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A new calculus for two-dimensional vortex dynamics


This article provides a user’s guide to a new calculus for finding the instantaneous complex potentials associated with point vortex motion in geometrically complicated planar domains, with multiple boundaries, in the presence of background flows. The key to the generality of the approach is the use of conformal mapping theory together with a special transcendental function called the Schottky–Klein prime function. Illustrative examples are given.

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  1. 1

    Acheson D.J.: Elementary Fluid Dynamics. Oxford University Press, Oxford (1990)

  2. 2

    Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)

  3. 3

    Baker H.F.: Abelian functions: Abel’s theorem and the allied theory of theta functions. Cambridge University Press, Cambridge (1897)

  4. 4

    Crowdy D.G., Marshall J.S.: Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains. Proc. R. Soc. A. 461, 2477–2501 (2005)

  5. 5

    Crowdy D.G., Marshall J.S.: The motion of a point vortex around multiple circular islands. Phys. Fluids 17, 056602 (2005)

  6. 6

    Crowdy D.G.: Calculating the lift on a finite stack of cylindrical aerofoils. Proc. R. Soc. A. 462, 1387–1407 (2006)

  7. 7

    Crowdy D.G.: Analytical solutions for uniform potential flow past multiple cylinders. Eur. J. Mech. B/Fluids 25(4), 459–470 (2006)

  8. 8

    Crowdy D.G., Marshall J.S.: The motion of a point vortex through gaps in walls. J. Fluid Mech. 551, 31–48 (2006)

  9. 9

    Crowdy D.G., Marshall J.S.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006)

  10. 10

    Crowdy D.G.: Explicit solution for the potential flow due to an assembly of stirrers in an inviscid fluid. J. Eng. Math. 62(4), 333–344 (2008)

  11. 11

    Crowdy D.G., Surana A., Yick K.-Y.: The irrotational flow generated by two planar stirrers in inviscid fluid. Phys. Fluids 19, 018103 (2007)

  12. 12

    Crowdy D.G., Surana A.: Contour dynamics in complex domains. J. Fluid Mech. 593, 235–254 (2007)

  13. 13

    Crowdy D.: Multiple steady bubbles in a Hele–Shaw cell. Proc. R. Soc. A. 465, 421–435 (2009)

  14. 14

    Crowdy D.G., Marshall J.S.: Computing the Schottky–Klein prime function on the Schottky double of planar domains. Comput. Methods Funct. Theory 7(1), 293–308 (2007)

  15. 15

    Ferrari, C.: Sulla trasformazione conforme di due cerchi in due profili alari, Memoire della Reale Accad. della Scienze di Torino, Ser. II 67 (1930)

  16. 16

    Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. American Mathematical Society, Providence (1969)

  17. 17

    Hejhal, D.A.: Theta Functions, Kernel Functions and Abelian Integrals, vol. 129. American Mathematical Society, Providence (1972)

  18. 18

    Johnson E.R., McDonald N.R.: The motion of a vortex near two circular cylinders. Proc. R. Soc. Lond. Ser. A 460, 939–954 (2004)

  19. 19

    Lagally, M.: Die reibungslose Strmung im Aussengebiet zweier Kreise. Z. Angew. Math. Mech. 9, 299–305, 1929 (English translation: The frictionless flow in the region around two circles, N.A.C.A., Technical Memorandum No 626, (1931))

  20. 20

    Lin C.C.: On the motion of vortices in two dimensions I: existence of the Kirchhoff–Routh function. Proc. Natl. Acad. Sci. 27, 570–575 (1941)

  21. 21

    Lin C.C.: On the motion of vortices in two dimensions II: some further investigations on the Kirchhoff–Routh function. Proc. Natl. Acad. Sci. 27, 576–577 (1941)

  22. 22

    Milne-Thomson L.M.: Theoretical Hydrodynamics. Dover, New York (1996)

  23. 23

    Nehari Z.: Conformal Mapping. Dover, New York (1952)

  24. 24

    Prosnak, W.J.: Computation of fluid motions in multiply connected domains. Wissenschaft & Technik, Frankfurt (1987)

  25. 25

    Pullin D.I.: Contour dynamics methods. Ann. Rev. Fluid Mech. 24, 89–115 (1992)

  26. 26

    Saffman P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992)

  27. 27

    Sedov L.I.: Two-Dimensional Problems in Hydrodynamics and Aerodynamics. Interscience Publishers, New York (1965)

  28. 28

    Surana A., Crowdy D.G.: Vortex dynamics in complex domains on a spherical surface. J. Comput. Phys. 227(12), 6058–6070 (2008)

  29. 29

    Whittaker E.T., Watson G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1927)

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Correspondence to Darren Crowdy.

Additional information

In remembrance of Philip Geoffrey Saffman (1931–2008).

Communicated by H. Aref

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Crowdy, D. A new calculus for two-dimensional vortex dynamics. Theor. Comput. Fluid Dyn. 24, 9–24 (2010). https://doi.org/10.1007/s00162-009-0098-5

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  • Two dimensional flow
  • Complex potential
  • Multiply connected


  • 47.15.ki
  • 47.15.km
  • 47.32.C