as: $$\nabla_{2D} \alpha 
=
  \nabla \alpha - {\left( \hat{n}_{w} \cdot \nabla \alpha \right)} \hat{n}_{w}$$ where $\hat{n}_{w}$ again is the normal of the wall pointing outward. The equilibrium Young’s angle $\theta_{0}$ can be defined uniquely for any grid cell along the wall, so an arbitrary wettability pattern can be created. $\Delta A_{w}$ and $\Delta V$ are properties of the mesh that can be accessed directly in OpenFOAM. The contact line tension source term is solved explicitly along with the surface tension source term.

The Navier slip condition is implemented as: $$\vec{v}
=
  l_{s} {\left( {I} - \hat{n}_{w}^{2} \right)} \nabla \vec{v}$$ where $l_{s}$ is the slip length ${I}$ is the identity matrix, and $\hat{n}_{w}^{2}$ is the dyadic product of the wall normal, $\hat{n}_{w}$, with itself. Using this formulation, the velocity perpendicular to the wall is always set to zero.

Results
=======

The first section of the results focuses on the static behavior of the implementation of the Generalized Navier Boundary Condition presented above. The second section covers the dynamic contact angle behavior. To speed up simulation times all results are for 2D droplets in a planar geometry, and using the symmetry of the system only half of the droplet is simulated. All 