Abstract
An exact solution is presented for the hydromagnetic natural convection boundary layer flow past an infinite vertical flat plate under the influence of a transverse magnetic field with magnetic induction effects included. The transformed ordinary differential equations are solved exactly, under physically appropriate boundary conditions. Closed-form expressions are obtained for the non-dimensional velocity (u), non-dimensional induced magnetic field component (B x ) and wall frictional shearing stress i.e. skin friction function (τ x ) as functions of dimensionless transverse coordinate (η), Grashof free convection number (G r ) and the Hartmann number (M). The bulk temperature in the boundary layer (Θ) is also evaluated and shown to be purely a function of M. The Rayleigh flow distribution (R) is derived and found to be a function of both Hartmann number (M) and the buoyant diffusivity parameter (ϑ *). The influence of Grashof number on velocity, induced magnetic field and wall shear stress profiles is computed. The response of Rayleigh flow distribution to Grashof numbers ranging from 2 to 200 is also discussed as is the influence of Hartmann number on the bulk temperature. Rayleigh flow is demonstrated to become stable with respect to the width of the boundary layer region and intensifies with greater magnetic field i.e. larger Hartman number M, for constant buoyant diffusivity parameter ϑ *. The induced magnetic field (B x ), is elevated in the vicinity of the plate surface with a rise in free convection (buoyancy) parameter G r , but is reduced over the central zone of the boundary layer regime. Applications of the study include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.
Similar content being viewed by others
Abbreviations
- B :
-
magnetic field induction vector
- B 0 :
-
magnetic flux density
- B x ′:
-
induced magnetic field component along plate
- C p :
-
specific heat at constant pressure
- E :
-
electric field vector
- E c :
-
Eckert number
- g :
-
gravitational acceleration
- G r :
-
Grashof number
- J :
-
current density vector
- K :
-
thermal conductivity
- M :
-
Hartmann number
- P r :
-
Prandtl number
- T w :
-
temperature of the wall
- T 0 :
-
temperature of the surrounding fluid
- u′:
-
velocity component
- x′:
-
coordinate parallel to plate
- y′:
-
coordinate transverse to plate
- α 1 :
-
thermal diffusivity
- β :
-
coefficient of thermal expansion
- δ :
-
thickness of the boundary layer
- η :
-
width of the symmetric boundary layer region (non-dimensional y′-coordinate)
- η e :
-
magnetic diffusivity or viscosity
- ρ :
-
fluid density
- μ e :
-
magnetic permeability
- ν :
-
kinematic coefficient of viscosity
- σ :
-
electrical conductivity
- μ :
-
dynamic coefficient of viscosity
- θ :
-
dimensionless temperature
- ϑ * :
-
buoyant diffusivity parameter
- ψ :
-
temperature difference between a general location and free stream (=T−T 0)
- ψ w :
-
temperature difference between the wall (plate) and the free stream (=T w −T 0)
References
Chanty JMG (1994) Magnetized plasma flow over an insulator at high magnetic Reynolds number. AIAA J Thermophys Heat Transf 8(4):795–797
Munipalli R, Aithal S, Shankar V (2003) Effect of wall conduction in proposed MHD enhanced hypersonic vehicles. In: 12th AIAA international space planes and hypersonic systems and technologies conference, Norfolk, VA, 15–19 December 2003
Soundalgekar VM, Takhar HS (1977) On MHD flow and heat transfer over a semi-infinite plate under transverse magnetic field. Nucl Eng Des 42(2):233–236
Soundalgekar VM, Akolkar SP (1980) Effects of external circuit on heat transfer in MHD channel flow of rarefied gas. Nucl Eng Des 58(1):75–78
Yagawa G, Masuda M (1982) Finite element analysis of magnetohydrodynamics and its application to lithium blanket design of a fusion reactor. Nucl Eng Des 71(1–2):121–136
Ahluwalia RK, Doss ED (1980) Convective heat transfer in MHD channels and its influence on channel performance. AIAA J Energy 4(3):126–134
Takhar HS, Bég OA (1997) Effects of transverse magnetic field, Prandtl number and Reynolds number on non-Darcy mixed convective flow of an incompressible viscous fluid past a porous vertical flat plate in saturated porous media. Int J Energy Res 21:87–100
Helmy KA (1999) MHD unsteady free convection flow past a vertical porous plate. Z Angew Math Mech 78:225–270
Bhargava R, Kumar L, Takhar HS (2003) Numerical solution of free convection MHD micropolar fluid flow between two parallel porous vertical plates. Int J Eng Sci 41(2):123–136
Damseh RA (2006) Magnetohydrodynamics-mixed convection from radiate vertical isothermal surface embedded in a saturated porous medium. ASME J Appl Mech 73(1):54–59
Naroua H (2006) Computational solution of hydromagnetic free convective flow past a vertical plate in a rotating heat generating fluid with Hall and ionslip currents. Int J Numer Methods Fluids 53(10):647–658
Zueco J (2007) Network simulation method applied to radiation and viscous dissipation effects on MHD unsteady free convection over vertical porous plate. Appl Math Model 31(9):2019–2033
Ghosh SK, Pop I (2006) A new approach on MHD natural convection boundary layer flow past a flat plate of finite dimensions. Heat Mass Transf 42(7):587–595
Snyder WT, Erlasan AH (1969) Energy dissipation in MHD duct flow. AIAA J 7(1):150–151
Denno KI, Fouad AA (1972) Effects of the induced magnetic field on the magnetohydrodynamic channel flow. IEEE Trans Electron Dev 19(3):322–331
Megakhed AA (1974) Effect of induced magnetic field and heat transfer on nonstationary magnetohydrodynamic flow around a porous plate. Magnetohydrodynamics 10(1):48–52
Helliwell JB (1979) Radiative magnetogasdynamic Couette flow with variable parameters. Z Angew Math Phys 30(5):811–824
Ghosh SK (1993) Unsteady hydromagnetic flow in a rotating channel with oscillating pressure gradient. J Phys Soc Jpn 62(11):3893–3903
Takhar HS, Kumari M, Nath G (1993) Unsteady free convection flow under the influence of a magnetic field. Arch Appl Mech 63(4–5):313–321
Takhar HS, Chamkha AJ, Nath G (1999) Unsteady flow and heat transfer on a semi-infinite flat plate with an aligned magnetic field. Int J Eng Sci 37(13):1723–1736
Pop I, Ghosh SK, Nandi DK (2001) Effects of the Hall current on free and forced convective flows in a rotating channel in the presence of an inclined magnetic field. Magnetohydrodynamics 37(4):348–359
Bég OA, Bakier AY, Prasad V, Zueco J, Ghosh SK (2009) Nonsimilar, laminar, steady, electrically-conducting forced convection liquid metal boundary layer flow with induced magnetic field effects. Int J Therm Sci 48(8):1596–1606
Chen T-M (2008) Radiation effects on magnetohydrodynamic free convection flow, Technical Note. AIAA J Thermophys Heat Transf 22(1):125–128
Soundalgekar VM, Gupta SK, Birajdar NS (1979) Effects of mass transfer and free convection currents on MHD Stokes’ problem for a vertical plate. Nucl Eng Des 53(3):339–346
Lewandowski WM (1991) Natural convection heat transfer from plates of finite dimension. Int J Heat Mass Transf 34:875–885
Isachenko VP, Osipova VA, Sukomel AS (1969). In: Heat transfer. Mir, Moscow, pp 341–451
Landau LD, Lifschitz EM (1960) Electrodynamics of continuous media. International course in theoretical physics. Pergamon, London
Eringen AC, Maugin GA (1989) Electrodynamics of continua, vol II. Springer, New York
Meyer RC (1958) On reducing aerodynamic heat-transfer rates by magneto-hydrodynamic techniques. J Aerospace Sci 25:561–566
Datta N, Jana RN (1977) Hall effects on hydromagnetic flow and heat transfer in a rotating channel. IMA J Appl Math 19(2):217–229
Bég OA, Zueco J, Takhar HS (2009) Unsteady magnetohydrodynamic Hartmann-Couette flow and heat transfer in a Darcian channel with Hall current, ionslip, viscous and Joule heating effects: network numerical solutions. Commun Nonlinear Sci Numer Simul 14:1082–1097
Naroua H, Takhar HS, Ram PC, Bég TA, Bég OA, Bhargava R (2007) Transient rotating hydromagnetic partially-ionized heat-generating gas dynamic flow with Hall/Ionslip current effects: finite element analysis. Int J Fluid Mech Res 34(6):493–505
Cramer KR, Pai S-I (1973) Magnetofluid dynamics for engineers and applied physicists. MacGraw-Hill, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghosh, S.K., Anwar Bég, O. & Zueco, J. Hydromagnetic free convection flow with induced magnetic field effects. Meccanica 45, 175–185 (2010). https://doi.org/10.1007/s11012-009-9235-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-009-9235-x