# Damping of Liquid Sloshing Using Hydrophobic Walls in the Off-Impulse Regime
## 2 Experimental Procedure  
### Experimental Setup  
The experiments were performed at the Geophysical Fluid Dynamics Laboratory of the University of Guadalajara, Mexico. Figure 1a shows a scheme of our experimental setup, which is similar to that used by [12]. It consists of two transparent acrylic containers with the same dimensions. The wall roughness is negligible, and of the lateral walls are perpendicular to the free surface of the water. The experimenter controlled their horizontal periodic motion through a DC motor. Both containers were on a moving stage on rails, and oscillated with a given amplitude \(A_{0}\) and determined frequency \(f\). Thus, their harmonic lateral movement is described by \(A(t)=A_{0}\cos(2\pi ft)\). An LED panel was placed on the mobile platform behind the two containers to enhance visualization of the containers. In front of the containers, a high-speed camera fixed on the moving stage recorded the water's superficial waves at 240 fps. Each container was partially filled with colored water. A green vegetable-based dye was used for the water in the hydrophobic-walled container and a red dye for the water in the non-hydrophobic container. All the walls of the first container, including its bottom, were coated with a thin layer of commercial Vaseline. In order to illustrate the different hydrophobic/hydrophilic conditions of the containers, Figure 1a shows a drop of red-dyed water that was left on a flat dry acrylic surface, while Figure 1b shows an identical green-colored drop on the same acrylic surface previously coated with Vaseline. The contact angles [14] are pointed out in each of the photographs. The photographs clearly show that the apparent contact angle \(\theta_{1}\) of the drop on the uncoated surface is greater than the angle \(\theta_{2}\) of the drop on the Vaseline-coated surface, confirming the hydrophobicity of the green container. The panel in Figure 2b points out the different parameters of our experiment: \(W\) is the width of the container, \(L\) is its length, \(H\) is the depth of water before forcing, and \(\eta_{0}\) is the maximum/minimum level of the free surface concerning its equilibrium. The free surface period and decaying time are controlled by the depth of the liquid \(H\) and the dimension of the container [15]. Measurements were made for eight different dimensions, each characterized by an aspect ratio \(\lambda=W/L\). Table 1 shows the values for \(\lambda\), \(W\), \(L\), and \(H\) used in our experiments. Moreover, the natural frequencies of the superficial were calculated using the following equation [16]:  
\[f_{0}=\frac{1}{2\pi}\sqrt{g\frac{\pi}{L}\tanh(\pi\frac{H}{L})} \tag{1}\]  
where \(g\) is the acceleration of gravity. We confirmed the surface tension had no significant impact (lower than \(1\%\)) on the \(f_{0}\) values. Figure 1: (**a**) Schematic of the complete experimental setup, which is similar to that used by [12]. The platform slides on the rails with a minimum of friction; the rails are shown by the horizontal gray lines. Two containers with the same dimensions oscillate on the platform; the water in the uncoated container is dyed red, while the water in the container with hydrophobic walls is dyed green. The motor on the left allows the oscillation frequency to be fixed through the power supply. (**b**) Detailed view of the two containers along with their main parameters: water level \(H\), container length \(L\), container width \(W\), and oscillation amplitude \(\eta_{0}\). Figure 2: Contact angles \(\theta_{1}\) and \(\theta_{2}\) of static drops of water placed on hydrophilic and hydrophobic surfaces. In (**a**), the drop is placed on a dry, solid, and flat surface without Vaseline, with \(\theta_{1}>90^{\circ}\). In (**b**), an identical drop is placed on a Vaseline-coated acrylic surface, with \(\theta_{2}\sim 90^{\circ}\). The spatial scale is represented by the horizontal solid black line, which refers to the drop’s extension on the hydrophobic surface.