Thin-liquid-film flow on three-dimensional topographically patterned rotating cylinders

The coating of rotating discrete objects with surface topography is a problem commonly encountered in manufacturing processes. Surface topography may induce undesired disturbances in the coating, leading to coatings of non-uniform thickness. To study this problem, we model the flow of thin liquid coatings in three dimensions on topographically patterned cylinders that rotate about their horizontal axes. An evolution equation describing variations in the coating thickness as a function of the axial coordinate, the angular coordinate, and time is solved numerically using a variable time-step finite-difference scheme. In the limit of a rapidly rotating cylinder, we neglect the effects of gravity and find that liquid accumulates at either pattern crests or pattern troughs. Using a long-wave analysis, we derive an expression for the critical Weber number that separates these regimes. If gravity is reincorporated, the accumulation of liquid at crests or troughs may cause the coating to sag under its weight, leading to the formation of droplets or rings whose spacing at large rotation rates is controlled by the balance between centrifugal and surface-tension forces. At lower rotation rates, where gravitational forces dominate, simulation results indicate that cylinder topography tends to alter the rate at which droplets form, but does not necessarily systematically affect the spacing between droplets. Flow visualization experiments yield results that agree quantitatively with predictions of the simulations and long-wave analysis. We observe the most uniform coatings in experiments at moderate rotation rates, where disturbances in the coating thickness develop slowly. This indicates that to obtain nearly uniform coatings in practice, the coating must be solidified faster than disturbances can develop.