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Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid

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Abstract

The point vortex system is usually considered as an idealized model where the vorticity of an ideal incompressible two-dimensional fluid is concentrated in a finite number of moving points. In the case of a single vortex in an otherwise irrotational ideal fluid occupying a bounded and simply-connected two-dimensional domain the motion is given by the so-called Kirchhoff–Routh velocity which depends only on the domain. The main result of this paper establishes that this dynamics can also be obtained as the limit of the motion of a rigid body immersed in such a fluid when the body shrinks to a massless point particle with fixed circulation. The rigid body is assumed to be only accelerated by the force exerted by the fluid pressure on its boundary, the fluid velocity and pressure being given by the incompressible Euler equations, with zero vorticity. The circulation of the fluid velocity around the particle is conserved as time proceeds according to Kelvin’s theorem and gives the strength of the limit point vortex. We also prove that in the different regime where the body shrinks with a fixed mass the limit dynamics is governed by a second-order differential equation involving a Kutta–Joukowski-type lift force. To prove these results, in a first step we reformulate the dynamics of the body in order to make more explicit different kind of interactions with the fluid. Precisely we establish that the Newton–Euler equations of translational and rotational dynamics of the body can be seen as a 3-dimensional ODE with coefficients solving an auxiliary problem for the fluid. When the circulation around the body is zero, this equation is a geodesic equation for a metric associated with the well-known “added inertia” phenomenon; with a nonzero circulation, an additional term similar to the Lorentz force of electromagnetism appears. Then, in the zero-radius limit, surprising relations between leading and subprincipal orders of various terms and modulation variables show up and allow us to establish a normal form with a gyroscopic structure. This leads to uniform estimates on the body’s dynamics thanks to a modulated energy, and therefore allows us to describe the transition of the dynamics in the limit.

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Notes

  1. The simple connectedness is a simplifying assumption but is actually not essential in the analysis.

  2. Also called external conformal radius or transfinite diameter in other contexts [37].

Abbreviations

B(q):

Magnetic-type field acting on the solid

\(C^{{{\mathcal {S}}_0}}\) :

Value of \(\psi ^{{{\mathcal {S}}_0}}_{{ -1}}\) on \(\partial \mathcal S_0\)

\(\mathrm{d}s\) :

Arc length

\(e_1, e_2\) :

Unit vectors of the canonical basis

E(q):

Electric-type field acting on the solid

\(\mathcal {E} (q,p)\) :

Total energy

\(\mathcal {E}_{\varepsilon } (q,p)\) :

Total energy of the shrinking solid

\(\mathcal F_0\) :

Domain initially occupied by the fluid

\(\mathcal F(q)\) :

Fluid domain associated with the solid position q

F(q, p):

Total force acting on the solid

\(F^{{{\mathcal {S}}_0}}_{\vartheta } \) :

Force term when \(\varOmega = \mathbb {R}^{2}\)

G :

Newtonian potential

h :

Position of the center of mass

\(I_\varepsilon \) :

Diagonal matrix \(\mathrm{diag}(\varepsilon ,1,1)\)

\(\mathcal {J}\) :

Solid’s moment of inertia

\(\mathcal {J}_{\varepsilon }\) :

Moment of inertia of the shrinking solid

\(K_{j}(q,\cdot )\) :

Normal trace of elementary rigid velocities

\(K_{j,\varepsilon } (q,\cdot )\) :

Normal trace of elementary rigid velocities on \(\partial \mathcal S_\varepsilon (q)\)

m :

Solid’s mass

\(m_\varepsilon \) :

Mass of the shrinking solid

\(M_{g}\) :

Genuine solid inertia

\(M_a\) :

Added inertia

M :

Total inertia of the solid

\(M^{{{\mathcal {S}}_0}}_{a}\) :

Added inertia of the solid when \( \varOmega = \mathbb {R}^{2}\)

\(M^{{{\mathcal {S}}_0}}_{{a},\vartheta }\) :

Conjugate matrix of \(M_{a}^{{{\mathcal {S}}_0}}\) by the rotation matrix of angle \(\vartheta \)

\(M^{{{\mathcal {S}}_0}}_{{a},\varepsilon }\) :

Added inertia of \({\mathcal S}_{0,\varepsilon }\) when \(\varOmega =\mathbb {R}^{2}\)

\({M}_\vartheta (\varepsilon )\) :

Rescaled total inertia

n :

Normal vector

\({p}_\varepsilon \) :

Solid velocity with rescaled angular velocity

\(P_a\) :

Added translation impulse

\(P^{{\mathcal {S}}_0}_{a, \vartheta } \) :

Translation impulses when \(\varOmega = \mathbb {R}^{2}\)

\(P_{{ 0}} (q, X) \) :

Harmonic polynomial extending \(\psi ^{{{\mathcal {S}}_0}}_{{ 0}} (q, \cdot ) \) in \(\mathcal S_0\)

\(P_{{ 1}} (q, X) \) :

Harmonic polynomial extending \(\psi ^{{{\mathcal {S}}_0}}_{{ 1}} (q, \cdot ) \) in \(\mathcal S_0\)

q :

Body position

\(q_\varepsilon \) :

Position of the shrinking solid

\(\mathcal Q\) :

Set of body positions without collision

\(\mathfrak Q\) :

Bundle of shrinking body positions without collision

\(\mathfrak Q^{\delta }\) :

Bundle of shrinking body positions at distance \(\delta \) from the boundary

\(\mathfrak Q_{\delta ,\varepsilon _{0}}\) :

Bundle of shrinking body positions at distance \(\delta \) from the boundary with \(\varepsilon < \varepsilon _{0}\)

\(R(\vartheta )\) :

\(2\times 2\) rotation matrix of angle \(\vartheta \)

\(\mathcal {R}(\vartheta )\) :

\(3\times 3\) rotation matrix of angle \(\vartheta \)

\(\mathcal S_0\) :

Domain initially occupied by the solid

\(\mathcal S(q)\) :

Solid domain associated with the solid position q

\(\mathcal S_{0,\varepsilon }\) :

Initial position of the shrinking solid

\(\mathcal S_{\varepsilon }(q)\) :

Position of the shrinking solid

u :

Fluid velocity

\(u^\varOmega \) :

Routh’ velocity

\(u_c\) :

Corrector velocity

U(q):

Potential energy

\(\gamma \) :

Circulation

\(\varGamma \) :

Christoffel symbols

\(\varGamma ^{\mathrm{rot}} \) :

Christoffel tensor related to the solid rotation

\(\varGamma ^{\partial \varOmega } \) :

Christoffel tensor omitting the solid rotation

\(\varGamma ^{{{\mathcal {S}}_0}}_{\vartheta }\) :

Christoffel symbols when \(\varOmega = \mathbb {R}^{2}\)

\(\varepsilon \) :

Typical size of the solid

\(\zeta \) :

Conformal center of \(\mathcal S_0\)

\(\zeta _\vartheta \) :

Conformal center of \(\mathcal S_0\) rotated of \(\vartheta \)

\(\vartheta \) :

Rotation angle of the solid

\(\xi _{j}\) :

Elementary rigid velocities

\(\varPi \) :

Fluid pressure

\(\rho _a\) :

Added angular impulse

\(\tau \) :

Tangential vector

\(\varphi _j(q,\cdot )\) (\(j=1,2,3\)):

Kirchhoff’s potentials

\(\varvec{\varphi }(q,\cdot )\) :

Vector containing the three Kirchhoff potentials

\(\varphi ^{{{\mathcal {S}}_0}}_{j}\) (\(j=1,2,3\)):

Kirchhoff’s potentials when \( \varOmega = \mathbb {R}^{2}\)

\(\varvec{\varphi }^{{\mathcal {S}}_0}\) :

Vector containing the three Kirchhoff potentials when \( \varOmega = \mathbb {R}^{2}\)

\(\varphi _{j,\varepsilon }(q,\cdot )\) (\(j=1,2,3\)):

Kirchhoff’s potentials of the shrinking body

\(\varvec{{\varphi }}_\varepsilon (q,\cdot )\) :

Vector containing the three Kirchhoff potentials

\(\overline{\varphi }_{j,\varepsilon } (q,\cdot )\) :

Functions harmonically conjugated to the Kirchhoff potentials \({\varphi }_{j,\varepsilon }(q,\cdot )\), up to a rotation

\(\overline{\varphi }^{{{\mathcal {S}}_0}}_{j} \) :

Functions harmonically conjugated to the Kirchhoff potentials \({\varphi }^{{{\mathcal {S}}_0}}_{j} \) when \(\varOmega = \mathbb {R}^2\)

\(\psi \) :

Circulatory part of the stream function

\(\psi ^\varOmega \) :

Routh’ stream function

\(\psi _c\) :

Corrector stream function

\(\psi _{\varepsilon }(q,\cdot )\) :

Circulatory part of the stream function of the shrinking solid

\(\psi ^{{{\mathcal {S}}_0}}_{k} (q, \cdot ) \) :

kth-order profile, defined in \(\mathbb {R}^{2} \setminus \mathcal S_0\)

\(\psi ^{\varOmega }_{k} (q, \cdot ) \) :

kth-order profile, defined in \(\varOmega \)

\(\varOmega \) :

Fixed domain occupied by the whole system

\(\varOmega _{\delta }\) :

Set of points at distance \(\delta \) from the boundary

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Acknowledgements

The authors thank the Agence Nationale de la Recherche, Project IFSMACS, Grant ANR-15-CE40-0010 for a partial financial support. The first and third authors were also partially supported by the Agence Nationale de la Recherche, Project DYFICOLTI, Grant ANR-13-BS01-0003-01, the second author by the Agence Nationale de la Recherche, Project OPTIFORM, Grant ANR-12-BS01-0007-04 and the third author by the Agence Nationale de la Recherche, Project BORDS, Grant ANR-16-CE40-0027-01. The third author was also partially supported by the Emergences Project “Instabilities in Hydrodynamics” funded by the Mairie de Paris and the Fondation Sciences Mathématiques de Paris, the Conseil Régional d’Aquitaine, grant 2015.1047.CP, by the H2020-MSCA-ITN-2017 program, Project ConFlex, Grant ETN-765579 and the Fondation Simone et Cino Del Duca.

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Authors and Affiliations

  1. CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Olivier Glass

  2. Université de Lorraine, CNRS, Inria, IECL, F-54000, Nancy, France

    Alexandre Munnier

  3. Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux, 351 cours de la Libération, 33405, Talence Cedex, France

    Franck Sueur

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  1. Olivier Glass
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Glass, O., Munnier, A. & Sueur, F. Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid. Invent. math. 214, 171–287 (2018). https://doi.org/10.1007/s00222-018-0802-4

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