# Directional spreading of a viscous droplet on a conical fibre

## TL;DR

This study models the capillary-driven spreading of a viscous droplet on a conical fiber, revealing how the shape and contact angles influence droplet motion from tip to base, aligning well with experiments.

## Contribution

The paper develops a mathematical framework combining lubrication theory and asymptotic expansions to predict droplet shape and movement on conical fibers, a novel approach for this problem.

## Key findings

- Droplet adopts a quasi-static shape during spreading.
- Predicted shape and velocity match experimental data.
- Droplet moves from tip to base due to contact angle asymmetry.

## Abstract

If a droplet is placed on a substrate with a conical shape it spontaneously starts to spread in the direction of a growing fibre radius. We describe this capillary spreading dynamics by developing a lubrication approximation on a cone and by the perturbation method of matched asymptotic expansions. Our results show that the droplet appears to adopt a quasi-static shape and the predictions of the droplet shape and spreading velocity from the two mathematical models are in excellent agreement for a wide range of slip lengths, cone angles and equilibrium contact angles. At the contact line regions, a large pressure gradient is generated by the mismatch between the equilibrium contact angle and the apparent contact angle that maintains the viscous flow. It is the conical shape of the substrate that breaks the front/rear droplet symmetry in terms of the apparent contact angle, which is larger at the thicker part of the cone than that at its thinner part. Consequently, the droplet is predicted to move from the cone tip to its base, consistent with experimental observations.