Simulation of Coating Layer Evolution and Drop Formation on Horizontal Cylinders

The lubrication form of the equations governing the flow of a thin liquid film on a horizontal right circular cylinder is derived. The equations are discretized and solved numerically using an alternating-direction implicit algorithm. Simulations demonstrate that the transition from a uniform coating to a final configuration of distinct drops follows a similar evolution for a wide range of cylinder radii. Initially gravity-driven drainage from the top and sides of the cylinder dampens the formation of any axial disturbances; only when this drainage slows do longitudinal waves begin to develop along the bottom of the cylinder. These waves grow rapidly and a series of alternating primary and satellite drops form during the transition from a linear to a nonlinear wave growth regime. This is followed by a slow drainage between adjacent drops as the drop pattern approaches an equilibrium state where surface tension forces exactly balance gravitational forces in each discrete drop. For large cylinder radii, these drops are localized on the bottom of the cylinder, while, for sufficiently small cylinder radii, these drops may wrap around the entire circumference of the cylinder. Integral measures of the evolving coating profile, such as the total energy and viscous dissipation rate, clearly show these growth phases. The equilibrium shape of large-amplitude pendant drops and the maximum sustainable drop volume for various cylinders are also considered.