## 3 Results and Discussion
### Droplet Impact on Horizontally Moving Micro-Pillar Surfaces  
#### 3.2.1 Droplet Impact Behavior  
Figure 8 shows the velocity distribution during a droplet impacting on a moving surface with different horizontal velocities. When a droplet impacts a moving surface with the partial penetration state, its dynamic behavior can be divided into the following stages, as shown in Figure 8a. Firstly, during the spreading stage, the kinetic energy of the droplet is gradually converted into surface energy, causing the spreading diameter of the droplet to continuously increase. Simultaneously, the droplet penetrates into the gaps between the micro-pillars, compressing the air within these gaps (\(t^{*}=1.14\)). When the kinetic energy of the droplet in the vertical direction drops to zero (\(t^{*}=1.70\)), the penetration stage of the droplet ends. Subsequently, the surface energy is reconverted into kinetic energy, and the droplet begins to rapidly retract and gradually detach from the micro-pillar surface. If the droplet's kinetic energy is sufficiently large, it will eventually completely bounce off the surface (\(t^{*}=3.49\)). As \(We_{t}\) increases, the horizontal momentum of the droplet increases, causing the droplet to exhibit horizontal displacement relative to its initial impact position upon detachment, as shown in Figure 8b. Additionally, it can be observed that in the upstream region within the micro-pillar gaps, there are upward-tilting velocity vectors, while in the downstream region, there are downward-tilting velocity vectors. This phenomenon reveals the velocity changes during the droplet's rolling process (as shown in Figure 8b, \(t^{*}=1.70\)). Figure 7: Variation in the maximum spreading diameter of a droplet impacting on a stationary wall with respect to \(We_{n}\) and the fitted correlation. Droplets undergo contact, spreading, retraction, and rebound processes when impacting both moving and stationary surfaces [31]. However, when a droplet impacts a horizontally moving micro-pillar surface, a unique rolling phenomenon occurs, where the upstream portion of the droplet detaches from the micro-pillar surface while the downstream portion continues to penetrate into the micro-pillar gaps, as shown in Figure 9. This phenomenon arises because the droplet possesses both upward and horizontal momentum during the rebound stage. Due to the presence of air at the bottom, the downstream portion of the droplet can continue to penetrate downward, while the upstream portion is subjected to surface tension and capillary forces, causing it to contract upward. This results in the droplet rolling along the horizontal direction. The magnitude of \(We_{t}\) influences the degree of rolling within the micro-pillar gaps, with greater rolling leading to more significant energy dissipation. Therefore, an increase in \(We_{t}\) not only exacerbates the asymmetry of the droplet [56], making it more likely to bounce off the surface, but the presence of the rolling phenomenon also suppresses the droplet's bouncing behavior. The final effect of \(We_{t}\) is a compromise of the above two influences. The pressure peak is closely related to the droplet's motion behavior [57]. The first pressure peak generated at the moment of droplet impact causes the droplet to rapidly change its motion state, initiating spreading or penetration. The second pressure peak is associated with the droplet's recoil and jetting phenomena, influencing the rebound  
Figure 8: Velocity distribution on the \(x\)-\(y\) cross-section during droplet impact on a moving surface (\(We_{t}=1.8\)). (**a**) \(We_{t}=0.8\); (**b**) \(We_{t}=6.4\). Figure 9: Droplet contour lines at different times in the cases of \(We_{t}=1.8\) and \(We_{t}=6.4\). direction and speed of the droplet. As shown in Figure 10, in the model of this study, the pressure peaks during droplet impact on both stationary and moving micro-pillar surfaces occur at the top of the micro-pillars.