Optimal leveling of flow over one-dimensional topography by Marangoni stresses

A thin viscous film flowing over a step down in topography exhibits a capillary ridge preceding the step. In applications, a planar liquid surface is often desired and hence there is a need to level the ridge. This paper investigates optimal leveling of the ridge by means of a Marangoni stress such as might be produced by a localized heater creating temperature variations at the film surface. The differential equation for the free surface based on lubrication theory and incorporating the effects of topography and temperature gradients is solved numerically for steps down in topography with different temperature profiles. Both rectangular “top-hat” and parabolic profiles, chosen to model physically realizable heaters, were found to be effective in reducing the height of the capillary ridge. Leveling the ridge is formulated as an optimization problem to minimize the maximum free-surface height by varying the heater strength, position, and width. With the optimized heaters, the variation in surface height is reduced by more than 50% compared to the original isothermal ridge. For more effective leveling, we consider an asymmetric n-step temperature distribution. The optimal n-step heater in this case results in (n+1) ridges of equal size; 2- and 3-step heaters reduce the variation in surface height by about 70% and 77%, respectively. Finally, we explore the potential of coolers and step temperature profiles for still more effective leveling.