The Journal of Geometric Mechanics (JGM)

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

Pages: 39 - 66, Volume 6, Issue 1, March 2014      doi:10.3934/jgm.2014.6.39

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Henry Jacobs - Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom (email)
Joris Vankerschaver - Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom (email)

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.

Keywords:  Fluid mechanics, solid mechanics, fluid-structure interaction, Lie groups.
Mathematics Subject Classification:  76T99, 53Z05, 22A22, 22E70.

Received: December 2012;      Revised: February 2014;      Available Online: April 2014.