Three dimensional sloshing of stratified liquid in a cylindrical tank

The sloshing of stratified liquid with density varying with depth in a three dimensional cylindrical tank is considered in the framework of linearized theory. The flow of stratified liquid is no longer irrotional and the governing equation is no longer Laplacian. The stream function is also invalid for three dimensional sloshing, which is different from two dimensional sloshing. We adopt the governing equations and boundary conditions in terms of a pressure function and a density function to instead of velocity potential for uniform liquid. Separation of variables and Laplace transformation methods are used to solve the governing equations for the constant Brunt-Väisälä frequency. The Residue theorem is applied to calculate the inverse Laplace transformation and the resonant behavior is also analyzed. It is found that the natural frequencies are weakened by the stratification of the liquid density. New natural frequencies have appeared which do not exist for uniform liquid. We find that these new frequencies have some special characters; they make the motion history of free surface irregular, the amplitudes caused by new frequencies are remarkable, nearly two-thirds of the amplitude caused by natural frequencies and all new frequencies are less than Brunt-Väisälä frequency N.