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Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions

New shallow-water equations (SWEs), for sloshing in three dimensions (two horizontal and one vertical) in a vessel which is undergoing rigid-body motion in 3-space, are derived. The rigid-body motion of the vessel (roll–pitch–yaw and/or surge–sway–heave) is modelled exactly and the only approximations are in the fluid motion. The flow is assumed to be inviscid but vortical, with approximations on the vertical velocity and acceleration at the surface. These equations improve previous shallow-water models. The model also extends to three dimensions the essence of the Penney–Price–Taylor theory for the highest standing wave. The surface SWEs are simulated using a split-step alternating direction implicit finite-difference scheme. Numerical experiments are reported, including comparisons with existing results in the literature, and simulations with vessels undergoing full 3-D rotations.