Abstract
In this thesis we investigate the dynamics of coupled liquid sloshing systems, which consist of a one-dimensional array of vessels, partially filled with fluid, being connected together by nonlinear springs. The fluid motion induces a hydrodynamic force on the side walls of the vessel, which induces the vessel to move. The vessel movement is also controlled via a restoring force of the attached springs, which in turn causes the fluid to alter its motion. The simplest coupled liquid sloshing system consists of one vessel connected to a side wall via a spring. We review this example and investigate the techniques documented in the literature to study the linear problem of such a system. Then we extend this linear theory to multivessel systems, in particular focussing on the 2-vessel system which introduces the notion of modes being in-phase or out-of-phase with each other. In the general N-vessel system we also identify the existence in parameter space of internal resonances, where different modes oscillates at the same frequency. Such a resonance provides a mechanism for energy exchange between modes in the nonlinear system. The nonlinear dynamics of the coupled liquid sloshing system are studied by employing a symplectic integration scheme based on a variational principle, to the shallow-water form of the governing equations. We present results for the shallow-water scheme in the linear, weakly nonlinear and fully nonlinear regimes. The shallow-water numerical scheme has trouble dealing with breaking waves, so we perform a feasibility study where we attempt to use a similar approach for a non-shallow fluid system. The governing equations are derived and results presented in the linear amplitude regime for the 1-vessel system. Finally, we develop variational principles for the coupled liquid sloshing system in the Eulerian framework based on the principle of constrained variations to derive the governing fluid equations and free-surface boundary conditions, from a natural Lagrangian functional. We use this constrained variational approach to derive the fully rotational 2D Euler equations and its stream function formulation.