Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T23:27:26.729Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady electrically driven flows

Published online by Cambridge University Press:  24 October 2008

R. T. Waechter
R. T. Waechter
Affiliation:
School of Physical Sciences, Flinders University, Bedford Park, South Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

Certain simple steady electrically driven flows under transverse applied magnetic fields, for which exact solutions are available, are considered. Electric potentials are prescribed on fixed perfectly conducting electrodes and the resulting Lorentz force j × B0 gives rise to a velocity field in the fluid. It is assumed that the magnetic Reynolds number of the flow is small (Rm ≪ 1) so that the induced magnetic field may be neglected. In particular, we examine in detail the simplest flat plate configuration which is correctly posed as a two-part mixed boundary value problem of the Wiener–Hopf type. At large Hartmann number, the fluid vorticity and the current flow transverse to the applied magnetic field are suppressed. Thin current transition layers, which emanate from an electrode edge and are aligned with the applied magnetic field, provide a smooth transition from one region of almost constant current to another. The thickness of these layers is found to be of the same order as the thickness of certain velocity transition layers occurring in other situations in magnetohydrodynamics.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 3 , July 1968 , pp. 871 - 894
Copyright
Copyright © Cambridge Philosophical Society 1968

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

REFERENCES

(1)Bouwkamp, C. J.A note on singularities occurring at sharp edges in electromagnetic diffraction theory. Physica 12 (1946), 467–474.CrossRefGoogle Scholar
(2)Braginskii, S. I.Magnetohydrodynamics of weakly conducting liquids. Soviet Physics J.E.T.P. 37 (1960), 1005–1014.Google Scholar
(3)Eckhaus, W. and de Jager, E. M.Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type. Arch. Rational Mech. Anal. 23 (1966), 26–86.CrossRefGoogle Scholar
(4)Hunt, J. C. R.Magnetohydrodynamic flows in rectangular ducts. J. Fluid Mech. 21 (1965), 577–590.CrossRefGoogle Scholar
(5)Hunt, J. C. R. and Stewartson, K.Magnetohydrodynamic flow in rectangular ducts. II. J. Fluid Mech. 23 (1965), 563–581.CrossRefGoogle Scholar
(6)Jones, D. S.The theory of electromagnetism (Pergamon Press, 1964).Google Scholar
(7)Maue, A. W.Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integral-gleichung. Z. Physik, 126 (1949), 601–618.CrossRefGoogle Scholar
(8)Moffatt, H. K.Electrically driven steady flows in magnetohydrodynamics. Proc. Eleventh Inter. Congress Appl. Mech., Munich (1964), 946–953.Google Scholar
(9)Noble, B.Methods based on the Wiener–Hopf technique for the solution of partial differential equations (Pergamon Press, 1958).Google Scholar
(10)Waechter, R. T. Asymptotic methods for large Hartmann number in magnetohydrodynamics, Ph.D. Dissertation, Cambridge (1966).Google Scholar
(11)Watson, G. N.A treatise on the theory of Bessel functions (Cambridge University Press, 1922).Google Scholar
(12)Yakubenko, A. Y.Certain problems concerning movement of conducting fluid in a plane channel, Zh. Prik. Mekh. i Tekh. Fiz. 6 (1963) (Russian). English translation Nat. Aeronaut. Space Adm., Washington, TTF-241 (1964).Google Scholar