# Damping of Liquid Sloshing Using Hydrophobic Walls in the Off-Impulse Regime
## 1 Introduction  
The sloshing phenomenon generally occurs in partially filled tanks with accelerated movement, e.g., liquid cargo ships in non-fully loaded navigation, moving oil tank transporters, elevated water towers perturbed by ground motion, and road vehicle tanks [1, 2]. Special applications include aerospace fuel tanks, the stability of which is very sensitive to uncontrolled oscillations [3]. This nonlinear phenomenon can lead to spillage of the transported liquid, or in the worst case damage to the container [4]. If the forcing lasts long enough, the sloshing intensity can cause significant compressive forces on the container walls and may be a source of destabilization or failure, especially if the forcing frequency is equal to the system's natural frequency. Therefore, rapid stabilization or wave damping is crucial to avoiding disasters. Several techniques have been developed to control sloshing and reduce the stresses on container walls. For example, through a series of controlled and detailed experiments, the work of [5] demonstrated the effectiveness of four different vertical baffle configurations to dampen the effects of sloshing and suppress the pressure on walls due to breaking waves. With the same purpose, [PERSON]. [6] studied the effect of damping on the sloshing pressure under resonance conditions through numerical analysis by placing two baffles in a partially filled rectangular tank. Different heights and/or distances between the baffles were tested andprovided good results. Following the same objective, [PERSON]. [7] proposed a new configuration that consisted of placing a horizontal perforated plate inside a container, which significantly reduces the wave amplitude and controls the sloshing. In [8], the authors further improved this method by placing horizontal baffles of different length, porosity, and depth. They found that sloshing tended to decrease when porosity was increased, and that the energy loss was mainly due to eddies generated when the fluid passed through the pores. Therefore, the effectiveness of energy dissipation by baffles in containers is clear. Moreover, it has been shown that the oscillation amplitude of the free surface depends on the viscosity. In [2], the authors found through experiments conducted in a 70% filled tank that higher liquid viscosity results in a greater reduction in sloshing pressure and splashing. They showed that for highly viscous fluids, friction with the container walls dissipates most of the energy. In addition, [PERSON]. [9] used numerical and experimental modeling to demonstrate that for highly viscous liquids the energy is mainly dissipated by the wall boundary layers. Therefore, sloshing is significantly lower with glycerin than with water. Viscosity can also be used to dampen sloshing when used as a thin layer on the free surface of water. In this way, [PERSON]. [10] performed experiments with two non-miscible liquid layers, that is, a layer of water in a partially filled reservoir covered by an upper layer of oil with a higher viscosity. When the depth of the water was 10 cm and the thickness of the oil layer was equal to 5 cm, the free surface was 13% more stable than with a single layer of water, while when the upper layer was 2.5 cm deep it was 6% more stable. The 'anti-sloshing' technique proposed the use of floating liquid foams for the same purpose. A number of studies [11; 12] have experimentally addressed the influence on sloshing of a liquid foam layer. Results have shown that a sufficiently thick foam layer (five layers of bubbles) increases the damping coefficient and reduces the amplitude of free-surface oscillations. Using an analytical model, the authors demonstrated that the dissipation of sloshing energy is mainly due to the bubbles closest to the walls. Based on the work of [12], the authors of [4] proposed placing solid spherical foams of uniform diameter on top of the fluid surface to minimize sloshing.