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The influence of a magnetic field on free convection in a finite vertical channel

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Abstract

An approximate analytical solution is presented for developing free convection flows of electrically conducting fluids between finite vertical channels which are subjected to a uniformly applied transverse magnetic field. Specifically, the basic approximation lies in the linearization of the governing boundary layer type of equations. It is demonstrated that the application of a transverse magnetic field reduces the induced flow rate in the channel and the heat transfer to the fluid.

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Abbreviations

b′ :

channel width

B′ :

applied magnetic field

c′ p :

specific heat at constant pressure

g′ :

gravitational body force per unit mass

Gr :

Grashof number, b′ 4 β′g′(\(T_\infty^\prime\)\(T_w^\prime \))/l′ν′ 2

k′ :

thermal conductivity of fluid

l′ :

channel length

Nu :

Nusselt number defined by (19)

M :

Hartmann number, (σ′B′ 2 b′ 2/ρ′ν′)1/2

p :

dimensionless pressure, p′ν′ 2/ρ′b′ 4 β′ 2 g′ 2(\(T_w^\prime \)\(T_\infty ^\prime \))2

p′ :

pressure

Pr :

Prandtl number, ν′ρ′c′ p/k′

T :

dimensionless temperature, (T′\(T_\infty ^\prime \))/(\(T_w^\prime \)\(T_\infty ^\prime \))

T′ :

temperature

u :

dimensionless fluid velocity in the flow direction, u′ν′/b′ 2 β′′(\(T_w^\prime \)\(T_\infty ^\prime \))

u′ :

fluid velocity in the flow direction

v :

dimensionless fluid velocity in the transverse direction, v′l′ν′/b′ 3 β′g′(\(T_w^\prime \)\(T_\infty ^\prime \))

v′ :

fluid velocity in the transverse direction

x :

dimensionless flow direction coordinate, x′/l′

x′ :

flow direction coordinate

y :

dimensionless transverse direction coordinate, y′/b′

y′ :

transverse direction coordinate

α n :

characteristic values defined by (17)

β n :

characteristic values defined by (18)

ν′ :

kinematic viscosity of fluid

ζ :

dimensionless independent variable, x/u i Gr

ρ′ :

density

σ′ :

electrical conductivity

e:

channel exit

i:

channel inlet

w:

wall condition

∞:

ambient condition

References

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Meric, R.A. The influence of a magnetic field on free convection in a finite vertical channel. Appl. Sci. Res. 33, 459–469 (1977). https://doi.org/10.1007/BF00411825

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Keywords

  • Heat Transfer
  • Nusselt Number
  • Free Convection
  • Transverse Magnetic Field
  • Hartmann Number