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

Olivier Glass (1) , Alexandre Munnier (2) , Franck Sueur (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 EDP - Equations aux dérivées partielles
3 IMB - Institut de Mathématiques de Bordeaux

The vortex point 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 pointwise massless 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 vanishing 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. In order to understand the solid dynamics one has to evaluate the pressure field on the boundary of the solid, that is, where the singularity is concentrated at the limit. Moreover the Newton equations driving the particle's dynamics involve a singular perturbation problem in time whereas the fluid state may be seen as solving an auxiliary problem. The fluid velocity can indeed be recovered by an elliptic-type problem where time appears only as a parameter. Since the analysis involves singular perturbation problem both in space and in time and that nothing excludes a priori some sharp energy transfer from the fluid toward the rigid immersed particle it is crucial to recast the solid dynamics under normal forms. This form makes appear the gyroscopic structure of these singularities and allows one to obtain uniform estimates on the dynamics thanks to some energy-type quantities modulated by the limit dynamics, and therefore to describe the transition of the dynamics in the limit. In order to get such normal forms we first establish that the Newton equations can be seen as a geodesic equation, with a metric associated with the well-known ``added inertia'' phenomenon, under an applied force similar to the Lorentz force which can be seen as an extension of the celebrated Kutta-Joukowski lift force. Then, in the limit, surprising relations between leading and subprincipal orders of various terms and the modulation variables show up and allow us to establish a normal form with a more gyroscopic structure. As a byproduct of our analysis we also prove that in the different regime where the body shrinks with a fixed mass the limit equation is a second-order differential equation involving a Kutta-Joukowski-type lift force, which extends the result of [Glass O., Lacave C., Sueur F., On the motion of a small body immersed in a two dimensional incompressible perfect fluid, Bull. Soc. Math. France Volume 142, fascicule 3, 489-536, 2014. ] to the case where the domain occupied by the solid-fluid system is bounded.

Domaines: Equations aux dérivées partielles [math.AP]

Olivier Glass, Alexandre Munnier, Franck Sueur. Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid. 2016. ⟨hal-00950544v2⟩