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The laminar plume above a line heat source in a transverse magnetic field

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Abstract

The dynamics of a buoyant plume rising above a horizontal line heat source in a transverse, horizontal magnetic field is investigated. Similarity is shown to occur when the magnetic field strength varies as the −2/5 power of vertical distance from the source. The plume depends on two parameters — the Prandtl number (Pr) and the Lykoudis number (Z L). Families of exact closed form solutions are derived for Pr=5/9 and Pr≥2. A family of numerical integrations for Pr=0.01 (typical of liquid metals) is also reported. The magnetic field is shown to affect the profiles of velocity and temperature by altering the similarity functions, the coefficients, and the value of the independent similarity variable corresponding to a fixed physical position. An approximate closed form solution valid for low Pr and high Z L is presented. Possible experimental tests of the theory are proposed.

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Abbreviations

B :

magnetic field strength

B 0 :

constant introduced in (9)

C :

constant introduced in (35)

C 1 :

constant of integration

C p :

specific heat at constant pressure

D :

constant introduced in (35)

f :

non-dimensional similarity stream function

g :

acceleration due to gravity

Gr :

Grashof number

I :

non-dimensional heat flux integral

J :

integral defined by (34)

M :

Hartmann number

m :

exponent introduced in (9)

jackie :

mass flow rate per unit length

N :

constant introduced in (8)

n :

exponent introduced in (8)

Pr :

Prandtl number

Q :

heat source strength per unit length

T :

temperature

T ∞ :

temperature of undisturbed fluid

u :

vertical velocity

v :

horizontal velocity

x :

vertical coordinate

y :

horizontal coordinate

z :

horizontal coordinate

Z L :

Lykoudis number

α :

volume coefficient of thermal expansion

η :

similarity variable

θ :

temperature excess above undisturbed ambient

θ 0 :

centerline temperature excess above undisturbed ambient

κ :

thermal diffusivity

μ :

dynamic viscosity

ν :

kinematic viscosity

ρ :

fluid density

σ :

electrical conductivity

φ :

normalized temperature similarity function

ψ :

stream function

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Author information

Authors and Affiliations

  1. Heat Transfer-Fluid Dynamics Section, Oak Ridge National Laboratory, P.O. Box Y, 37830, Oak Ridge, Tennessee, USA

    Donald D. Gray

Authors
  1. Donald D. Gray
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Additional information

Research sponsored by the U.S. Energy Research and Development Administration under interagency agreement with Union Carbide Corporation.

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Gray, D.D. The laminar plume above a line heat source in a transverse magnetic field. Appl. Sci. Res. 33, 437–457 (1977). https://doi.org/10.1007/BF00411824

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  • DOI: https://doi.org/10.1007/BF00411824

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