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 solute concentration 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 critical value of ϵ2Pe beyond which the VA approximation breaks down is often much larger than unity and depends on the evaporation rate. Furthermore, applying the VA approximation outside of its regime of validity results in drastically different film-height and solute-distribution predictions depending on the evaporation model. Scaling relations are derived from physical arguments to show how the critical value of ϵ2Pe depends on the evaporation rate under each evaporation model.