We examine the transient film flow under the action of gravity over solid substrates with three-dimensional topographical features. Our focus is placed on the coating of a periodic array of rectangular cuboid trenches. The Navier-Stokes equations are solved using the volume-of-fluid method, fully taking into account the flow in both the liquid and gas phases. Using this scheme, we are able to determine the different wetting patterns that may arise depending on parameters such as the various geometrical characteristics of the trench, the lateral distance between them, the substrate wettability and the liquid viscosity. We present flow maps that describe the conditions under which the liquid film may successfully coat the patterned substrate, resulting in the so-called Wenzel state, or air may become entrapped inside the topography of the substrate. In the latter case, we describe in detail the position and shape of the air inclusions, how they are formed and the conditions under which coating can approach the ideal Cassie-Baxter state. We investigate in detail the effect of the sidewalls, typically ignored when considering the case of ideal 2D trenches (i.e., trenches extending to infinity in the lateral direction), through the enhancement of the viscous resistance inside the trench and the effect of capillarity in the case of narrow trenches. We also examine the coating behavior for a wide range of liquids and show that successful coating is favored for liquids with moderate viscosities. Finally, we perform simulations for the coating of two successive trenches in the flow direction and show that in the case of 3D trenches, the differences between the coating of the first and subsequent trenches are not significant.
