A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion

TL;DR

This paper develops a variational principle for modeling two-dimensional incompressible rotational fluid flow with free surfaces in a moving vessel, coupling fluid dynamics with vessel motion through a novel mathematical framework.

Contribution

It introduces a new variational formulation that couples fluid flow with vessel motion, including innovative treatment of pressure boundary conditions and free surface variations.

Findings

Derivation of a variational principle for coupled fluid-vessel system

Automatic generation of coupled equations for vessel path and fluid motion

Novel treatment of pressure boundary conditions in variational setting

Abstract

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler-Poincar\'e variations, the derivation of free surface variations, and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically.