content_image/356270.jpg
in equilibrium, characterized by vanishing order parameter gradients. Taking the divergence of the pressure tensor gives the force density acting on the fluid 

<img src='content_image/356270.jpg'>

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. 

## B. System geometry and boundary conditions 

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 

