## 2 Phenomenological Model
### Droplet Behavior and Runback Water Modeling
In the case of partially frozen beads, coalescence will involve only the residual liquid volume, leaving the existing solid beads unchanged. For hydrophilic substrates (\(\theta_{s}<90^{\circ}\)), coalescence will occur if the distance \(d\) between the centers on the surface is lower than the sum of the base area radii:  
\[d\leq r_{1}sin\theta_{s}+r_{2}sin\theta_{s} \tag{11}\]  
while under hydrophobic conditions it will occur according to the geometrical consideration in Figure 4, as  
\[\sqrt{d^{2}+\left(r_{1}-r_{2}\right)^{2}cos^{2}\theta_{s}}\leq r_{1}+r_{2} \tag{12}\]  
Droplets are subjected to external forces, namely gravity, aerodynamic drag and pressure gradient if exposed to a fluid flow. Drag is evaluated via a drag coefficient computed for spherical caps and from the local flow velocity. The local velocity, given the small height of the droplets, is computed from the local shear stress, assuming a linear profile normal distance from the wall. These forces deform the droplet shape, inducing hysteresis due to the difference between the advancing \(\theta_{a}\) and receding \(\theta_{s}\) contact angles. The actual contact angle will be a function of the azimuthal angle along the contact line, as reported in [25; 26]. The balance of drag, gravitational force projected along the surface, and surface tension defines whether the droplet will stay still or will be shed away. Since aerodynamic drag increases with \(r^{2}\), gravity contribution with \(r^{3}\), and surface tension only with \(r\), we can find a maximum radius allowing for immobile drops. Then, the next step is the modeling of the moving droplet. A power balance between aerodynamic force work, surface tension work, and viscous dissipation (sum of laminar viscous dissipation within the droplet and wedge dissipation) is performed. Then, the next step is the modeling of the moving droplet. Following [18; 19], a power balance between aerodynamic force work, gravity force work, surface tension work and viscous dissipation (sum of laminar viscous dissipation within the droplet and wedge dissipation) is performed,  
\[\Phi_{D}+\Phi_{g}-\Phi_{\sigma}-\Phi_{\mu}=0 \tag{13}\]  
\[\Phi_{D}=\mathbf{F}_{D}\cdot\mathbf{u}_{d}=\frac{1}{2}\rho_{a}c_{D}|\mathbf{u}_{a}-\mathbf{u}_ {d}|(\mathbf{u}_{a}-\mathbf{u}_{d})\cdot\mathbf{u}_{d} \tag{14}\]  
\[\Phi_{g}=\mathbf{F}_{g}\cdot\mathbf{u}_{d}=\rho_{I}V_{d}\mathbf{g}\cdot\mathbf{u}_{d} \tag{15}\]  
where \(\mathbf{u}_{a}\) is the local airflow velocity and \(c_{D}\) is the drag coefficient, which can be estimated as the one of a flow past a sphere, as done by [27]. The further condition that the droplet moving direction agrees with the one of the resulting force \(\mathbf{F}_{g}+\mathbf{F}_{D}\) is required in order to close the problem. The derivation of surface tension work \(\Phi_{\sigma}\) and viscous dissipation  
Figure 4: Coalescence condition for the hydrophobic substrate. (given by the sum of laminar viscous dissipation within the droplet and wedge dissipation) are given in [18; 19]. The procedure is suited for both planar and curved surfaces, provided that the substrate radius of curvature is larger than the droplet one.