Nonlinearity of Finite-Amplitude Waves in Rectangular Containers

This paper investigates the two-dimensional nonlinear finite-amplitude sloshing dynamics of an irrotational, incompressible fluid in a rectangular container. In specific, the paper addresses the influence of surface tension represented by a coefficient, β, and the ratio between the fluid height and the container’s width, represented by h/L, on the nonlinear normal sloshing modes. To achieve this objective, we first study the influence of, β, and, h/L, on the modal frequencies and generate a map in the (h/L, β) parameters’ space to highlight regions of possible nonlinear internal resonances up to the fifth mode. The map is used to characterize the regions where a single uncoupled nonlinear mode is sufficient to capture the response of surface waves. For these regions, we study the influence of surface tension on the effective nonlinearity of the first four modes and illustrate its considerable influence on the softening/hardening behavior of the uncoupled nonlinear modes. Subsequently, we investigate the response of the sloshing waves near internal resonance of the two-to-one type. We show that, in the vicinity of these internal resonances, the steady-state sloshing response can either contain a contribution from the two interacting modes (coupled-mode response) or only the higher frequency mode (un-coupled high-frequency mode response). We show that the regions where the coupled-mode uniquely exists has a clear dependence on the surface tension.