Numerical modeling of liquid withdrawal from gravure cavities in coating operations; the effect of cell pattern

A particular free-surface flow of industrial importance is the unsteady motion of a thin layer of viscous liquid on an engraved solid surface or substrate. A model is presented and numerical studies are performed for the cell-emptying behaviour when an engraved roller is used to transfer a liquid coating onto a moving belt or web. The three-dimensional unsteady liquid motion is calculated where the flow domain is bounded above by a stress-free surface and bounded below by a moving substrate with one of several different patterns of indentations. The goal is to ascertain, theoretically, how the choice of pattern affects the performance of the substrate. The physical model is simplified through use of the long-wave or lubrication approximation that is appropriate to liquid flow in thin layers. Three different patterns of cells: square, diamond and hexagonal, are considered. In each case the cell shape is taken to be paraboloidal. Early-time cell emptying is found to be quite independent of pattern. There is a degree of re-filling however, which is pattern-dependent. In this work, a high degree of symmetry is exploited to reduce the size of the numerical computations. Only Newtonian liquids are considered.