Nonlinear theory of free coating onto a vertical surface

Abstract

Abstract The work of Landau and Levich (1942) initiated detailed theoretical and experimental study of the flow of the thin liquid film entrained by a steady withdrawal of a sheet from a bath of liquid. The existing theories, which differ considerably, are all based on linearization of the problem and give single relationships between film thickness T and a capillary number Ca . The present paper demonstrates theoretically that differing physical properties for each liquid result not in a single curve but in a family of curves for T vs. Ca and that the complete set of previous experimental work required and fitted the family of curves. Previous theories could fit only a portion of this experimental data. The solution requires a nonlinear theory to include inertial terms and two‐dimensional flow and thus treat the parameter of liquid physical properties. This is obtained by applying a nonlinear approach based on the direct method of Galerkin. Owing to the nonlinear character of the solution, the new theory has the advantage of accurately determining the shape and size of the upper meniscus profile. Excellent agreement with the complete set of available experimental data is obtained.