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

TL;DR

This paper develops a geometric framework using Lagrange-Poincare reduction to describe fluid-structure interactions, encompassing both inviscid and viscous regimes, offering a novel perspective on coupled elastic body and fluid dynamics.

Contribution

It introduces a geometric approach to fluid-structure interaction modeling, unifying inviscid and viscous regimes within the Lagrange-Poincare formalism, and incorporates viscosity as an external force.

Findings

Derived equations for elastic body and perfect fluid interaction.

Extended the framework to include viscosity and no-slip boundary conditions.

Provided an alternative geometric formulation to traditional fluid-structure models.

Abstract

In this paper, we derive the equations of motion for an elastic body interacting with a perfect fluid via the framework of Lagrange-Poincare 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-Poincare equation. Expressing fluid-structure interactions with Lagrange-Poincare theory provides an alternative to the traditional description of the Navier-Stokes equations on an evolving domain.