# Damping of Liquid Sloshing Using Hydrophobic Walls in the Off-Impulse Regime
## 4 Free Surface Damping
### Waves Visualization  
In this section, we describe a series of video snapshots of the tested hydrophobic and hydrophilic containers in order to compare them and discuss the influence of applying an oil-based coating to the walls.  
Figure 9 shows six snapshots extracted from the video for \(\lambda=1.00\) 1 s after the forcing was suddenly stopped (see the inset of Figure 7). A time interval equal to  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\lambda\) & \(\tau_{uncated}^{-1}\) (Hz) & \(\tau_{coated}^{-1}\) (Hz) &
\begin{tabular}{c} **Relative** \\ **Difference (\%)** \\ \end{tabular} \\ \hline
0.16 & 1.78 & 2.22 & 20 \\ \hline
0.37 & 1.53 & 2.00 & 23 \\ \hline
0.45 & 1.07 & 2.12 & 50 \\ \hline
0.69 & 0.57 & 3.84 & 85 \\ \hline
1.00 & 0.53 & 2.50 & 78 \\ \hline
1.18 & 1.25 & 1.53 & 19 \\ \hline
1.44 & 1.11 & 2.00 & 44 \\ \hline
2.20 & 0.98 & 1.58 & 38 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Damping time for different values of \(\lambda\). The relative difference is provided in %.  
Figure 8: Damping values \(\tau f_{0}\) deduced from the best fits for each \(\lambda\) value presented in Table 1. Data points in green represent the hydrophobic containers, while those in red represent the uncoated ones.  
\(0.166\) s \(\simeq 1/2/f_{0}\) separates each pair of images. Following the same methodology, the red color indicates hydrophilic walls and the green color hydrophobic ones. In panel (a), it can be observed that the red wave is slightly larger in amplitude and steeper than the green wave. However, the horizontal width is greater at the top of the green wave's crest. This can be explained by greater adhesion of the fluid to the walls of the container under the hydrophilic condition. Note that both waves are in phase at this instant. In panel (b), the crests of the two waves again reach roughly the same amplitude on the opposite wall. The two waves also have roughly the same shape, although the green wave is again wider than the red one. The waves are still in phase. In panel (c), the waves are completely phase-shifted, and a shape similar to a breaking wave can be detected in the green container. Because of the hydrophobic wall in this container, the fluid cannot adhere to the wall; as a result, its shape is curved, which is contrary to the behavior of the fluid in the hydrophilic container. In panel (d), the waves are again phase-shifted, and an even greater difference in amplitude and shape is observed. In the red hydrophilic container, the waveform is now quite different from panel (a), with most of the liquid stuck to the wall. However, in the green hydrophobic container the wave is significantly attenuated. In panel (e), the wave in the hydrophilic container oscillates with the same amplitude as in panel (a), but now on the opposite wall, whereas the amplitude of the wave in the hydrophobic container continues to decrease. This suggests that the effect of the impact load on the walls is significantly reduced by the hydrophobic coating. Finally, the behavior of the two waves in panel (f) is similar to that in panel (e), except with a shifted phase.  
This behavior was found to be similar in all of our experiments, with the fluid adhering more readily to the uncoated walls of the hydrophilic container than to the walls with hydrophobic coating. The shape of the free surface in the latter container also remained smoother. These features show the importance of the boundary layer and hydrophobic effects, which appear as a source of energy dissipation.  
Figure 9: Snapshots taken after forcing was suddenly stopped for \(\lambda=1.00\). The fluid in the container with hydrophilic walls is dyed red, while the fluid in the container with hydrophobic coating is dyed green. There is a time difference of \(0.166\) s between each pair of images.