Theoretical Characterization of Liquid Sloshing in Containers

This paper theoretically characterizes the sloshing phenomena at the free liquid surface in partially or filled stationary containers based on the analogy with surface gravity waves. The oscillations of the liquid surface and the associated dynamics of the liquid-air interface results in interfacial wave displacement of two types progressive (in-phase) and standing (out of phase) modes. This study involves a rectangular stationary fluid tank partially filled with the tested fluids and subjected to the dominant progressive wave at the liquid-air interface. The sloshing strength is decided based on the temporal progression of the generated wave spectrum obtained utilizing small-amplitude wave theory and linear stability analysis in combination. The expressions for determining the angular frequency, the phase speed, and the corresponding temporal growth rate of the progressive disturbances is derived. The tested fluids include standard liquid water and commercially preferable fluids ethylene glycol and glycerol. This model developed based on the irrotationality of fluid motion neglects the viscous influence and comprises the gravity and the surface tension mimicking a practical scenario. Utilizing deep-water assumptions, the effect of surface tension has been determined for wavenumbers in the range of 490-1250 rad/m. The independent evolution demonstrates the influence of surface tension on the temporal growth rate. Water with the maximum surface tension has a higher growth rate than a low surface tension fluid ethylene glycol among the tested fluids. The effect of surface tension is to escalate the temporal growth rate leading to instability and increased liquid sloshing rate.