Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes

Henry Jacobs; Joris Vankerschaver

doi:10.3934/jgm.2014.6.39

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2014, Volume 6, Issue 1: 39-66. Doi: 10.3934/jgm.2014.6.39

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Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes

Henry Jacobs^1, and
Joris Vankerschaver^1,

1.
Department of Mathematics, Imperial College London, London SW7 2AZ

Received: December 2012

Revised: February 2014

Published: March 2014

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we derive the equations of motion for an elastic body interacting with a perfect fluid via the framework of Lagrange-Poincaré reduction. We model the combined fluid-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 fluid-structure interactions where the fluid evolves by the inviscid fluid equations. Along the way, we describe various geometric structures appearing in fluid-structure interactions: principal connections, Lie groupoids, Lie algebroids, etc. We finish by introducing 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 fluid as an externally forced Lagrange-Poincaré equation. Expressing fluid-structure interactions with Lagrange-Poincaré theory provides an alternative to the traditional description of the Navier-Stokes equations on an evolving domain.

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Mathematics Subject Classification: 76T99, 53Z05, 22A22, 22E70.

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