Fluid-structure interaction in the lagrange-poincaré formalism: The navier-stokes and inviscid regimes

Abstract

In this paper, we derive the equations of motion for an elastic body interacting with a perfect uid via the framework of Lagrange-Poincaré reduc-tion. We model the combined uid-structure system as a geodesic curve on the total space of a principal bundle on which a diffeomorphism group acts. After reduction by the diffeomorphism group we obtain the uid-structure interac- tions where the uid evolves by the inviscid uid equations. Along the way, we describe various geometric structures appearing in uid-structure interactions: principal connections, Lie groupoids, Lie algebroids, etc. We ffnish by intro-ducing viscosity in our framework as an external force and adding the no-slip boundary condition. The result is a description of an elastic body immersed in a Navier-Stokes uid as an externally forced Lagrange-Poincaré equation. Expressing uid-structure interactions with Lagrange-Poincaré theory provides an alternative to the traditional description of the Navier-Stokes equations on an evolving domain. © American Institute of Mathematical Sciences.