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Regular and Chaotic Dynamics

, Volume 23, Issue 5, pp 530–550 | Cite as

Finite-time Collapse of Three Point Vortices in the Plane

  • Vikas S. Krishnamurthy
  • Mark A. Stremler
Article

Abstract

We investigate the finite-time collapse of three point vortices in the plane utilizing the geometric formulation of three-vortexmotion from Krishnamurthy, Aref and Stremler (2018) Phys. Rev. Fluids 3, 024702. In this approach, the vortex system is described in terms of the interior angles of the triangle joining the vortices, the circle that circumscribes that triangle, and the orientation of the triangle. Symmetries in the governing geometric equations of motion for the general three-vortex problem allow us to consider a reduced parameter space in the relative vortex strengths. The well-known conditions for three-vortex collapse are reproduced in this formulation, and we show that these conditions are necessary and sufficient for the vortex motion to consist of collapsing or expanding self-similar motion. The geometric formulation enables a new perspective on the details of this motion. Relationships are determined between the interior angles of the triangle, the vortex strength ratios, the (finite) system energy, the time of collapse, and the distance traveled by the configuration prior to collapse. Several illustrative examples of both collapsing and expanding motion are given.

Keywords

ideal flow vortex dynamics point vortices 

MSC2010 numbers

70F07 70K99 76B47 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Erwin Schrodinger International Institute for Mathematics and PhysicsViennaAustria
  2. 2.Department of Biomedical Engineering and MechanicsBlacksburgUSA

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