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Corner regions in the asymptotic solution of ε∇2u = ∂u/∂y with reference to MHD duct flow

Published online by Cambridge University Press:  24 October 2008

L. Pamela Cook ,
G. S. S. Ludford  and
J. S. Walker
L. Pamela Cook
Affiliation:
Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, N. Y.
G. S. S. Ludford
Affiliation:
Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, N. Y.
J. S. Walker
Affiliation:
Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, N. Y.
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Abstract

In the asymptotic solution of the title equation for simple boundary data on the rectangle 0 ≤ x ≤ l, 0 ≤ y ≤ 1 the bottom corners are not passive, as has previously been supposed, but in fact generate the parabolic side layers at x = o, l. The top corners, though passive, lead to unconventional elliptic quarter-plane problems. Implications for the MHD duct problem are indicated.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 1 , July 1972 , pp. 117 - 122
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Temperley, D. J. and Todd, L.The effects of wall conductivity in magnetohydrodynamic duct flow at high Hartmann numbers. Proc. Cambridge Philos. Soc. 69 (1971), 337.CrossRefGoogle Scholar
(2)Walker, J. S., Ludford, G. S. S. and Hunt, J. C. R.Three-dimensional MHD duct flows with strong transverse magnetic fields. Part 2. Variable-area rectangular ducts with conducting sides. J. Fluid Mech. 46 (1971).CrossRefGoogle Scholar
(3)Cook, L. Pamela. The asymptotic solution of ε∇2ω = ∂ω/∂y on a rectangle. Ph.D. Thesis (Cornell University, 1971).Google Scholar
(4)Cook, L. Pamela and Ludford, G. S. S.The behaviour as ε → 0 + of solutions to ε∇2ω = ∂ω/∂y in |y| ≤ 1 for discontinuous boundary data. SIAM J. Math. Anal. 2 (1971), 567.CrossRefGoogle Scholar
(5)Cook, L. Pamela and Ludford, G. S. S. The behaviour as ε → 0 + of solutions to ε∇2ω = ∂ω/∂y in |y| ≤ in the rectangle 0 ≤ x ≤ l, |y| 1. Submitted for publication.Google Scholar