Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T23:12:26.023Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

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

[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