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Solutions of a free boundary problem in a doubly connected domain via a circular-arc polygon

Published online by Cambridge University Press:  06 June 2014

J. S. MARSHALL
J. S. MARSHALL*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK email: jonathan.marshall1@imperial.ac.uk
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Abstract

This paper addresses a free boundary problem for a steady, uniform patch of vorticity surrounding a single flat plate of zero thickness and finite length. Exact solutions to this problem have previously been found in terms of conformal maps represented by Cauchy-type integrals. Here, however, it is demonstrated how, by considering an associated circular-arc polygon and using ideas from automorphic function theory, these maps can be expressed in a simple non-integral form.

Keywords

Free boundary problemMultiply connectedConformal mappingAutomorphic functions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 5 , October 2014 , pp. 579 - 594
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Crank, J. (1984) Free and Moving Boundary Problems, Oxford University Press, Oxford, UK.Google Scholar
[2]Craster, R. V. (1994) Two related free boundary problems. IMA J. Appl. Math. 52, 253–270.CrossRefGoogle Scholar
[3]Craster, R. V. (1997) The solution of a class of free boundary problems. Proc. R. Soc. A 453, 607–630.CrossRefGoogle Scholar
[4]Crowdy, D. G. (1999) A class of exact multipolar vortices. Phys. Fluids 11, 2556–2564.CrossRefGoogle Scholar
[5]Crowdy, D. G. (2004) Exact solutions for uniform vortex layers attached to corners and wedges. Euro. J. Appl. Math. 15, 643–650.CrossRefGoogle Scholar
[6]Crowdy, D. G. (2008) The Schwarz problem in multiply connected domains and the Schottky–Klein prime function. Complex Var. Elliptic Equ. 53 (3), 221–236.CrossRefGoogle Scholar
[7]Crowdy, D. G. & Marshall, J. S. (2006) Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6 (1), 59–76.CrossRefGoogle Scholar
[8]Davis, P. J. (1974) The Schwarz Function and its Applications, The Mathematical Association of America, Washington, DC.CrossRefGoogle Scholar
[9]Ford, L. R. (1972) Automorphic Functions, Chelsea, New York, NY.Google Scholar
[10]Goluzin, G. M. (1969) Geometric Theory of Functions of a Complex Variable, American Mathematical Society, Providence, RH.CrossRefGoogle Scholar
[11]Howison, S. D. (1987) Complex variables in industrial mathematics. In: Neunzert, H. (editor), Proceedings of the 2nd European Symposium on Mathematics in Industry, Oberwolfach, Germany, Kluwer Academic, Stuttgart, Germany, pp. 155–166.Google Scholar
[12]Howison, S. D. & King, J. R. (1989) Explicit solution to six free boundary problems in fluid flow and diffusion. IMA J. Appl. Math. 42, 155–175.CrossRefGoogle Scholar
[13]Johnson, E. R. & McDonald, N. R. (2006) Vortical source-sink flow against a wall: The initial value problem and exact steady states. Phys. Fluids 18, 076601.CrossRefGoogle Scholar
[14]Johnson, E. R. & McDonald, N. R. (2007) Steady vortical flow around a finite plate. Q. J. Mech. Appl. Math. 60 (1), 65–72.CrossRefGoogle Scholar
[15]Johnson, E. R. & McDonald, N. R. (2009) Necking in coating flow over periodic substrates. J. Eng. Math. 65, 171–178.CrossRefGoogle Scholar
[16]Marshall, J. S. (2012) Steady uniform vortex patches around an assembly of walls or flat plates. Q. J. Mech. Appl. Math. 65 (1), 27–60.CrossRefGoogle Scholar
[17]McDonald, N. R. & Johnson, E. R. (2009) Gap-leaping vortical currents. J. Phys. Ocean. 39, 2665–2674.CrossRefGoogle Scholar
[18]Polubarinova-Kochina, P. Ya. (1962) Theory of Groundwater Movement, Princeton University Press, Princeton, NJ.Google Scholar
[19]Tuck, E. O., Bentwick, M. & van der Hoek, J. (1983) The free boundary problem for gravity-driven unidirectional viscous flows. IMA J. Appl. Math. 30, 191–208.CrossRefGoogle Scholar