# Long-Term Dynamics of Water Droplet Impact on Rotating Hydrophilic Disk
## 3 Results and Discussions
### Spreading Dynamics on Rotating Disks  
#### 3.2.1 Spreading Factor  
Figure 7 shows the evolution of the spreading factor for droplets impacting the disk at different Weber numbers when the disk is stationary or rotating. Each experiment was repeated three times, and error bands were plotted alongside the ensemble-averaged spreading factor curve. The narrow error bands, compared to the difference between adjacent curves caused by changes in \(We\), suggest that the variations in the curves are not due to measurement errors but to actual differences in \(We\). Figure 7a depicts the spreading factor evolution for droplets impacting a stationary disk at varying Weber numbers. As \(We\) increases, the initial rising of the spreading factor \(\beta\) becomes steeper, indicating faster spreading. The maximum spreading factor increases with \(We\), reflecting the greater conversion of inertia into surface energy. When \(We\) increases from 20 to 200, the maximum spreading factor rises from 1.90 to 3.50, and the time for the droplet to reach this maximum decreases slightly from 3.75 ms to 3.25 ms. Notably, at low \(We\) values (e.g., \(We=20\) and 33), the droplet diameter undergoes minor oscillation in its equilibrium state at the disk center, whereas for higher \(We\), e.g., \(We>67\), these oscillations disappear. Figure 5: Droplet snapshots at \(We=166\) and \(Bo=10.5\). Figure 6: Distribution of droplets’ final states. At low rotating speeds (e.g., \(Bo=0.3\)), as shown in Figure 7b, the evolution of the spreading factors closely resembles that on a stationary disk. Increasing \(We\) results in a higher maximum spreading factor and faster spreading velocity due to enhanced inertial effects. Specifically, as \(We\) rises from 20 to 200, the maximum spreading factor increases from 1.92 to 3.60, while the time to reach the maximum spreading factor decreases from 3.6 ms to 3.2 ms. At \(Bo=0.3\), the centrifugal force has minimal impact, allowing the droplet to complete its spreading and retraction phases, eventually stabilizing as either a single droplet or a ring-like droplet. At \(We=20\) and 33, the droplet exhibits slight oscillations, which disappear at higher \(We\). At \(Bo=10.5\), the spreading characteristics are similar to those at \(Bo=0\) and 0.3, reaffirming that the spreading phase is dominated by inertia. However, at \(Bo=10.5\), the retraction phase is significantly reduced, especially at high \(We\), as the increasing centrifugal force after maximum spreading prevents effective droplet retraction. Figure 8 presents the temporal evolution of the spreading factor for droplets within \(20\leq We\leq 200\), across different values of \(Bo\) ranging from 0.3 to 10.5. The results indicate that the maximum spreading factor on rotating disks is slightly higher than that on stationary disks for the same \(We\). However, disk rotation speed has a minimal impact on the maximum spreading factor or the spreading process when the Weber number is held constant. The variation in spreading time is also minor, with a difference of only 0.2 ms across different bond numbers at \(We=20\), and 0.4 ms for \(We=200\). At low Weber numbers, such as \(We=20\) and 33, the droplet oscillates at the disk center after completing the retraction stage. The oscillation period and amplitude are stable for a given \(We\) but decrease as the impact velocity increases. For instance, at \(We=20\), the oscillation period is 10 ms with an amplitude of approximately 0.15, while at \(We=33\), the period shortens to 8 ms, and the amplitude reduces to about 0.05. Figure 8a,b show more subdued oscillations at \(We=33\). Notably, the oscillation disappears entirely at \(We=67\). As \(We\) increases from 20 to 200 across various \(Bo\) values, the maximum spreading factor rises from 1.92 to 3.76. While the disk's rotation has little influence on the spreading phase, it significantly affects the retraction phase during which the azimuthal velocity component is important.