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Natural convection in a rectangular enclosure in the presence of a magnetic field with uniform heat flux from the side walls

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Summary

A numerical study is presented for magnetohydrodynamic free convection of an electrically conducting fluid in a two-dimensional rectangular enclosure in which two side walls are maintained at uniform heat flux condition. The horizontal top and bottom walls are thermally insulated. A finite difference scheme comprising of modified ADI (Alternating Direction Implicit) method and SOR (Successive-Over-Relaxation) method is used to solve the governing equations. Computations are carried out over a wide range of Grashof number, Gr and Hartmann number, Ha for an enclosure of aspect ratio 1 and 2. The influences of these parameters on the flow pattern and the associated heat transfer characteristics are discussed. Numerical results show that with the application of an external magnetic field, the temperature and velocity fields are significantly modified. When the Grashof number is low and Hartmann number is high, the central streamlines are elongated and the isotherms are almost parallel representing a conduction state. For sufficiently large magnetic field strength the convection is suppressed for all values of Gr. The average Nusselt number decreases with an increase of Hartmann number and hence a magnetic field can be used as an effective mechanism to control the convection in an enclosure.

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Abbreviations

Ar :

aspect ratio,H/L

B 0 :

induction magnetic field

H 0 :

magnetic field,H 0=B 0/μ m

g :

gravitational acceleration

Gr :

Grashof number,gβq″(L/k)L 3/v 2

H :

height of the enclosure

Ha :

Hartmann number,\(\mu _m H_0 L\sqrt {\sigma /\mu }\)

k :

thermal conductivity

Nu :

local Nusselt number

\(\overline {Nu}\) :

average Nusselt number

p :

pressure

Pr :

Prandtl number, ν/α

q″:

heat flux

t :

time

T :

dimensionless temperature, (θ−θ0)/q″(L/k)

u :

vertical velocity

U :

dimensionless vertical velocity,uL/ν

v :

horizontal velocity

V :

dimensionless horizontal velocity,vL/ν

x :

vertical coordinate

X :

dimensionless vertical coordinate,x/L

y :

horizontal coordinate

Y :

dimensionless horizontal coordinate,y/L

α:

thermal diffusivity

β:

thermal expansion coefficient

θ:

temperature

θ0 :

reference temperature

ρ:

density

ν:

kinematic viscosity

μ:

viscosity

μ m :

magnetic permeability

σ:

electrical conductivity

Ψ:

stream function

ψ:

dimensionless stream function, Ψ/ν

τ:

dimensionless time,tν/L 2

ω:

vorticity

ξ:

dimensionless vorticity, ωL 2/ν

ΔX :

grid spacing inX-direction

ΔY :

grid spacing inY-direction

Δτ:

time increment

∇ 2 :

Laplacian operator

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Author notes

  1. M. Venkatachalappa

    Present address: UGC-DSA Centre in Fluid Mechanics, Department of Mathematics, Central College, Bangalore University, 560001, Bangalore, India

  2. C. K. Subbaraya

    Present address: Adichunchanagiri Institute of Technology, 577102, Chikmagalur, India

Authors and Affiliations

  1. UGC-DSA Centre in Fluid Mechanics, Department of Mathematics, Central College, Bangalore University, 560001, Bangalore, India

    C. K. Subbaraya

Authors
  1. M. Venkatachalappa
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  2. C. K. Subbaraya
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Venkatachalappa, M., Subbaraya, C.K. Natural convection in a rectangular enclosure in the presence of a magnetic field with uniform heat flux from the side walls. Acta Mechanica 96, 13–26 (1993). https://doi.org/10.1007/BF01340696

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