On thin evaporating drops: When is the -law valid?

Abstract

We study the evolution of a thin, axisymmetric, partially wetting drop as it evaporates. The effects of viscous dissipation, capillarity, slip and diffusion-dominated vapour transport are taken into account. A matched asymptotic analysis in the limit of small slip is used to derive a generalization of Tanner's law that takes account of the effect of mass transfer. 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 d 2 -law. However, for a sufficiently large rate of evaporation, our analysis predicts that a (slightly different) 'd 13/7 -law' is more appropriate. Our asymptotic results are validated by comparison with numerical simulations.

Kinetic effects regularize the mass-flux singularity at the contact line of a thin evaporating drop

“…This would allow us to obtain predictions for the evaporation time, the evolution of the drop volume (or, equivalently, the dynamic contact angle or drop thickness), and, in the case of a moving contact line, the evolution of the contact-set radius within this model. Previous theoretical work has obtained such predictions for the lens evaporation model (with a mass-flux singularity at the contact line) [11,12,41,42] and for other evaporation models [7,11,18,[43][44][45][46][47][48][49]. In particular, it would be informative to compare the predictions of this previous work to the corresponding predictions for the model considered here.…”

“…Thus, the vertical extent of the drop is much smaller than the radius of the circular contact set of the drop; since the latter is the relevant lengthscale for the transport of liquid vapour, the gas phase occupies the region z * > 0 to leading order in the limit of a thin drop. We assume that the dynamics of the vapour may be reduced to a diffusion equation for the vapour concentration c * , with constant diffusion coefficient D. We further assume that the timescale of vapour diffusion is much shorter than the timescale of the liquid flow (a common assumption in the literature [9,10,12]). Thus, transport of the vapour is governed to leading order in the thin-film limit by Laplace's equation, with…”

“…To leading order, the vapour transport equation (10) and the mixed-boundary conditions (12) and (13) become…”

“…A commonly used model of a drop evaporating into an inert gas is the 'lens' model [2,5,[8][9][10][11][12]. The lens model is based on the assumptions that the drop is axisymmetric, the vapour concentration field is stationary, and the vapour immediately above the liquid-gas interface is at thermodynamic equilibrium, with the equilibrium vapour concentration being constant.…”

“…The expression (1) for the mass flux has an inverse-square-root singularity at the contact line. Since this singularity is integrable, the total mass flux out of the drop is not singular, and physically reasonable predictions for the evolution of the drop volume are obtained even without regularization of the mass-flux singularity [10,12]. However, the need to supply a diverging mass flux means that there is a singularity in the depth-averaged radial velocity of the liquid flow within the drop [10,12].…”