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We study the dynamic wetting of a self-propelled viscous droplet using thetime-dependent lubrication equation on a conical-shaped substrate for differentcone radii, cone angles and slip lengths. The droplet velocity is found toincrease with the cone angle and the slip length, but decrease with the coneradius. We show that a film is formed at the receding part of the droplet, muchlike the classical Landau-Levich-Derjaguin (LLD) film. The film thickness $h_f$is found to decrease with the slip length $\lambda$. By using the approach ofmatching asymptotic profiles in the film region and the quasi-static droplet,we obtain the same film thickness as the results from the lubrication approachfor all slip lengths. We identify two scaling laws for the asymptotic regimes:$h_fh''_o \sim Ca^{2/3}$ for $\lambda\ll h_f$ and $h_f h''^{3}_o\sim(Ca/\lambda)^2$ for $\lambda\gg h_f$, here $1/h''_o$ is a characteristic lengthat the receding contact line and $Ca$ is the capillary number. We compare theposition and the shape of the droplet predicted from our continuum theory withmolecular dynamics simulations, which are in close agreement. Our results showthat manipulating the droplet size, the cone angle and the slip length providesdifferent schemes for guiding droplet motion and coating the substrate with afilm.

Film deposition of a self-propelled droplet on a cone with slip