  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.
