# Long-Term Dynamics of Water Droplet Impact on Rotating Hydrophilic Disk
## 2 Experimental Description  
The experiments of droplets impacting a rotating disk are conducted and measured through the devices sketched in Figure 1. The experimental system is composed of a droplet generation apparatus, an electric displacement tool, a rotating disk controlled by an AC servo motor (INOVANCE MS1H2-10C30 CB, Shenzhen, China), an LED light source (BEDIRO BD400-SS, Shaoxing, China) with a light diffuser and a high-speed camera (IDT NX3-S3, Newark, NJ, USA) connected to a computer. The servo motor controls the disk to rotate stably in the counterclockwise direction with a rotating speed varying between 0 and 5000 rpm (revolutions per min). The rotating speed bias is within 0.8 rpm. The displacement device consists of three directional components, allowing for precise adjustments in the \(x\), \(y\), and \(z\) directions. This enables the setting of both the falling height and the impact point. The disk surface is cleaned with ethanol firstly and then distilled water before each experiment to ensure the clarity of photos and measurement accuracy. The camera equipped with a 100 mm focal length lens is used to record experimental phenomena. The acquisition frame rate is set to 4000 fps with a resolution of 512 \(\times\) 512 pixels2, providing sufficient temporal and spatial resolution. The camera is positioned above the rotating disk, at a small angle from the vertical direction. During post-processing, the original image is corrected to obtain a top-down view. Footnote 2: [[https://www.sciencedirect.com/science/article/pier/pier/pier.pdf](https://www.sciencedirect.com/science/article/pier/pier/pier.pdf)]([https://www.sciencedirect.com/science/article/pier/pier/pier.pdf](https://www.sciencedirect.com/science/article/pier/pier/pier.pdf))  
The impact process of droplets is primarily governed by the interplay between surface tension, viscous, and inertial forces. To generalize the dynamic evolution of droplets, two dimensionless numbers are introduced: the Weber number (\(We\)) and the rotational Bond number (\(Bo\)). The Weber number characterizes the ratio of inertial force to surface tension, while the rotational Bond number represents the ratio of centrifugal force to surface tension. Figure 1: Experimental set-up. These parameters provide insight into how different forces influence droplet behavior during impact. \[We=\frac{\rho V_{\text{im}}D_{0}}{\sigma}, \tag{1}\]  
\[Bo=\frac{\rho\omega^{2}V}{\sigma}=\frac{1}{6}\frac{\pi\rho\omega^{2}D_{0}^{2}}{ \sigma}, \tag{2}\]  
where \(\rho,D_{0},V,V_{\text{im}}\) represent the droplet's density, initial diameter, volume, and impact velocity, respectively. \(\sigma\) denotes the air-liquid surface tension. \(\omega\) is the rotating angular velocity of the disk expressed in rad/s. \(N\) is the corresponding rotating speed in rpm (revolutions per minute), thus \(N=\frac{30}{\pi}\omega\). The droplet's spreading and retraction process is characterized by the spreading factor, defined as the ratio of the instantaneous diameter \(D\) to the initial one \(D_{0}\). The spreading diameter \(D\) was determined using an image-processing technique based on the gray-level difference method described in [PERSON] et al. (2018). In the present experimental setup, the measurement error for the diameter is typically 2 pixels, corresponding to an uncertainty of 0.1 mm. \[\beta=\frac{D}{D_{0}}. \tag{3}\]  
The working fluid used in the experiments is deionized water, with its physical properties listed in Table 1. The values were obtained directly or through linear interpolation from the data provided in [PERSON] et al. (2018). Droplets generated by the present generation system have an initial diameter of 2.5 \(\pm\) 0.1 mm. All experiments were conducted in a controlled environment at a room temperature of 25 \({}^{\circ}\)C.  
A stainless disk with a diameter of 137 mm and a surface roughness \(Ra=0.6\,\upmu\)m (measured by Jitai Keyi TR200 Roughness Meter, Beijing, China) is used throughout the study.