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

New shallow-water equations, for sloshing in two dimensions (one horizontal and one vertical) in a vessel which is undergoing rigid-body motion in the plane, are derived. The planar motion of the vessel (pitch-surge-heave or roll-sway-heave) is exactly modelled 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 for sloshing. The model also contains the essence of the Penney-Price-Taylor theory for the highest standing wave. The surface shallow water equations are simulated using a robust implicit finite-difference scheme. Numerical experiments are reported, including simulations of coupled translation-rotation forcing, and sloshing on a ferris wheel. Asymptotic results confirm that rotations should be of order h/L where h is the mean depth and L the vessel length, but translations can be of order unity, in the shallow water limit