## 3 Results
(**d–f**) Droplet shapes on ridges with (**a**) \(\beta\) = 0.56, (**b**) 0.66, and (**c**) 0.75, under a constant \(AR\) of 2.0. \(U_{x}\) and \(U_{y}\) represent the averaged jumping velocities of the merged droplets in the \(x\) and \(y\) directions, respectively. \(\Delta x\) denotes the offset distance of the drop’s center from the center of the ridge. distance generally increases. For instance, compared to weak asymmetry, with \(\beta\) = 0.84 (\(\theta_{R}\) = 135\({}^{\circ}\)), the lowest \(\beta\) results in an approximately 2.7-fold increase in offset distance (for \(AR\) = 2.0). Additionally, for the lowest ridge (\(AR\) = 1.0), an approximately 8-fold increase in height is predicted. Therefore, ensuring uniform wettability distribution on the ridges seems crucial for droplet vertical mobility. When the jumping droplets can be moved again upon re-approaching the surface, due to collisions with solid structures or other droplets, their horizontal movement can occur. The process of droplets re-approaching the surface closely resembles that of collisions with the surface at low speeds. The current prediction of the asymmetric wettability phenomena on macrostructures potentially provides vertical jumping performance in practical applications. The observed changes in droplet bounce and trajectory variation due to wettability highlight the characteristics of droplet coalescence jumping influenced by wettability. These features can be explained by the horizontal and vertical velocity components of the droplet. Figure 5 depicts the predicted horizontal average velocity (\(U_{x}\)) of the merged droplets over time. For analysis, specific droplet shapes at certain time intervals are included below the graph. Consequently, increased wettability asymmetry induces significant changes in \(U_{x}\), particularly during contraction, when the droplet detaches from the ridge surface around 10-15 ms. At this moment, a significant difference in \(U_{x}\) is observed due to the disparity in droplet detachment from the left and right sides of the ridge. This difference in detachment maintains a significant disparity in \(U_{x}\), even after 15 ms. For instance, compared to weak asymmetry with \(\beta\) = 0.84, a 7.2-fold and a 4.5-fold increase in \(U_{x}\) are predicted at \(\beta\) = 0.56 and 0.66, respectively. Moreover, the maximum \(U_{x}\) is predicted around 20-25 ms, coinciding with the maximum height of the jumping. These results suggest that adjusting the distribution of ridge wettability is necessary to control the horizontal velocity of detaching droplets. The vertical velocity component is crucial in droplet coalescence jumping, as it directly influences self-cleaning and heat transfer performance [12]. Figure 6 shows the temporal variation and maximum values of \(U_{y}\) under wettability asymmetry and the ridge's aspect ratio. Figure 6a,b represent the flat surface without ridges using dashed lines and a star symbol, respectively. From the start of droplet coalescence at 0 s, \(U_{y}\) gradually increases. During the contraction phase, where the droplet shape transitions from elliptical to contracted, \(U_{y}\) sharply increases (5-10 ms). The maximum value of \(U_{y}\) is reached fastest on surfaces without ridges, and next fastest at the lowest \(\beta\). As the wettability ratio increases, the time taken to reach the maximum vertical velocity increases. \(U_{y}\) starts decreasing around 10-15 ms, when the droplet detaches from the ridge surface. The rate of decrease  
Figure 4: Jumping feature of the coalesced droplet: the maximal height (filled symbol; left axis) and the offset distance (open symbol; right axis), investigated as a function of \(\theta_{R}\), under three \(AR\)s of the ridges. in \(U_{y}\) (i.e., the slope of the graph) is highest on surfaces without ridges and next highest at the lowest \(\beta\). As the difference between \(\theta_{R}\) and \(\theta_{L}\) decreases, the rate of decrease in \(U_{y}\) relatively diminishes.