# Damping of Liquid Sloshing Using Hydrophobic Walls in the Off-Impulse Regime
## 4 Free Surface Damping  
### Damping Time  
As mentioned above, the main objective of this work is to determine the influence of hydrophobic walls on the damping of the superficial waves after the container's oscillation is suddenly stopped. Figure 6 shows the oscillations of the water level for both the uncoated container (red line) and the hydrophobic container (green line). It can be observed that the superficial waves in the container with hydrophobic walls are damped faster than the waves in the uncoated container. To quantify the damping time after the forcing was suddenly stopped, we used the following equation to fit the experimental points:  
\[\eta(t)=\eta_{0}\exp(-t/\tau)\cos(2\pi f_{0}t) \tag{2}\]  
where \(\eta_{0}\) corresponds to the initial amplitude and \(f_{0}\) is the natural frequency of the superficial waves. To experimentally estimate the natural frequency of the superficial waves, the Fourier transform of the signal shown in Figure 6 was calculated. The spectrum for \(\lambda=1.00\) is shown in Figure 7.  
The first peak represents the external forcing frequency (\(f=1.18\) Hz) just before the excitation was stopped. The second peak corresponds to the natural frequency of the system (\(f=3.8\) Hz), which is consistent with Equation (1) and Table 1. The amplitude of the free surface waves in the uncoated container is observed to be greater than in the hydrophobic coated container by approximately 33%. The inset of Figure 7 shows an example of the evolution of the free surface elevation \(\eta\) as a function of time for both containers after forcing was turned off. The dashed blue line is the external envelope of the damping process, i.e., \(\eta_{0}\exp(-t/\tau)\), where \(\eta_{0}\) and \(1/\tau\) are two fitting parameters. The best fitted values were \(\tau^{-1}=2.50\) Hz for the container with hydrophobic coating and 0.53 Hz for the container with uncoated walls. The higher value of \(\tau^{-1}\) for the hydrophobic container indicates greater damping. In this case with \(\lambda=1.00\), the hydrophobic coating on the walls leads to about five times better dissipation of the superficial waves than in the uncoated container. Figure 5: Oscillation amplitude \(\eta_{0}\) of the water level for \(\lambda=1.00\) as a function of the forcing frequency \(f\) (left-hand axis). Green bars: hydrophobic container walls; red bars: uncoated container walls. The blue solid line represents the ratio between both amplitudes in %, with an average of approximately 33%. Applying the same process to all available containers (see Table 1), it is possible to estimate \(\tau\) for the coated and uncoated configurations. Our results are reported in Figure 8 in the form of nondimensional parameters; the \(x\)-axis is \(\lambda\), while the \(y\)-axis corresponds to \(f_{0}\tau\). The latter quantity corresponds to the number of periods necessary for the system to be damped. The red curve is always above the green curve, which means that the uncoated containers require more time than the hydrophobic containers to attenuate the superficial waves. Figure 6: Time history of the water level after periodic forcing was suddenly stopped for \(\lambda=1.00\). The green line represents the container with hydrophobic coating and the red line represents the uncoated container. Figure 7: Fourier spectrum of the free surface displacement in Figure 6 (\(\lambda=1.00\)). Inset: the signal in Figure 6, where the dashed blue lines are exp\((-t/\tau)\). It should be mentioned that an additional test was carried out on a pair of containers with larger dimensions of \(W=21\) cm and \(L=47.5\) cm, corresponding to an aspect ratio \(\lambda=0.44\). It is interesting to note that the damping ratio was almost equal (8% greater) compared to that reported for \(\lambda=0.45\) in Table 2. This result leads us to believe that \(\lambda\) is indeed the correct nondimensional parameter to consider.