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–Poincaré 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.
This paper investigates the coupled motion between the dynamics of N vessels coupled together in a one-dimensional array by springs, and the motion of the inviscid fluid sloshing within each vessel. We develop a fully-nonlinear model for the system relative to a moving frame such that the fluid in each vessel is governed by the Euler equations and the motion of each vessel is modelled by a forced spring equation. By considering a linearization of the model, the characteristic equation for the natural frequencies of the system is derived, and analysed for a variety of non-dimensional parameter regimes. It is found that the problem can exhibit a variety of resonance situations from the 1 : 1 resonance to (N + 1)-fold 1 : · · · : 1 resonance, as well as more general r : s : · · · : t resonances for natural numbers r, s, t. This paper focuses in particular on determining the existence of regions of parameter space where the (N + 1)-fold 1 : · · · : 1 resonance can be found.
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