# Long-Term Dynamics of Water Droplet Impact on Rotating Hydrophilic Disk
###### Abstract  
Ice accretion from the impingement of supercooled water droplets on the rotating components of aero-engines reduces engine efficiency and poses significant in-flight safety risks. In the present study, we experimentally investigate the impact of water droplets on the center of a rotating disk to gain insights into the icing mechanisms on these components. The effects of impact velocity and disk rotation speed on dynamic behaviors are systematically explored by visualizing the phenomena and quantitatively analyzing the evolution of droplet diameters during long time durations. Three distinct regimes of impact dynamics are identified based on the final states: stable rotation, stable ring, and ring ejection. The experimental results reveal that the spreading phase is primarily governed by inertial effects, with minimal influence from disk rotation, while the latter significantly affects the retraction phase. The maximum spreading factor increases with the impact velocity and shows little dependence on rotation, and the spreading time remains nearly unchanged. Scaling laws for the maximum and equilibrium spreading factors as functions of the Weber number and rotational Bond number are established. While the maximum spreading factor increases with impact velocity on static  
This topic has been extensively studied in the literature, providing a solid foundation for understanding droplet behavior on solid surfaces [20; 21]. After hitting the solid surface, droplets can exhibit different outcomes such as deposition, splash, or rebounding, depending on the impact conditions [22]. Key factors influencing droplet dynamics include droplet properties (e.g., size, surface tension, viscosity [23]), impact velocity, and surface properties (e.g., wettability [24; 25], roughness [26; 27], temperature [28; 29], curvature [30; 31]). In comparison to the wealth of studies on static surfaces, research on droplet impact on rotating surfaces is relatively sparse. Among these studies, the focus has often been on the behavior of droplets impacting superhydrophobic surfaces, particularly regarding the reduction of contact time due to rotational effects. [PERSON] et al. [32] investigated the dynamics of droplets striking the center of high-speed rotating superhydrophobic walls and found that the spreading dynamics were primarily governed by inertia. During the retraction phase, the outward centrifugal force became significant, causing a doughnut-shaped droplet to form, which notably reduced contact time compared to that on stationary superhydrophobic surfaces. [PERSON] et al. [33] observed that rotation reduced both the contact time and the collision force of droplets impacting the center and eccentric regions of rotating disks. [PERSON] et al. [34] derived scaling laws for the maximum spreading diameter as functions of the Weber and capillary numbers, as well as for dimensionless contact time as a function of the capillary number for bouncing droplets after eccentric impacts on rotating superhydrophobic surfaces. [PERSON] et al. [35] focused on the splashing regime during eccentric droplet impact on a spinning superhydrophobic disk, identifying thresholds between bouncing and splashing regimes. Droplets impacting rotating hydrophilic surfaces have also garnered attention due to their frequent occurrence in real-world applications. [PERSON] et al. [36] is one of the primary studies on this problem where the authors examined the effect of surface rotating speed on droplet impact behavior, identifying three distinct outcomes, including complete drop adherence to the surface, partial reflection, and total reflection. [PERSON] et al. [37] analyzed the influence of impact eccentricity on droplet spreading on copper surfaces. [PERSON] et al. [38] explored the dynamics of ethanol droplets striking the center of rotating glass or silicon substrates through both numerical simulations and experiments.