# Dynamics of rigid bodies in a two dimensional incompressible perfect fluid

## TL;DR

This paper models the motion of rigid bodies in a 2D incompressible perfect fluid, reformulating the dynamics as a second-order ODE that captures fluid-structure interactions, including vorticity and circulation effects.

## Contribution

It introduces a novel reformulation of rigid body dynamics in fluid as a second-order ODE with coefficients depending on vorticity and circulation, highlighting geometric and gyroscopic effects.

## Key findings

- Reformulation of Newton's laws as a second-order ODE for rigid bodies.
- Identification of geodesic and gyroscopic effects in fluid-structure interactions.
- Analysis of interactions involving vorticity, circulation, and potential flows.

## Abstract

We consider the motion of several rigid bodies immersed in a two-dimensional incompress-ible perfect fluid, the whole system being bounded by an external impermeable fixed boundary. The fluid motion is described by the incompressible Euler equations and the motion of the rigid bodies is given by Newton's laws with forces due to the fluid pressure. We prove that, for smooth solutions, Newton's equations can be recast as a second-order ODE for the degrees of freedom of the rigid bodies with coefficients depending on the fluid vorticity and on the circulations around the bodies, but not anymore on the fluid pressure. This reformulation highlights geodesic aspects linked to the added mass effect, gyroscopic features generalizing the Kutta-Joukowski-type lift force, including body-body interactions through the potential flows induced by the bodies' motions, body-body interactions through the irrotational flows induced by the bodies' circulations, and interactions between the bodies and the fluid vorticity.