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On thin evaporating drops: When is the $d^{2}$ -law valid?

  • M. A. Saxton (a1), J. P. Whiteley (a2), D. Vella (a1) and J. M. Oliver (a1)


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.


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