We study the dynamic wetting of a self-propelled viscous droplet using the time-dependent lubrication equation on a conical-shaped substrate for different cone radii, cone angles and slip lengths. The droplet velocity is found to increase with the cone angle and the slip length, but decrease with the cone radius. We show that a film is formed at the receding part of the droplet, much like the classical Landau–Levich–Derjaguin film. The film thickness hf is found to decrease with the slip length λ. By using the approach of matching asymptotic profiles in the film region and the quasi-static droplet, we obtain the same film thickness as the results from the lubrication approach for all slip lengths. We identify two scaling laws for the asymptotic regimes: hfh′′o∼Ca2/3 for λ≪hf and hfh′′3o∼(Ca/λ)2 for λ≫hf; here, 1/h′′o is a characteristic length at the receding contact line and Ca is the capillary number. We compare the position and the shape of the droplet predicted from our continuum theory with molecular dynamics simulations, which are in close agreement. Our results show that manipulating the droplet size, the cone angle and the slip length provides different schemes for guiding droplet motion and coating the substrate with a film.
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