When liquid filled containers are excited vertically, it is known that, for some combinations of frequency and amplitude, the free surface undergoes unbounded motion leading to instability, called parametric instability or parametric resonance, while for other combinations the free surface remains plane. In this paper, the stability of the plane free surface is investigated theoretically when the vessel is a vertical axisymmetric container. The effect of coupled horizontal excitation on the stability is examined. The dynamics of sloshing flows under specified excitations are simulated numerically using fully nonlinear finite element method based on non-linear potential flow theory. A mixed Eulerian-Lagrangian technique combined with 4th-order Runge-Kutta method is employed to advance the solution in time. A regridding technique based on cubic spline is applied to the free surface for every finite time step to avoid possible numerical instabilities. 1. Introduction The motion of the unrestrained free surface of the liquid due to external excitation in the liquid filled containers is known as sloshing. Sloshing is likely to be seen whenever we have a liquid with a free surface in the presence of gravity. At equilibrium the free surface of the liquid is static, when the container is perturbed; an oscillation is set up in the free surface. The phenomenon of liquid sloshing occurs in a variety of engineering applications such as sloshing in liquid propellant launch vehicles, liquid oscillation in large storage tanks by earthquake, sloshing in the nuclear reactors of pool type, nuclear fuel storage tanks under earthquake, and the water flow on the deck of ship. Such liquid motion is potentially dangerous problem to engineering structures and environment leading to failure of engineering structures and unexpected instability. Thus, understanding the dynamic behaviour of liquid free surface is essential. As a result the problem of sloshing has attracted many researchers and engineers motivating to understand the complex behaviour of sloshing and to design the structures to withstand its effects. Liquid sloshing can be stimulated by a variety of container excitations. The container excitation can be horizontal, vertical, or rotational. Under horizontal excitations the liquid free surface experiences normal sloshing; the sloshing frequency will be equal to excitation frequency. When the external excitation frequency is equal to fundamental slosh frequency, the free surface undergoes resonance. Extensive research has been done on sloshing response under pure
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