# Long-Term Dynamics of Water Droplet Impact on Rotating Hydrophilic Disk
## 3 Results and Discussions
### Phenomenal Description
#### 3.1.2 On Rotating Disks  
To investigate the impact dynamics of droplets at different initial velocities on rotating disks with varying speeds, cross-experiments were conducted by changing both the impact velocity and the disk's rotating speed. Figure 3 shows the behavior of a water droplet impacting a rotating disk at \(We=20\) and \(Bo=0.3\). Under these conditions, the droplet's behavior closely mirrors that observed on a stationary disk. After completing the spreading and retraction phases, the droplet continues to oscillate with small amplitude at the center of the disk. This oscillation remains stable, with no indication of stopping even after 50 ms under the given experimental conditions. A similar oscillation was also observed in the droplet's final state at \(We=33\) and \(Bo=0.3\). The droplet's behavior at \(We=166\) and \(Bo=0.3\) is shown in Figure 4. Compared to the case at \(We=20\), the maximum spreading diameter increases significantly due to the higher Weber number. At the maximum spreading, a distinct fingering disturbance appears along the rim edge, as seen in the subfigure at \(t=3\) ms. This disturbance begins to fade after 4 ms during the retraction phase, which progresses more rapidly at high Weber numbers than at lower ones. Eventually, the droplet stabilizes at the center of the rotating disk without oscillation and continues to rotate as a single object. Unlike the phenomenon shown in Figure 3, the central lamella of the droplet at \(We=166\) and \(Bo=0.3\), as shown in Figure 4, breaks from its thinnest region during the final retraction stage around \(t=12\) ms. Following this breakup, the rim's inner edge  
Figure 3: Snapshots of a water droplet impacting a rotating disk at \(We=20\) and \(Bo=0.3\). recoils outward while the outer edge recoils inwards, causing the droplet to assume a crater shape. As the central area dries, the droplet transitions into a ring-like shape, stabilizing at \(t=30\) ms. This ring-like droplet remains attached to the hydrophilic surface. Similar ring-like droplets were previously observed during the retraction phase on rotating superhydrophobic disks, but in those cases, the droplets eventually bounced off the disk [32]. By increasing \(Bo\) to 10.5 while maintaining \(We\) at 166, we observed the impact phenomenon depicted in Figure 5. Compared to the case at \(Bo=0.3\), the impact outcome shows no significant difference, and the maximum spreading diameter remains relatively constant. Clear fingering disturbances quickly appear along the rim edge after impact, but at the higher \(Bo\), these ripples display a rotational tendency, as illustrated in the sub-images at \(t=2\) ms and 3 ms. The ripples spiral outward from the droplet rim, driven by the combined effects of inertia and centrifugal force. Their direction is clockwise, consistent with the disk's counterclockwise rotation. Several red arrows have been added to these two sub-images to enhance visualization. Another notable difference is that droplet retraction is greatly reduced at this high bond number, as the centrifugal force outweighs surface tension during the retraction phase. After 6 ms, as illustrated in Figure 5, there is almost no retraction of the droplet. The central lamella becomes very thin and dries up by 22.5 ms, while the rim exhibits protrusions and fingering disturbances, eventually leading to the droplet being ejected in a finger-like manner. The three distinct types of impact outcomes, based on their final states across the considered \(We\) and \(Bo\) ranges, are summarized in Figure 6. Droplets that stabilize after the spreading and retraction phases and rotate at the same speed as the disk are represented by filled circular markers, defining the stable rotation regime. Hollow circular markers correspond to cases where ring-like-shaped, stable droplets form, characterizing the stable ring regime.