Suppression and reversal of drop formation on horizontal cylinders due to surfactant convection

Abstract

When a thin liquid film is applied to the surface of a horizontal cylinder, gravity will cause a drainage of liquid from the top and sides of the cylinder towards the cylinder bottom. If surfactant is present on the surface of the film, this will cause a convection of surfactant resulting in a higher concentration of surfactant on the cylinder bottom compared to the top and sides of the cylinder. The result is a surface tension gradient, which is equivalent to a surface shear stress, and acts to oppose the drainage of the coating layer due to gravity. For sufficiently small cylinders, this cannot only slow the drainage but reverse the flow, causing a net flux of liquid upward from the bottom of the cylinder towards the top of the cylinder. If this flux is sufficiently strong, a “collar” of liquid forms around the cylinder. In this paper, we develop a mathematical model, based on the lubrication approximations, of the gravitational, surface tension, and surface tension gradient forces, and their effects on the evolution of a thin liquid film coating a horizontal circular cylinder. Using finite differences and an alternating direction implicit technique, numerical simulations show that even for comparatively weak surfactants, surface tension gradient effects greatly affect the flow history and must be included to accurately model the evolution of the film. They cannot only slow the drainage of liquid towards a pendant drop on the bottom of the cylinder, but reverse the flux, resulting in a thicker coating on the top of the cylinder compared to the surfactant-free case. Results from the simulation are presented over a wide range of the dimensionless parameters which characterize the problem.