in equilibrium, characterized by vanishing order parameter gradients. Taking the divergence of the pressure tensor gives the force density acting on the fluid which clarifies the origin of the chemical term in equation (2). In this paper we will consis- tently use B = − A , from which it follows that the remaining equilibrium properties of the system are given by γ = ( − 8 κA/ 9) 1 2 , ξ = ( − κ/ 2 A ) 1 2 and φ ∗ = ± 1. We choose a system geometry to mimic the constant flux configuration of gravity driven thin films, as shown in figure 2. We choose a rectangular domain of linear dimensions L × W × H . Initially, the thin film occupies the volume V$_{0}$ = L$_{0}$ × W × h$_{c}$ . The thin film is forced in the x direction while the surrounding fluid is left to evolve passively. This is achieved by setting the gravitational force as ρ g = ρg$_{x}$ 1 $_{2}$( φ + 1) ˆ x . Similarly, the viscosity is varied using the mixing rule η ( φ ) = 〈 η 〉 (1 + δηφ ), where 〈 η 〉 is the mean viscosity and δη is the viscosity contrast. The viscosity of the thin film is then η$_{+1}$ = 〈 η 〉 (1 + δη ), while the viscosity of the surrounding fluid is η − 1 = 〈 η 〉 (1 − δη ). To complete the model, boundary conditions for v and φ have to be provided. We choose the system geometry so that the solid substrate is parallel to the x − y plane and located at z = 0 as shown in figure 2. The impenetrability of the solid is ensured by fixing v$_{z}$ ( x, y, z = 0) = 0, while stick boundary conditions are imposed at the wall, v$_{x}$ ( x, y, z = 0) = v$_{y}$ ( x, y, z = 0) = 0 . We fix periodic boundary conditions in the y direction, i.e. , v ( x, y = W, z ) = v ( x, y = 0 , z ), φ ( x, y = W, z ) = φ ( x, y = 0 , z ). In the x direction a constant mass inflow is kept by fixing ∂$_{x}$ v ( x = { 0 , L } , y, z ) = 0 and ∂$_{x}$φ ( x = { 0 , L } , y, z ) = 0 . At the upper boundary, a shear free boundary condition is imposed by fixing ∂$_{z}$ v ( x, y, z = H ) = 0 , while a vanishing concentration gradient normal to the boundary is fixed by imposing ∂$_{z}$φ ( x, y, z = H ) = 0 . In equilibrium, a drop of fluid sits on the solid describing the shape of a spherical cap that intersects the solid boundary with an equilibrium contact angle θ$_{e}$ (for vanishing θ$_{e}$ the spherical cap is replaced by a film that covers the whole substrate). The equilibrium contact angle is determined by Young’s Law − γ$_{SL}$$_{+1}$ − γ cos θ$_{e}$ + γ$_{SL}$$_{−}$$_{1}$ = 0, where γ$_{SL}$$_{i}$ is the surface 9 B. System geometry and boundary conditions − ∇ · P = − ∇ P − φ ∇ µ,