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Motion of a particle immersed in a two dimensional incompressible perfect fluid and point vortex dynamics

Franck Sueur

TL;DR

This work rigorously derives point vortex dynamics as the zero-radius limit of a rigid body moving in a two-dimensional incompressible, perfect fluid under the Euler equations, covering unbounded and bounded domains, irrotational and vorticity-containing flows, and both massive and massless inertia regimes. The authors formulate explicit ODEs for the solid’s degrees of freedom by introducing added inertia, an a-connection, Kirchhoff potentials, a harmonic circulatory field, and the Kutta–Joukowski lift, with two independent proofs (complex-analytic and real-analytic) of the body-frame reformulation. Energy considerations are central: a renormalized kinetic energy is conserved, and, in bounded domains, Munnier’s geodesic framework extends to nonzero circulation, yielding a forcing term $F(q,q')$ that couples the solid to the fluid via $E(q)$ and $B(q)$. In the presence of vorticity, a Yudovich-type global theory is developed in the body frame, along with macroscopic normal forms and zero-radius limits, establishing a coherent passage to point vortex dynamics even when the ambient flow carries vorticity. The results thus provide a rigorous bridge between fluid-structure interaction and classical point vortex models, including explicit inertia, gyroscopic, and energy structures that survive the shrinking limit and inform the emergent vortex dynamics.

Abstract

In these notes, we expose some recent works by the author in collaboration with Olivier Glass, Christophe Lacave and Alexandre Munnier, establishing point vortex dynamics as zero-radius limits of motions of a rigid body immersed in a two dimensional incompressible perfect fluid in several inertia regimes.

Motion of a particle immersed in a two dimensional incompressible perfect fluid and point vortex dynamics

TL;DR

This work rigorously derives point vortex dynamics as the zero-radius limit of a rigid body moving in a two-dimensional incompressible, perfect fluid under the Euler equations, covering unbounded and bounded domains, irrotational and vorticity-containing flows, and both massive and massless inertia regimes. The authors formulate explicit ODEs for the solid’s degrees of freedom by introducing added inertia, an a-connection, Kirchhoff potentials, a harmonic circulatory field, and the Kutta–Joukowski lift, with two independent proofs (complex-analytic and real-analytic) of the body-frame reformulation. Energy considerations are central: a renormalized kinetic energy is conserved, and, in bounded domains, Munnier’s geodesic framework extends to nonzero circulation, yielding a forcing term that couples the solid to the fluid via and . In the presence of vorticity, a Yudovich-type global theory is developed in the body frame, along with macroscopic normal forms and zero-radius limits, establishing a coherent passage to point vortex dynamics even when the ambient flow carries vorticity. The results thus provide a rigorous bridge between fluid-structure interaction and classical point vortex models, including explicit inertia, gyroscopic, and energy structures that survive the shrinking limit and inform the emergent vortex dynamics.

Abstract

In these notes, we expose some recent works by the author in collaboration with Olivier Glass, Christophe Lacave and Alexandre Munnier, establishing point vortex dynamics as zero-radius limits of motions of a rigid body immersed in a two dimensional incompressible perfect fluid in several inertia regimes.

Paper Structure

This paper contains 31 sections, 36 theorems, 284 equations.

Table of Contents

  1. Introduction
  2. Case of an unbounded irrotational flow
  3. Reduction to an ODE. Statement of Theorem \ref{['pasdenom']}
  4. Explicit definition of the ODE coefficients $\mathcal{M}_{a,\, \theta}$, $\Gamma_{a,\theta}$ and $B_{ \theta}$
  5. Kirchhoff potentials
  6. Added inertia
  7. Added inertia connection
  8. Harmonic field
  9. Conformal center
  10. Kutta-Joukowski field
  11. Reformulation as an ODE in the body frame. Statement of Theorem \ref{['reup']}
  12. Reformulation of the solid equations
  13. Decomposition of the velocity field
  14. A complex-analytic proof of Theorem \ref{['reup']}
  15. A real-analytic proof of Theorem \ref{['reup']}
  16. ...and 16 more sections

Key Result

Theorem 2.2

There exists a $C^{\infty}$ mapping $\theta \in \mathbb R \mapsto ({\mathcal{M}}_{a, \theta} , B_{\theta} ) \in S^{+}_3 (\mathbb{R}) \times \mathbb{R}^3$ depending only on $\mathcal{S}_0$ such that the equations Euler1-Euler3b are equivalent to the following ODE for $q=(h , \theta)$: where $\Gamma_{a,\theta}$ denotes the a-connection associated with ${\mathcal{M}}_{a, \theta}$, the fluid ve

Theorems & Definitions (70)

  • Definition 1.1: Massive and massless particles
  • Conjecture : $\mathcal{C}$
  • Definition 2.1: a-connection
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • proof
  • Proposition 2.6
  • Lemma 2.7
  • ...and 60 more