# Long-Term Dynamics of Water Droplet Impact on Rotating Hydrophilic Disk
## 3 Results and Discussions
### Spreading Dynamics on Rotating Disks
#### 3.2.4 Spreading Time and Retraction Time  
We decompose the droplet dynamics before reaching the equilibrium state into the spreading phase and the retraction phase. This enables us to investigate the effect of disk rotation on the two phases separately. The spreading time \(t_{\text{d}}\) is defined as the duration required for the droplet to reach the maximum spreading factor from the impact instant \(t=0\) ms. The retraction time \(t_{\text{r}}\) is defined as the duration from the moment of the maximum spreading to the end of the retraction phase. The spreading and retraction time as a function of \(We\) for different \(Bo\) is shown in Figure 12. It is observed that the variation of \(t_{\text{d}}\) is less than 1 ms with respect to \(We\) for the same \(Bo\), while the variation of \(t_{\text{d}}\) versus \(Bo\) for the same \(We\) is less than 0.4 ms. Therefore, the spreading time hardly depends on \(We\) and \(Bo\). The independence of \(t_{\text{d}}\) on \(We\) for \(We\geq 20\) was also observed in previous studies on stationary disks [50]. The reason why the disk rotation has little effect on the spreading time is because the spreading is dominated by the inertial effect, in agreement with previous observations for droplets on superhydrophobic rotating surfaces [32; 34].  
When the droplet impinges the stationary wall, \(t_{\text{r}}\) increases with respect to the Weber number, from 13.75 ms at \(We=20\) to 37 ms at \(We=200\). However, the retraction time decreases with increasing \(We\) in the range of [67; 200] for a given \(Bo>0\). This is like the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(Bo\) & **0** & **0.3** & **1.2** & **2.6** & **4.7** & **7.3** & **10.5** \\ \hline Value of \(c\) & 0.00318 & 0.00349 & 0.00442 & 0.00636 & 0.00827 & 0.00854 & 0.0104 \\ Standard error of \(c\) & 0.00031 & 0.00010 & 0.00021 & 0.00081 & 0.00017 & 0.00056 & 0.00067 \\ Value of \(d\) & 1.329 & 1.471 & 1.563 & 1.621 & 1.565 & 1.688 & 1.632 \\ Standard error of \(d\) & 0.045 & 0.017 & 0.035 & 0.014 & 0.022 & 0.103 & 0.095 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fitting parameters \((c,d)\).  
Figure 11: The relationship of the equilibrium spreading factor \(\beta_{\text{eq}}\) with \(We\) and \(Bo\). (**a**) \(\beta_{\text{eq}}\) as a function of \(Bo\) for different \(We\). (**b**) \(\beta_{\text{eq}}\) as a function of \(We\) for different \(Bo\).  
Figure 12: Variation of spreading time (**a**) \(t_{\text{d}}\) and (**b**) retraction time \(t_{\text{r}}\) with respect to \(We\) and \(Bo\).  
decrease in the retraction time with respect to \(We\) observed for droplets impacting rotating superhydrophobic surfaces as reported in [PERSON] et al. [32] and [PERSON] et al. [34]. As shown in Figure 12b, \(t_{\rm r}\) decreases with increasing \(Bo\) for a fixed Weber number. At \(We=67\), \(t_{\rm r}\) decreases from 26.25 ms to 8.85 ms by increasing \(Bo\) from 0 to 10.5. At \(We=200\), \(t_{\rm r}\) decreases more significantly, from 37 ms at \(Bo=0\) to 5.25 ms at \(Bo=10.5\). On one hand, the maximum spreading area of the droplets is larger at high Weber numbers, representing larger surface energy, which consequently induces a higher retraction velocity. On the other hand, the equilibrium spreading factor \(\beta_{\rm eq}\) increases with \(Bo\) at a given \(We\), thus the drop of the spreading factor from \(\beta_{\rm max}\) (only dominated by \(We\)) to \(\beta_{\rm eq}\) decreases with \(Bo\) at a given \(We\).