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Journal of Nonlinear Science

, Volume 22, Issue 5, pp 849–885 | Cite as

Bilinear Relative Equilibria of Identical Point Vortices

Article

Abstract

A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, conveniently taken to be the x- and y-axes of a Cartesian coordinate system, is introduced and studied. In the general problem we have m vortices on the y-axis and n on the x-axis. We define generating polynomials q(z) and p(z), respectively, for each set of vortices. A second-order, linear ODE for p(z) given q(z) is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm’s comparison theorem, is that if p(z) satisfies the ODE for a given q(z) with its imaginary zeros symmetric relative to the x-axis, then it must have at least nm+2 simple, real zeros. For m=2 this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that, given q(z)=z 2+η 2, where η is real, there is a unique p(z) of degree n, and a unique value of η 2=A n , such that the zeros of q(z) and p(z) form a relative equilibrium of n+2 point vortices. We show that \(A_{n} \approx\frac{2}{3}n + \frac{1}{2}\), as n→∞, where the coefficient of n is determined analytically, the next-order term numerically. The paper includes extensive numerical documentation on this family of relative equilibria.

Keywords

Ideal fluids Vortex dynamics Point vortices Relative equilibria Polynomials 

Mathematics Subject Classification

76B47 34M99 34B24 15A15 

Notes

Acknowledgements

H.A. thanks Peter Clarkson for correspondence regarding the integrable ODE mentioned in Sect. 6, and he thanks Kevin O’Neil for several comments regarding issues relevant to this work. This work was supported in part by a Niels Bohr Visiting Professorship at the Technical University of Denmark sponsored by the Danish National Research Foundation. Peter Beelen was partially supported by DNRF (Denmark) and NSFC (China), grant No. 11061130539.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Engineering Science & MechanicsVirginia TechBlacksburgUSA
  2. 2.Department of MathematicsTechnical University of DenmarkKgs. LyngbyDenmark
  3. 3.Center for Fluid DynamicsTechnical University of DenmarkKgs. LyngbyDenmark

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