1212.1144 (Henry O. Jacobs et al.)

On the use of Lie groupoids in fluid-structure interactions    [PDF]

Henry O. Jacobs, Joris Vankerschaver

Thanks to the work of V.I. Arnold [Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfaites] (1966) we know that an ideal fluid can be understood as a Lagrangian mechanical system on a Lie group of volume preserving diffeomorphisms. Moreover, one can reduce by the particle relabeling symmetry of the system to arrive at equations of motion on the Lie algebra of divergence free vector fields. In this article we will investigate the equations of motion for a flexible body immersed in an incompressible ideal fluid. We will find that the configuration manifold for such a system is the source fiber of a Lie groupoid. Moreover, we will reduce by the particle relabeling symmetry of the system to obtain equations of motion on the Lie algebroid in close analogy with the work of V.I. Arnold. As this groupoid acts transitively on itself, we will be able to execute this reduction as an instance of Lagrange-Poincare reduction. We will also address the case of Navier-Stokes fluids by including a dissipative viscous force and amending the original Lie groupoid to include a no-slip condition.

View original: http://arxiv.org/abs/1212.1144