Regular and Chaotic Dynamics

, Volume 23, Issue 5, pp 569–579 | Cite as

Stationary Configurations of Point Vortices on a Cylinder

  • Dariya V. SafonovaEmail author
  • Maria V. Demina
  • Nikolai A. Kudryashov


In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation Γ1 and Γ22 = −μΓ1) are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.


point vortices stagnation points stationary configuration vortices on a cylinder polynomial solution of differential equation 

MSC2010 numbers

33D45 76M23 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Borisov, A.V. and Mamaev, I. S., Mathematical Methods in the Dynamics of Vortex Structures, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).zbMATHGoogle Scholar
  2. 2.
    Kadtke, H.B. and Campbell, L. J., Method for Finding Stationary States of Point Vortices, Phys. Rev. A, 1987, vol. 36, no. 1, pp. 4360–4370.CrossRefGoogle Scholar
  3. 3.
    Aref, H., Relative Equilibria of Point Vortices and the Fundamental Theorem of Algebra, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2011, vol. 467, no. 2132, pp. 2168–2184.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Tkachenko, V.K., Studying the Violations of the Superfluidity of Helium in Broad Capillaries under the Influence of Heat Flow, Thesis PhD, Moscow: Institute for Physical Problems, 1964.Google Scholar
  5. 5.
    Aref, H., Vortices and Polynomials, Fluid Dynam. Res., 2007, vol. 39, nos. 1–3, pp. 5–23.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    O’Neil, K. A., Symmetric Configurations of Vortices, Phys. Lett. A, 1987, vol. 124, no. 9, pp. 503–507.MathSciNetCrossRefGoogle Scholar
  8. 8.
    O’Neil, K. A., Relative Equilibrium and Collapse Configurations of Four Point Vortices, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 117–126.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    O’Neil, K. A., Clustered Equilibria of Point Vortices, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 555–561.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    O’Neil, K. A., Minimal Polynomial Systems for Point Vortex Equilibria, Phys. D, 2006, vol. 219, no. 1, pp. 69–79.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Demina, M.V. and Kudryashov, N.A., Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 562–576.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Demina, M.V. and Kudryashov, N.A., Vortices and Polynomials: Non-Uniqueness of the Adler–Moser Polynomials for the Tkachenko Equation, J. Phys. A, 2012, vol. 45, no. 19, 195205, 12 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Demina, M.V. and Kudryashov, N.A., Point Vortices and Classical Orthogonal Polynomials, Regul. Chaotic Dyn., 2012, vol. 17, no. 5, pp. 371–384.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    O’Neil, K. A., Continuous Parametric Families of Stationary and Translating Periodic Point Vortex Configurations, J. Fluid Mech., 2007, vol. 591, pp. 393–411.MathSciNetzbMATHGoogle Scholar
  15. 15.
    O’Neil, K. A., Singular Continuation of Point Vortex Relative Equilibria on the Plane and Sphere, Nonlinearity, 2013, vol. 26, no. 3, pp. 777–804.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    O’Neil, K. A., Relative Equilibrium and Collapse Configurations of Heterogeneous Vortex Triple Rings, Phys. D, 2007, vol. 236, no. 2, pp. 123–130.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    O’Neil, K. A., Stationary States of Identical Point Vortices and Vortex Foam on the Sphere, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2013, vol. 469, no. 2150, 20120622, 12 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Aref, H. and Brøns, M., On Stagnation Points and Streamline Topology in Vortex Flows, J. Fluid Mech., 1998, vol. 370, pp. 1–27.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Smirnov, V. I., Course of Higher Mathematics: Vol. 3, Part 2. Textbook for High Schools, St. Peterburg: BHV, 2010.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Dariya V. Safonova
    • 1
    Email author
  • Maria V. Demina
    • 1
  • Nikolai A. Kudryashov
    • 1
  1. 1.Department of Applied MathematicsNational Research Nuclear University MEPhIMoscowRussia

Personalised recommendations