Abstract
We consider the transport of vapour caused by the evaporation of a thin, axisymmetric, partially wetting drop into an inert gas. We 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. Provided that the vapour concentration is finite, our model leads to a finite mass flux in contrast to the contact-line singularity in the mass flux that is observed in more standard models that neglect kinetic effects. 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. An explicit expression is derived for the mass flux through the free surface of the drop. A matched-asymptotic analysis is used to further investigate the regularization of the mass-flux singularity 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 regularize the mass-flux singularity. The inner problem is solved explicitly using the Wiener–Hopf method and a uniformly valid composite expansion is derived for the mass flux in this asymptotic limit.
Highlights
The evaporation of a liquid drop on a solid substrate has many important biomedical, geophysical, and industrial applications
We find that there is an outer region away from the contact line where the equilibrium assumption is recovered from our constitutive law and an inner region near the contact line where kinetic effects regularize the mass-flux singularity
We note the physical significance of two extreme cases: Pek = 0 corresponds to the case of no mass transfer, while Pek = ∞ corresponds to the case in which the vapour immediately above the free surface is at thermodynamic equilibrium, so that c = 1 on z = 0, 0 ≤ r < 1
Summary
The evaporation of a liquid drop on a solid substrate has many important biomedical, geophysical, and industrial applications. For a constant equilibrium vapour concentration, a kinetics-based model has the major advantage that, to leading order in the thin-film limit, the vapour transport problem depends on the liquid flow solely through the geometry of the contact set (and not through the drop thickness). We shall exploit the simplicity of a kinetics-based model with a constant equilibrium vapour concentration to perform a mathematical analysis of the model and investigate the way in which kinetic effects regularize the mass-flux singularity Another possible constitutive law for the equilibrium vapour concentration is Kelvin’s equation; this takes into account the variation in vapour pressure due to the curvature of the liquid–gas interface [22]. We adopt a linear, kinetics-based constitutive law for the mass flux across the liquid–gas interface, inspired by the Hertz–Knudsen relation (2); we assume that the equilibrium vapour concentration is constant.
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