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Letters in Mathematical Physics

, Volume 6, Issue 1, pp 1–16 | Cite as

Motions of vortex patches

  • Jacob Burbea
Article

Abstract

An evolution equation describing the motion of vortrex patches is established. The existence of steady solutions of this equation is proved. These solutions arem-fold symmetric regions of constant vorticity ω0 and are uniformly rotating with angular velocity Ω in the range
$$\tilde \Omega _{m - 1}< \tilde \Omega \leqslant \tilde \Omega _m (\tilde \Omega = \Omega /\omega _0 ,m \geqslant 2)$$
where\(\tilde \Omega _m = (m - 1)/2m\). We call this class, ofm-fold symmetric rotating regionsD, the class
of them-waves of Kelvin. Any
may be regarded as a simply connected region which is a stationary configuration of the Euler equations in two dimensions. If
then any magnification, rotation or reflection is also in
with the same angular velocity Ω ofD. The angular velocity\(\Omega _m = \tilde \Omega _m \omega _0 \) corresponds only to the circle solution, which is a trivial member of every class
,m⩾2. The class
corresponds to the rotating ellipses of Kirchoff. Other properties of the class
are established.

Keywords

Reflection Vortex Statistical Physic Vorticity Angular Velocity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Hill, M.J.M., ‘On the Motion of Fluid, Part of Which is Moving Rotationally and Part Irrotationally’,Phil. Trans. 175, 363–411 (1884).Google Scholar
  2. 2.
    Hoffman, K.,Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N. J., 1962.Google Scholar
  3. 3.
    Keller, H.B. and Langford, W.F., ‘Iterations, Perturbations, and Multiplicities for Nonlinear Bifurcation Problems’,Arch. Pational Mech. Anal. 48, 82–108 (1972).Google Scholar
  4. 4.
    Lamb, H.,Hydrodynamics, Dover Publications, New York, 1945.Google Scholar
  5. 5.
    Love, A.E.H., ‘On the Stability of Certain Vortex Motions’,Proc. Lond. Math. Soc. (1)25, 18–42 (1893).Google Scholar

Copyright information

© D. Reidel Publishing Company 1982

Authors and Affiliations

  • Jacob Burbea
    • 1
  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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