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Preparation of Free-Surface Hyperbolic Water Vortices
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A transformation between stationary point vortex equilibria.

Vikas S Krishnamurthy1, Miles H Wheeler2, Darren G Crowdy3

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Proceedings. Mathematical, Physical, and Engineering Sciences
|September 14, 2020
PubMed
Summary
This summary is machine-generated.

A novel transformation unifies diverse point vortex equilibria in fluid dynamics. This method generates new equilibria with adjustable parameters, creating infinite hierarchies or finite sequences of solutions.

Keywords:
Adler–Moser polynomialsBurchnall–Chaundypoint vortex equilibria

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Area of Science:

  • Fluid Dynamics
  • Mathematical Physics

Background:

  • Stationary point vortex equilibria are fundamental in understanding fluid dynamics.
  • Existing methods for generating these equilibria are often specific and lack unification.

Purpose of the Study:

  • To introduce a new transformation for generating stationary point vortex equilibria.
  • To demonstrate the unifying power of this transformation across known and new hierarchies.

Main Methods:

  • A novel transformation is applied to point vortex equilibria with specific circulation values.
  • The transformation introduces a free complex parameter, acting as an integration constant.
  • Iteration of the transformation generates complex hierarchies of equilibria.

Main Results:

  • The transformation produces new equilibria with adjustable parameters.
  • It generates infinite hierarchies or finite sequences of equilibria.
  • Starting from a single vortex, it recovers known hierarchies (Adler-Moser polynomials) and a new class of Loutsenko polynomials.

Conclusions:

  • The new transformation unifies disparate results in point vortex equilibrium research.
  • It provides a systematic method for generating complex vortex configurations.
  • The transformation's applicability to Loutsenko polynomials is a novel finding.