A Quantitative Analysis of the Vertical-Averaging Approximation for Evaporating Thin Liquid Films

Thin liquid films play a central role in coating processes and other industrial and natural applications. Efficient optimization of these processes requires an understanding of capillary leveling, Marangoni flow, evaporation, and related phenomena. Although mathematical models are useful for gaining such understanding, it can be difficult to extract physical insight as the number of phenomena considered increases, so simplifying assumptions such as the vertical-averaging (VA) approximation for multicomponent films are often employed. In this work, we consider two-component films consisting of a solute and volatile solvent, and use lubrication theory to examine the performance of the VA approximation for three common evaporation models: constant, one-sided, and diffusion-limited. Whereas the VA approximation typically assumes ε²Pe<<1, where ε is the aspect ratio and Pe is the Péclet number, we find that the VA approximation works well when ε²Pe<<t_p, where t_p is the shortest time scale of phenomena that significantly depend on vertical gradients. Applying the VA approximation outside of the regime in which it is valid results in drastically different film-height and solute-distribution predictions depending on the evaporation model.