Meccanica

, Volume 45, Issue 2, pp 175–185

Hydromagnetic free convection flow with induced magnetic field effects

Article

DOI: 10.1007/s11012-009-9235-x

Cite this article as:
Ghosh, S.K., Anwar Bég, O. & Zueco, J. Meccanica (2010) 45: 175. doi:10.1007/s11012-009-9235-x

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 (Bx) and wall frictional shearing stress i.e. skin friction function (τx) as functions of dimensionless transverse coordinate (η), Grashof free convection number (Gr) 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 (Bx), is elevated in the vicinity of the plate surface with a rise in free convection (buoyancy) parameter Gr, 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.

Keywords

MHD free convection Magnetic induction Boundary layer control Bulk temperature Frictional drag Hyperbolic functions Rayleigh flow 

Notation

B

magnetic field induction vector

B0

magnetic flux density

Bx

induced magnetic field component along plate

Cp

specific heat at constant pressure

E

electric field vector

Ec

Eckert number

g

gravitational acceleration

Gr

Grashof number

J

current density vector

K

thermal conductivity

M

Hartmann number

Pr

Prandtl number

Tw

temperature of the wall

T0

temperature of the surrounding fluid

u

velocity component

x

coordinate parallel to plate

y

coordinate transverse to plate

Greek

α1

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)

ηe

magnetic diffusivity or viscosity

ρ

fluid density

μe

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 (=TT0)

ψw

temperature difference between the wall (plate) and the free stream (=TwT0)

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsNarajole Raj CollegeMidnapore (West)India
  2. 2.Magnetohydrodynamics and Heat Transfer Group, Mechanical Engineering Department, Sheaf BuildingSheffield Hallam UniversitySheffieldUK
  3. 3.ETS Ingenieros Industriales Campus Muralla del Mar, Departamento de Ingeniería Térmica y FluidosUniversidad Politécnica de CartagenaCartagena (Murcia)Spain

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