Unified Scaling Law and Regime Transitions for Eccentric Drop-On-Drop Impacts on Curved Substrates.

The dynamics of an eccentric droplet impacting a sessile droplet on flat, convex, and concave superhydrophobic surfaces are investigated using three-dimensional Lattice Boltzmann simulations. We analyze the coupled effects of Weber number, impact eccentricity, and substrate curvature on collision outcomes. Results show that convex curvature amplifies spreading and accelerates droplet shedding through divergent centrifugal effects, whereas concave curvature imposes geometric confinement that converges momentum inward, thereby effectively suppressing splashing. A regime map identifies four distinct rebound modes, with curvature significantly shifting transition boundaries. Furthermore, a theoretical scaling law is developed based on energy conservation, incorporating an effective driving Weber number and a curvature correction factor. This model successfully collapses data onto a unified master curve for maximum spreading, predicting the interplay between inertia and geometry. A deviation is observed only in the high-eccentricity "glancing" regime, where shear-driven elongation prevails over pressure-driven spreading. These findings offer theoretical guidance for optimizing droplet-based technologies on nonplanar surfaces.