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When a paint film is applied to the underside of a horizontal substrate, instabilities will lead to the formation of pendant drops.  For sufficiently thick coatings, the drops may even detach from the coating before the coating dries. But even for thin paint films, with sufficiently high evaporation rates and viscosity, drop formation will lead to coating irregularities with thick regions (corresponding to the center of the drop) adjacent to much thinner regions (corresponding to troughs between drops.) In this paper we present a mathematical and numerical model based on the lubrication approximations for the drying of paint films on the underside of horizontal substrates. The paint is modeled as a multicomponent liquid with one non-volatile and one volatile component, termed the ``resin'' and ``solvent'' respectively. Our model includes the effects of surface tension and gravitational forces, and we show that surface tension gradient effects due to solvent evaporation must be considered to correctly model the post-application flow of certain paints. Simulations are given for an array of parameters, and they show that surface tension gradients can dramatically affect the post-application behavior of a model paint film, generally causing the suppression and reversal in drop growth, consequently leading to a more even final coating layer.

Suppression and Reversal of Drop Formation in a Model Paint Film