Modelling the contact-line dynamics of an evaporating drop

We study the evolution of a thin, axisymmetric, partially wetting drop as it evaporates. The effects of viscous dissipation, capillarity, slip, gravity, surface-tension gradients, and contact-angle hysteresis are taken into account in the regime in which the transport of vapour is dominated by diffusion. We find a criterion for when the contact-set radius close to extinction evolves as the square-root of the time remaining until extinction - the famous d2-law. However, for a sufficiently large rate of evaporation, our analysis predicts that a 'd13/7-law' is more appropriate. We also determine how each of the physical effects in our model influences the evolution of the drop and hence its extinction time. Our asymptotic results are validated by comparison with numerical simulations. We then revisit our model for the vapour phase and take kinetic effects into account through a linear constitutive law that states that the mass flux through the drop surface is proportional to the difference between the vapour concentration in equilibrium and that at the interface. We perform a local analysis near the contact line to investigate the way in which kinetic effects regularize the mass- flux singularity at the contact line. The problem is further analysed via a matched asymptotic analysis in the physically relevant regime in which the kinetic timescale is much smaller than the diffusive one. We find that the effect of kinetics is limited to an inner region near the contact line, in which kinetic effects enter at leading order and cause the vapour concentration at the free surface to deviate from its equilibrium value. We also derive an explicit expression for the mass flux through the free surface of the drop.