Abstract
The equations of two-dimensional parametric sloshing are derived for a fluid in an arbitrary shape tank. The parametric instability of the liquid free surface is reduced to the Mathieu equation. An energy-growth exponent λ (EGE) and the corresponding non-dimensional energy-growth coefficient (EGC) η are defined for measuring the instability intensity and determining the stability boundaries of the parametrically-excited fluid system. A stability criterion is proposed by using the EGE. On the basis of the numerical tests, the analytical expressions of EGE/EGC are developed for the fluid system that is governed by the Mathieu equation. Under the small excitations, the stability boundaries by using EGE/EGC agree well with those by the traditional theory. The unstable properties, including instability intensity, instability boundaries and competition of unstable patterns etc, of parametric sloshing are analyzed and discussed in details with EGE/EGC for the fluid system. The fluid system always selects such an unstable pattern that possesses a larger energy-growth exponent. A two-dimensional parametric sloshing experiment is respectively conducted for the fluid in the rectangular, circular and U-shaped tanks. The theoretical and experimental stability boundaries of the principal parametric sloshing are obtained and compared for the fluid in three different shape tanks. Some theoretical predictions of the unstable properties of parametric sloshing are verified through the experiment.
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