Transient flow of gravity-driven viscous films over substrates with rectangular topographical features

We study the transient, two-dimensional film flow over solid substrates with variable topography, a flow that has practical applications in microelectronics and microfluidics. The problem we address here is the advancing of a thin liquid film over square-shaped trenches with different depths and widths, under the influence of the gravitational force. We use the volume-of-fluid method to obtain completely different wetting patterns depending on the dimensions of the topography, the capillary and Reynolds numbers, and the contact angle. On one hand, we predict continuous coating, i.e., the formation of the Wenzel state, in which a thin liquid film covers the entire trench, while steady flow is established upstream and downstream this topographical feature. This is the desirable pattern, when perfect coating is sought, as in the manufacturing of microelectronic devices. Under different conditions, we predict that the film almost completely bypasses the trench, entrapping air inside it, i.e., forming the Cassie state. The coating quality is clearly poor in this case, but this pattern reduces the drag on the film, and therefore, it is desirable in the operation of super-hydrophobic surfaces for microfluidic applications. Between these two extreme configurations, we uncover a large variety of patterns, in which the film partially wets the trench forming an air inclusion all along its bottom or its upstream or downstream inner corners or the film may break up periodically. We produce comprehensive flow maps covering a wide range of relevant parameter values.