Regular and Chaotic Dynamics

, Volume 23, Issue 5, pp 519–529 | Cite as

Dipole and Multipole Flows with Point Vortices and Vortex Sheets

  • Kevin A. O’NeilEmail author


An exact method is presented for obtaining uniformly translating distributions of vorticity in a two-dimensional ideal fluid, or equivalently, stationary distributions in the presence of a uniform background flow. These distributions are generalizations of the well-known vortex dipole and consist of a collection of point vortices and an equal number of bounded vortex sheets. Both the vorticity density of the vortex sheets and the velocity field of the fluid are expressed in terms of a simple rational function in which the point vortex positions and strengths appear as parameters. The vortex sheets lie on heteroclinic streamlines of the flow. Dipoles and multipoles that move parallel to a straight fluid boundary are also obtained. By setting the translation velocity to zero, equilibrium configurations of point vortices and vortex sheets are found.


point vortex vortex sheet equilibrium dipole 

MSC2010 numbers

76B47 37F10 34M15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D., Vortex Crystals, in Advances in Applied Mechanics: Vol. 39, E. van derGiessen, H. Aref (Eds.), San Diego: Acad. Press, 2003, pp. 1–79.Google Scholar
  2. 2.
    Burton, G.R., Rearrangements of Functions, Saddle Points and Uncountable Families of Steady Configurations for a Vortex, Acta Math., 1989, vol. 163, nos. 3–4, pp. 291–309.zbMATHGoogle Scholar
  3. 3.
    Cao, D., Liu, Zh., and Wei, J., Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two, Arch. Ration. Mech. Anal., 2014, vol. 212, no. 1, pp. 179–217.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Crowdy, D.G., Exact Solutions for Rotating Vortex Arrays with Finite-Area Cores, J. Fluid Mech., 2002, vol. 469, pp. 209–235.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Crowdy, D. and Marshall, J., Growing Vortex Patches, Phys. Fluids, 2004, vol. 16, no. 8, pp. 3122–3130.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dritschel, D., The Stability and Energetics of Corotating Uniform Vortices, J. Fluid Mech., 1985, vol. 157, pp. 95–134.CrossRefzbMATHGoogle Scholar
  7. 7.
    Elcrat, A.R. and Miller, K. G., Rearrangements in Steady Multiple Vortex Flows, Comm. Partial Differential Equations, 1995, vol. 20, nos. 9–10, pp. 1481–1490.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Elcrat, A., Fornberg, B., Horn, M., and Miller, K., Some Steady Vortex Flows past a Circular Cylinder, J. Fluid Mech., 2000, vol. 409, pp. 13–27.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Elcrat, A., Ferlauto, M., and Zannetti, L., Point Vortex Model for Asymmetric Inviscid Wakes past Bluff Bodies, Fluid Dyn. Res., 2014, vol. 46, no. 3, 031407, 10 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gallizio, F., Iollo, A., Protas, B., and Zannetti, L., On Continuation of Inviscid Vortex Patches, Phys. D, 2010, vol. 239, nos. 3–4, pp. 190–201.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Miller, K. G., Stationary Corner Vortex Configurations, Z. Angew. Math. Phys., 1996, vol. 47, no. 1, pp. 39–56.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Moore, D. W., Saffman, P. G., and Tanveer, S., The Calculation of Some Batchelor Flows: The Sadovskii Vortex and Rotational Corner Flow, Phys. Fluids, 1988, vol. 31, pp. 978–990.CrossRefzbMATHGoogle Scholar
  13. 13.
    O’Neil, K.A., A Vortex Sheet/Point Vortex Dipole, in Proc. of the 24th International Congress of Theoretical and Applied Mechanics (ICTAM’2016, 21–26 Aug 2016, Montreal, Canada), 2 pp.Google Scholar
  14. 14.
    O’Neil, K.A., Collapse and Concentration of Vortex Sheets in Two-Dimensional Flow, Theor. Comput. Fluid Dyn., 2009, vol. 24, nos. 1–4, pp. 39–44.zbMATHGoogle Scholar
  15. 15.
    Saffman, P.G. and Scheffield, J. S., Flow over a Wing with an Attached Free Vortex, Studies in Appl. Math., 1976/77, vol. 57, no. 2, pp. 107–117.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Saffman, P.G. and Szeto, R., Equilibrium Shapes of a Pair of Equal Uniform Vortices, Phys. Fluids, 1980, vol. 23, no. 12, pp. 2339–2342.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Strebel, K., Quadratic Differentials, Ergeb. Math. Grenzgeb. (3), vol. 5, Berlin: Springer, 1984.Google Scholar
  18. 18.
    Turkington, B., On Steady Vortex Flow in Two Dimensions: 1, Comm. Partial Differential Equations, 1983, vol. 8, no. 9, 999–1030.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zannetti, L., Vortex Equilibrium in the Flow past Bluff Bodies, J. Fluid Mech., 2006, vol. 562, pp. 151–171.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of TulsaTulsaUSA

Personalised recommendations