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Hydromagnetic free convection flow with induced magnetic field effects

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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.

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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


velocity component


coordinate parallel to plate


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 (=TT 0)

ψ w :

temperature difference between the wall (plate) and the free stream (=T w T 0)


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Ghosh, S.K., Anwar Bég, O. & Zueco, J. Hydromagnetic free convection flow with induced magnetic field effects. Meccanica 45, 175–185 (2010).

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