A model for a spreading and melting droplet on a heated substrate

We develop a model to describe the dynamics of a spreading and melting droplet on a heated substrate. The model, developed in the capillary-dominated limit, is geometrical in nature and couples the contact line, trijunction, and phase-change dynamics. The competition between spreading and melting is characterized by a single parameter K-T that represents the ratio of the characteristic contact line velocity to the characteristic melting ( or phase-change) velocity. A key component of the model is an equation of motion for the solid. This equation of motion, which accounts for global effects through a balance of forces over the entire solid liquid interface, including capillary effects at the trijunction, acts in a natural way as the trijunction condition. This is in contrast to models of trijunction dynamics during solidi cation, where it is common to specify a trijunction condition based on local physics alone. The trijunction dynamics, as well as the contact angle, contact line position, and other dynamic quantities for the spreading and melting droplet, are predicted by the model and are compared to an isothermally spreading liquid droplet whose dynamics are controlled exclusively by the contact line. We nd that in general the differences between the dynamics of a spreading and melting droplet and that of an isothermally spreading droplet increase as K-T increases. We observe that the presence of the solid phase in the spreading and melting configuration tends to inhibit spreading relative to an isothermally spreading droplet of the same initial geometry. Finally, we nd that increasing the effect of spreading promotes melting.