Self-propulsion dynamics of small droplets on general surfaces with curvature gradient

TL;DR

This paper presents a theoretical model for the self-propulsion of small droplets on curved surfaces with curvature gradients, deriving a scaling law and validating it against experimental data.

Contribution

It introduces a new variational approach-based reduced model that accurately describes droplet motion on general curved surfaces, including scaling laws.

Findings

Derived a quantitative model for droplet self-propulsion.

Established a $s \,\sim\, t^{1/3}$ scaling law for droplet displacement.

Model aligns well with previous experimental results.

Abstract

We study theoretically the self-propulsion dynamics of a small droplet on general curved surfaces by a variational approach. A new reduced model is derived based on careful computations for the capillary energy and the viscous dissipation in the system. The model describes quantitatively the spontaneous motion of a liquid droplet on general surfaces. In particular, it recovers previous models for droplet motion on the outside surface of a cone. In this case, we derive a scaling law of the displacement $s\sim t^{1/3}$ of a droplet with respect to time $t$ by asymptotic analysis. Theoretical results are in good agreement with experiments in previous literature without adjusting the friction coefficient in the model.