Asymmetric freezing of a sliding droplet on an inclined surface

We investigate the asymmetric freezing of a liquid droplet sliding on an inclined cold surface using numerical simulations based on the lubrication approximation. The combined effects of gravity, capillarity and solidification kinetics on droplet motion, interfacial deformation and the resulting frozen morphology are examined through systematic variations in substrate inclination, wettability, effective Bond number and Stefan number. Our results show that sliding prior to and during the early stages of freezing plays a dominant role in governing the asymmetry of the frozen droplet. A tilted ice cusp forms at the droplet tip due to the competition between gravitational forces and capillary resistance, with its orientation and magnitude strongly dependent on substrate wettability and inclination. Greater inclination and increased wettability enhance asymmetry in droplet morphology. Further, highly wetting substrates favour capillary-driven retraction and induce transient liquid motion opposite to gravity during freezing. The evolution of contact-angle hysteresis at both the solid surface and the liquid–ice interface underscores the importance of early-time dynamics, when the unfrozen liquid remains mobile and gravitational effects are most pronounced. Decomposition of the liquid motion into capillary- and gravity-driven contributions provides physical insight into contact-line pinning, receding-edge thinning and the development of asymmetric liquid–ice contact angles. Increasing the Stefan number accelerates freezing, limits sliding-induced deformation, and reduces both the cusp angle and the post-freezing contact-angle contrast. Overall, this study establishes a physical framework for understanding the morphology of frozen droplets on inclined substrates.