FIG. 2: Schematic representation of the system geometry. A thin film of thickness, h$_{c}$ , width W , and initial length, L$_{0}$ , is forced by gravity in the x direction on an heterogeneous substrate composed of hydrophilic ( θ$_{e}$ = 0 ◦ ) and hydrophobic ( θ$_{e}$ = 90 ◦ ) domains. The film is surrounded by a fluid of much smaller viscosity. The total system occupies a box of size L × W × H . A. Governing equations In this section we present the diffuse interface model that governs the dynamics of a gravity driven thin film in contact with a solid substrate. Figure 2 shows a schematic rep- resentation of the system, which is composed of two immiscible liquid phases of different viscosities in contact with a solid wall. One phase corresponds to the driven thin film, while the other plays the role of a surrounding fluid of much smaller viscosity. Instead of con- sidering that these phases are separated by a sharp interface, we introduce a concentration variable, or order parameter, φ ( r ), which has constant values in the bulk of each phase, φ = 1 for the thin film, and φ = − 1 for the surrounding fluid. The concentration varies smoothly between these values across the interface. The dynamics of both phases follows by imposing that mass, momentum, and concentration are conserved in time. For an incompressible fluid, mass conservation gives the continuity equation ∇ · v = 0 , (1) where v ( r ) is the velocity of the fluid. Momentum conservation leads to the Navier-Stokes equations 7