A scaling theory of the free-coating flow on a plate withdrawn from a pool

A theory of free coating of a Newtonian liquid on a plate is developed based on scaling analysis of the flow. The analysis addresses the flow in the apical portion of the meniscus, where the film is entrained, and also of the bulk liquid conveyed from pool depths up to the surface. Film thickness as well as the characteristic curvature of the meniscus is predicted as a function of the capillary number Ca and the non-dimensional parameter $Po=μ(g∕ρσ3)1∕4$⁠. No term in the Navier-Stokes equations is arbitrarily neglected; thus, the theory is valid virtually for any Ca and Po. The theory entails two adjustable constants, determined by least-squares fitting with the experimental data of Kizito et al. [“Experimental free coating flows at high capillary and Reynolds number,” Exp. Fluids 27, 235 (1999)]. However, once determined, these constants are to be considered as “universal” for free-coating on a plate. Rich thickness-vs-Ca behavior as a function of Po is predicted with a steep ascent of thickness at a certain Ca threshold, which depends on Po. At diverging Ca, the film thickness scaled to $d0=μUρg12$ asymptotes to a finite value independent of Po, while in the small-Ca limit the classical Landau-Levich law is duly recovered. The theory provides full physical understanding of each aspect of this behavior, revealing the roles of capillary-, inertial-, and gravity forces in the various regimes. The theoretical predictions are in full qualitative and quantitative agreement with the whole set of experimental results of Kizito et al., spanning over three orders of magnitudes both of Ca and Po. A non-monotonous behavior of the characteristic meniscus curvature scaled to the reciprocal film thickness, with a growth followed by a drop as a function of Ca, is predicted, in qualitative accordance with earlier experimental observations and computational results.