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Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics

Fluid motion in tanks is usually described in space industry with the so-called Lomen hypothesis which assumes the vorticity is null in the moving frame. We establish in this contribution that this hypothesis is valid only for uniform rotational motions. We give a more general formulation of this coupling problem, with a compact formulation. We consider the mechanical modeling of a rigid body with a motion of small amplitude, containing an incompressible fluid in the linearized regime. We first establish that the fluid motion remains irrotational in a Galilean referential if it is true at the initial time. When continuity of normal velocity and pressure are prescribed on the free surface, we establish that the global coupled problem conserves an energy functional composed by three terms. We introduce the Stokes - Zhukovsky vector fields, solving Neumann problems for the Laplace operator in the fluid in order to represent the rotational rigid motion with irrotational vector fields. Then we have a good framework to consider the coupled problem between the fluid and the rigid motion. The coupling between the free surface and the ad hoc component of the velocity potential introduces a "Neumann to Dirichlet" operator that allows to write the coupled system in a very compact form. The final expression of a Lagrangian for the coupled system is derived and the Euler-Lagrange equations of the coupled motion are presented.