Hall current parameter (m), rotational parameter (K2), magnetic field inclination (Θ), and the electrical conductance ratios of the upper (⏱) and lower (⏲) plates. Solutions are derived using the method of complex variables. Asymptotic solutions are also presented for very high rotation parameter and Hartmann number of order equal to unity, for which Ekman-Hartmann boundary layers are identified at the plates. A parametric study of the evolution of velocity and induced magnetic field distributions is undertaken. It is shown that generally increasing Hall current effect (m) serves to accentuate the secondary (cross) flow but oppose the primary flow. An increase in rotational parameter (K2) is also found to counteract primary flow intensity. An elevation in the Grashof number i.e. free convection parameter (G) is shown to aid the secondary induced magnetic field component (Hz); however there is a decrease in magnitudes of the primary induced magnetic field component (Hx) with increasing Grashof number. Increasing inclination of the applied magnetic field (Θ, is also found to oppose the primary flow (u1) but conversely to strongly assist the secondary flow (w1). Both critical primary (Gcx) and secondary (Gcz)Grashof numbers are shown to be reduced with increasing inclination of the magnetic field (Θ), increasing Hall parameter (m) and rotational parameter (K2). Applications of the study arise in rotating MHD induction power generators and also astrophysical flows.
Cite this paper: nullS. Ghosh, O. Bég and A. Aziz, "A Mathematical Model for Magnetohydrodynamic Convection Flow in a Rotating Horizontal Channel with Inclined Magnetic Field, Magnetic Induction and Hall Current Effects," World Journal of Mechanics, Vol. 1 No. 3, 2011, pp. 137-154. doi: 10.4236/wjm.2011.13019.
References

[1]   K. Takenouchi, “Transient Magnetohydrodynamic Chan- nel Flow with Axial Symmetry at a Supersonic Speed,” Journal of the Physical Society of Japan, Vol. 54, No. 4, 1985, pp. 1329-1338. doi:10.1143/JPSJ.54.1329

[2]   R. C. Goforth and C. H. Kruger, “Investigation of Secondary Flows in Magneto-Hydrodynamic Channels,” AIAA Journal of Propulsion and Power, Vol. 9, No. 6, 1993, pp. 889-897. doi:10.2514/3.23704

[3]   S. Khan and J. N. Davidson, “Magneyohydrodynamic Coolant Flows in Fusion Reactor Blankets,” Annals Nuclear Energy, Vol. 6, No. 9-10, 1979, pp. 499-515. doi:10.1016/0306-4549(79)90023-9

[4]   H. Kumamaru and Y. Fujiwara, “Magnetohydrodynamic Flow in Rectangular Channel. Effect of Wall Thickness and Interaction of Parallel Channel Flows,” Journal of Nuclear Science Technology, Vol. 36, No. 1, 1999, pp. 110-113. doi:10.3327/jnst.36.110

[5]   Z. P. Aguilar, P. Arumugam and I. Fritsch, “Study of Magnetohydrodynamic Driven Flow through LTCC Channel with Self-Contained Electrodes,” Journal of Electroanalytical Chemistry, Vol. 591, No.2, 2006, pp. 201-209. doi:10.1016/j.jelechem.2006.04.019

[6]   O. Nath, S. N. Ojha and H. S. Takhar, “Self Similar MHD Shock Waves for a Rotating Atmosphere under the Influence of Gravitation,” Astrophysics and Space Science Journal, Vol. 200, No. 1, 1993, pp. 27-34. doi:10.1007/BF00658107

[7]   S. S. Niranjan, V. M. Soundalgekar and H. S. Takhar, “Free Convection Effects on MHD Horizontal Channel Flow with Hall Currents,” IEEE. Transactions on Plasma Science, Vol. 18, No. 2, 1990, pp. 177-183. doi:10.1109/27.131017

[8]   P. C. Ram, A. K. Singh and H. S. Takhar, “Effects of Hall and Ionslip Currents on Convective Flow in a Rotating Fluid with a Wall Temperature Oscillation,” Magnetohydrodynamics and Plasma Research, Vol. 5, No. 1, 1995, pp. 1-16.

[9]   A. S. Slaouti, H. S. Takhar and G. Nath, “Spin-up and Spin-down for a Rotating Disc in the Presence of Buoyancy Force and Magnetic Field and Buoyancy Force,” Acta Mechanica, Vol. 156, 2002, pp. 109-129. doi:10.1007/BF01188745

[10]   J. Zueco and O. Anwar Bég, “Network Numerical Analysis of Hydromagnetic Squeeze Film Flow Dynamics between Two Parallel Rotating Disks with Induced Magnetic Field Effects,” Tribology International, Vol. 43, No. 3, 2010, pp. 532-543. doi:10.1016/j.triboint.2009.09.002

[11]   D. E. Loper, “A Linear Theory of Rotating, Thermally Stratified, Hydromagnetic Flow,” Journal of Fluid Mechanics, Vol. 72, No. 1, 1975, pp. 1-16. doi:10.1017/S002211207500290X

[12]   H. Sato, “The Hall Effect in the Viscous Flow of Ionized Gas between Parallel Plates under Transverse Magnetic Field,” Journal of the Physical Society of Japan, Vol. 16, No. 7, 1961, pp. 1427-1433. doi:10.1143/JPSJ.16.1427

[13]   B. C. Chandrasekhara and N. Rudraiah, “Three Dimensional Magnetohydrodynamic Flow between Two Porous Disks,” Applied Scientific Research, Vol. 25, No. 1, 1972, pp. 179-192. doi:10.1007/BF00382294

[14]   B. S. Mazumder, “Effect of Wall Conductances on Hydromagnetic Flow and Heat Transfer in a Rotating Channel,” Acta Mechanica, Vol. 28, No. 1-4, 1977, pp. 85-99. doi:10.1007/BF01208791

[15]   H. S. Takhar and G. Nath, “Unsteady Flow over a Stretching Surface with a Magnetic Field in a Rotating Fluid,” Zeitschrift fur Angewwandte Mathematik und Physik, Vol. 49, No. 6, 1998, pp. 989-1001. doi:10.1007/s000330050135

[16]   H. S. Takhar, A. K. Singh and G. Nath, “Unsteady MHD Flow and Heat Transfer on a Rotating Disk in an Ambient Fluid,” International Journal of Thermal Sciences, Vol. 41, No. 2, 2002, pp. 147-155. doi:10.1016/S1290-0729(01)01292-3

[17]   A. R. Bestman and S. K. Adjepong, “Unsteady Hydromagnetic Free-Convection Flow with Radiative Heat Transfer in a Rotating Fluid: II—Compressible Optically Thin Fluid,” Astrophysics and Space Science Journal, Vol. 143, No. 2, 1988, pp. 217-224. doi:10.1007/BF00637135

[18]   M. Kumari, H. S. Takhar and G. Nath, “Vorticity Interaction in a Two-Dimensional Unsteady Stagnation Point Flow with an Applied Magnetic Field,” Mechanics Research Communications, Vol. 18, 1991, pp. 151-156. doi:10.1016/0093-6413(91)90061-Z

[19]   P. C. Ram and H. S. Takhar, “MHD Free Convection from an Infinite Vertical Plate in a Rotating Fluid with Hall and Ionslip Currents,” Fluid Dynamics Research, Vol. 11, No. 3, 1993, pp. 99-105. doi:10.1016/0169-5983(93)90009-Y

[20]   H. S. Takhar, P. C. Ram and S. S. Singh, “Unsteady MHD Flow of a Dusty Viscous Liquid in a Rotating Channel with Hall Currents,” International Journal of Energy Research, Vol. 17, No. 1, 1993, pp. 69-74. doi:10.1002/er.4440170109

[21]   A. K. Singh, N. C. Sacheti and P. Chandran, “Transient Effects on Magneto-Hydrodynamic Couette Flow with Rotation: Accelerated Motion,” International Journal of Engineering Science, Vol. 32, No. 1, 1994, pp. 133-139. doi:10.1016/0020-7225(94)90155-4

[22]   S. K. Ghosh and P. K. Bhattacharjee, “Magnetohydrodynamic Convection Flow in a Rotating Channel,” Archives of Mechanics, Vol. 52, No. 2, 2000, pp. 303-318.

[23]   S. K. Ghosh and D. K. Nandi, “Magnetohydrodynamic Fully Developed Combined Convection Flow between Vertical Plates Heated Asymmetrically,” Journal of Technical Physics, Vol. 41, No. 2, 2000, pp. 173-185.

[24]   S. Vempaty and P.. Satyamurty, “Rotating Hydromagnetic Flow in the Presence of a Horizontal Magnetic Field,” Zeitschrift für angewandte Mathematik und Physik, Vol. 55, No. 5, 2004, pp. 800-825. doi:10.1007/s00033-003-1123-y

[25]   A. Chakraborti, A. S. Gupta, B. K. Das and R. N. Jana, “Hydromagnetic Flow Past a Rotating Porous Plate in a Conducting Fluid Rotating about a Non-Coincident Parallel Axis,” Acta Mechanica, Vol. 176, No. 1-2, 2005, pp. 107-119. doi:10.1007/s00707-004-0144-8

[26]   O. A. Bég, H. S. Takhar, G. Nath and A. J. Chamkha, “Mathematical Modeling of Hydromagnetic Convection from a Rotating Sphere with Impulsive Motion and Buoyancy Effects,” Non-Linear Analysis: Modeling and Control Journal, Vol. 11, No. 3, 2006, pp. 227-245.

[27]   H. Naroua, H. S. Takhar, P. C. Ram, T. A. Bég, O. A. Bég and R. Bhargava, “Transient Rotating Hydromagnetic Partially-Ionized Heat-Generating Gas Dynamic Flow with Hall/Ionslip Current Effects: Finite Element Analysis,” International Journal of Fluid Mechanics Research, Vol. 34, No. 6, 2007, pp. 493-505. doi:10.1615/InterJFluidMechRes.v34.i6.10

[28]   S. K. Ghosh, O. A. Bég and J. Zueco, “Rotating Hydromagnetic Optically-Thin Gray Gas Flow with Thermal Radiation Effects,” Journal of Theoretical Applied Mechanics, Vol. 39, No. 1, 2009, pp. 101-120.

[29]   O. A. Bég, J. Zueco and H. S. Takhar, “Unsteady Magnetohydrodynamic Hartmann–Couette Flow and Heat Transfer in a Darcian Channel with Hall Current, Ionslip, Viscous and Joule Heating Effects: Network Numerical Solutions,” Communications in Nonlinear Science Numerical Simulation, Vol. 14, No. 4, 2009, pp. 1082-1097. doi:10.1016/j.cnsns.2008.03.015

[30]   M. Guria, B. K. Das, R. N. Jana, R. N. and S. K. Ghosh, “Effects of Wall Conductance on MHD Fully Developed Flow with Asymmetric Heating of the Wall,” International Journal of Fluid Mechanics Research, Vol. 34, No. 6, 2007, pp. 521-534. doi:10.1615/InterJFluidMechRes.v34.i6.40

[31]   G. S. Seth and S. K. Ghosh, “Unsteady Hydromagnetic Flow in a Rotating Channel in the Presence of Inclined Magnetic Field,” International Journal of Engineering Science, Vol. 24, No. 7, 1986, pp. 1183-1193. doi:10.1016/0020-7225(86)90013-3

[32]   S. K. Ghosh, “ A Note on Steady and Unsteady Hydromagnetic Flow in a Rotating Channel in the Presence of Inclined Magnetic Field,” International Journal of Engineering Science, Vol. 29, No. 8, 1991, pp. 1013-1016. doi:10.1016/0020-7225(91)90175-3

[33]   S. K. Ghosh and P. K. Bhattacharjee, “Hall Effects on Steady Hydromagnetic Flow in a Rotating Channel in the Presence of an Inclined Magnetic Field,” Czechoslovak Journal of Physics, Vol. 50, No. 6, 2000, pp. 759-767. doi:10.1023/A:1022839020051

[34]   I. Pop, S. K. Ghosh and D. K. Nandi, “Effects of the Hall Current on Free and Forced Convection Flows in a Rotating Channel in the Presence of an Inclined Magnetic Field,” Magnetohydrodynamics, Vol. 37, No. 4, 2001, pp. 348-359.

[35]   O. Anwar Bég, L. Sim, J. Zueco and R. Bhargava, “Numerical Study of Magnetohydrodynamic Viscous Plasma Flow in Rotating Porous Media with Hall Currents and Inclined Magnetic Field Influence,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 2, 2010, pp. 345-359. doi:10.1016/j.cnsns.2009.04.008

[36]   O. Anwar Bég, A. Y. Bakier, V. R. Prasad and S. K. Ghosh, “Nonsimilar, Laminar, Steady, Electrically-Conducting Forced Convection Liquid Metal Boundary Layer Flow with Induced Magnetic Field Effects,” Internatinal Journal of Thermal Sciences, Vol. 48, No. 8, 2009, pp. 1596-1606. doi:10.1016/j.ijthermalsci.2008.12.007

[37]   S. K. Ghosh, O. Anwar Bég, J. Zueco, “Hydromagnetic Free Convection Flow with Induced Magnetic Field Effects,” Meccanica, Vol. 45, No. 2, 2010, pp. 175-185. doi:10.1007/s11012-009-9235-x

[38]   K. R. Cramer and S-I. Pai, “Magnetofluid Dynamics for Engineers and Applied Physicists,” MacGraw-Hill, New York, 1973.

[39]   J. A. Shercliff, “A Textbook of Magnetohydrodynamics,” Pergamon, Oxford, 1965.

[40]   M. M. Rashidi, M. Keimanesh, O. Anwar Bég and T. K. Hung, “Magneto-Hydrodynamic Biorheological Transport Phenomena in a Porous Medium: A Simulation of Magnetic Blood Flow Control and Filtration,” International Journal for Numerical Methods in Biomedical Engineering, 2011.

[41]   O. Anwar Bég, V. R. Prasad, B. Vasu, N. Bhaskar Reddy, Q. Li and R. Bhargava, “Free Convection Heat and Mass Transfer from an Isothermal Sphere to a Micropolar Regime with Soret/Dufour Effects,” International Journal of Heat and Mass Transfer, Vol. 54, No. 1-3, 2011, pp. 9-18. doi:10.1016/j.ijheatmasstransfer.2010.10.005

[42]   O. Anwar Bég, J. Zueco and S. K. Ghosh, “Unsteady Hydromagnetic Natural Convection of a Short-Memory Viscoelastic Fluid in a Non-Darcian Regime: Network Simulation,” Chemical Engineering Communications, Vol. 198, No. 2, 2011, pp. 172-190.

[43]   O. Anwar Bég and O. D. Makinde, “Viscoelastic Flow and Species Transfer in a Darcian High-Permeability Channel,” Journal of Petroleum Science and Engineering, Vol. 76, No. 3-4, 2011, pp. 93–99. doi:10.1016/j.petrol.2011.01.008

[44]   O. Anwar Bég, J. Zueco, S. K. Ghosh and A. Heidari, “Unsteady Magneto-Hydrodynamic Heat Transfer in a Semi-Infinite Porous Medium with Thermal Radiation Flux: Analytical and Numerical Study,” Advances in Numerical Analysis, Vol. 2011, 2011, pp. 1-17. Article ID 304124.

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