Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T08:03:35.748Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady free convection above a point heat source and a horizontal line heat source in a vertical magnetic field

Published online by Cambridge University Press:  29 March 2006

A. M. Soward
A. M. Soward
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Cambridge University
Get access Rights & Permissions [Opens in a new window]

Abstract

An electrically conducting fluid is contained above a horizontal plane. A uniform vertical magnetic field is applied externally and the plane is maintained at a uniform temperature except for a point or a line heat source. Density variations are ignored except where they give rise to buoyancy forces.

(i) The point heat source. Non-linear effects are small sufficiently far from the source. The resulting buoyancy forces interact with the magnetic forces to maintain a radial inflow towards the heat source. This fluid then escapes vertically as a jet, its structure now depending on the additional influence of viscosity. The perturbations of the temperature distribution and the magnetic field due to the motion are obtained. Finally, the effects of these perturbations back on to the fluid velocity are considered. The most striking features of the perturbations are (a) the action of the jet as a line source of heat for the fluid in the outer regions, (b) the large (compared to other perturbations) eddy in the jet.

(ii) The line heat source. The temperature distribution and magnetic field are weakly perturbed only if the thermal and electrical conductivities are sufficiently small. Similar results are obtained, as in (i) above, provided ε (a dimensionless number characterising the strength of thermal convection: see (1.32), (3.24)) is less than ¼. However, even for small ε, the effects of thermal convection cannot be ignored. Hence, superimposed on the jet is an eddy (driven by buoyancy forces) whose flux of fluid increases indefinitely with its height above the plane. When ε > ¼, the results suggest that numerous eddies will be formed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 39 , Issue 4 , 15 December 1969 , pp. 753 - 780
Copyright
© 1969 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chandrasekar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1953 Higher Transcendental Functions (vol. I). New York: McGraw-Hill (Bateman Manuscript Project).
Fujii, T. 1963 Theory of the steady laminar natural convection above a horizontal line heat source and a point heat source Int. J. Heat Mass Transfer, 6, 597.Google Scholar
Hoult, D. P. 1965 Round laminar jet in an axial magnetic field Phys. Fluids, 8, 886.Google Scholar
Jungclaus, G. 1965 Herleitung allgemeiner magnetohydrodynamischer Grenzschicht-gleichungen mit expliziten Lösungen für das Längsfeld Acta Mechanica, 1, 265.Google Scholar
Schlichting, H. 1955 Boundary Layer Theory. London: Pergamon.
Toomre, J. 1967 Magnetohydrodynamic jets. Ph.D. Thesis, Cambridge University.
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. London: Academic.
Watson, G. N. 1958 A Treatise on the Theory of Bessel Functions. Cambridge University Press.