## 2 Numerical Methods  
This study analyzed the effect of asymmetry in ridge wettability on droplet coalescence jumping using the VOF numerical analysis method. The behavior of droplet coalescence jumping was investigated by varying the contact angle ratio and the aspect ratio of the ridges. Liquids and gases, namely water and air, were used at room temperature and atmospheric pressure, respectively. The two phases were denoted by subscripts 1 and 2, and the volume fraction was represented by \(\psi\). Numerical analysis was conducted based on previous studies that modeled collision phenomena on solid surfaces [27,28]. The three-dimensional unsteady mass and momentum equations were employed in the computational domain,  
\[\frac{\partial}{\partial t}(\rho)+\
abla\cdot(\rho\overrightarrow{v})=0, \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho\overrightarrow{v})+\
abla\cdot(\rho \overrightarrow{v}\overrightarrow{v})=-\
abla p+\
abla\cdot\left[\mu(\
abla \overrightarrow{v}+(\
abla\overrightarrow{v})^{T})\right]+\rho\overrightarrow {g}+2\sigma\rho\kappa\
abla\psi_{2}/(\rho_{1}+\rho_{2}), \tag{2}\]  
where \(\rho=\psi_{2}\rho_{2}+(1-\psi_{2})\rho_{1}\), \(\mu=\psi_{2}\mu_{2}+(1-\psi_{2})\mu_{1}\). The curvature of the interface was defined as \(\kappa=-(\
abla\cdot\overrightarrow{n})\), where \(\overrightarrow{n}\) is the unit vector normal to the interface, and \(\overrightarrow{g}\) is the gravitational acceleration. Volume forces were introduced based on a continuous surface tension model [29], and the volume fraction was calculated by employing the equation \(\partial\psi/\partial t+\overrightarrow{v}\cdot\
abla\psi=0\), with a surface tracking method based on [PERSON] and Kothe [30]. The discretization model for spatial and temporal derivatives was based on basic convection models and research by [PERSON] [31], respectively. Residuals were constrained to be less than \(10^{-5}\) to improve convergence of velocity and pressure fields. A time step of 1 us and a maximum internal iteration of 25 were applied. A minimum of 50 grid cells were assigned per droplet size to ensure analysis accuracy, and the computational domain introduced a rectangular channel (10 \(\times\) 8 \(\times\) 8 mm\({}^{3}\)) with ridges on the bottom surface. In this study, as shown in Figure 1a, the aspect ratio (\(AR\)) of the ridge was defined as the ratio of height (\(H\)) to width (\(W\)), denoted as \(H/W\), with \(W\) fixed at 0.2 mm. Three ridge aspect ratios of 1.0, 1.5, and 2.0 were introduced for analysis. Additionally, droplet coalescence jumping on a flat surface without ridges was analyzed to obtain comparative  
Figure 1: (**a**) Schematics of droplet coalescence and droplet jumping on a ridged surface with different apparent contact angles of \(\theta_{L}\) and \(\theta_{R}\). The wettability of the flat surface is represented as the contact angle of \(\theta_{S}\) (=\(\theta_{L}=160^{\circ}\)). The ratio of wettability is defined as \(\beta=\theta_{R}/\theta_{L}\) in the current study. (**b**) Shape evolution in the coalescence process. The single droplet has a diameter of the droplet \(D\), and the aspect ratio of the ridge is denoted by a ratio between the height and width (i.e., \(AR=H/W\)). data. Previous research reported that the advancing and receding contact angles of the droplets were nearly identical at 160\({}^{\circ}\), with negligible contact angle hysteresis [32]. Based on this, static contact angles on the surface (\(\theta_{S}\)) and left side (\(\theta_{L}\)) were set to 160\({}^{\circ}\) to predict jumping behavior. To impart asymmetric contact angles, a static contact angle condition of \(\theta_{R}\) = 90-160\({}^{\circ}\) was applied to the right side. For symmetrical wetting, a relationship of \(\beta\) = \(\theta_{R}\)/\(\theta_{L}\) = 1.0 (i.e., \(\theta_{R}\) = 160\({}^{\circ}\)) was satisfied. The droplet size and properties were defined as \(D\) = 2.0 mm, \(\rho\) = 998.2 kg/m\({}^{3}\), \(\sigma\) = 0.072 N/m, and \(\mu\) = 1 mPa-s, respectively.