Structure of geometrical nonlinearities in problems of liquid sloshing in tanks of non-cylindrical shape

Abstract

Structure of geometrical nonlinearities in mathematical model of liquid sloshing in tanks of non-cylindrical shape is under consideration. In contrast to the case of cylindrical reservoir, some new types of nonlinearities occur in mathematical statement of the problem. They are connected with four main reasons. First, they are determined by new normal modes, which correspond to non-cylindrical shape of the tank and take into account some nonlinear properties of the problem (for example, they follow tank walls above level of a free surface). Second, determination of the potential energy of the liquid includes tanks geometry in close vicinity of cross-section of undisturbed free surface of the liquid and tank walls. Third type of manifestation of geometrical nonlinearities is connected with compensation of elevation of liquid level due to non-cylindrical type of tank shape for providing law of mass conservation. The fourth type of nonlinearities is connected with simultaneous manifestation of physical and geometrical nonlinearities. Investigation showed that mostly manifestation of nonlinear properties of liquid sloshing, connected with geometrical nature, is predetermined by inclination and curvature of tank walls in close vicinity of contact of undisturbed liquid with tank walls. We illustrated some general properties of geometrical nonlinearities by the example of three cases of tanks, namely, cylindrical, conic, and paraboloidal tank, which is selected such that its walls have the same inclination near free surface of the liquid as conic tank, but in this case curvature is manifested supplementary.