to minimize numerical diffusion [@rusche2002].

The volume fraction is used to calculate phase-averaged densities, velocities, and viscosities, which are used in the momentum balance $$\frac{\partial \rho \vec{v}}{\partial t}
+ \nabla \cdot {\left( \rho \vec{v} \otimes \vec{v} \right)}
=
- \nabla p
+ \nabla \cdot {\left( \mu \nabla \vec{v} \right)}
+ \rho \vec{g}
+ \vec{f}_{{\text{st}}}
- \vec{f}_{{\text{cl}}}$$ and the continuity equation: $$\nabla \cdot \vec{v}
=
  0$$ In the above equations $\rho$ is the density, $\vec{v}$ is the velocity, $t$ time, $p$ is pressure $\mu$ is the viscosity, $g$ is gravity, $\vec{f}_{{\text{st}}}$ is the surface tension force, $\vec{f}_{{\text{cl}}}$ is the contact line tension force, and $\otimes$ is the dyadic product. The density $\rho$, velocity $\vec{v}$, and viscosity $\mu$, are all phase averaged using $\alpha$.

The surface tension force is calculated using the expression: $$\vec{f}_{{\text{st}}}
=
  \sigma_{{\text{st}}} \kappa \nabla{\alpha}$$ where $\sigma_{{\text{st}}}$ is the surface tension coefficient, $$\kappa
=
- {\left( \nabla \cdot \vec{n} \right)}$$ is the curvature of the interface, and $$\vec{n}
=
  \frac{\nabla \alpha}{{\left|{\nabla \alpha}\right|}}$$ is the normal of the interface [@brackbill1992].

Using $\alpha$, the line tension force defined in equation \[eqn:volumeForceCor\] is rewritten as: $$\vec{f}_{{\text{cl}}}
=
  \sigma_{{\text{st}}} \cos{\theta_{0}} \;
  g {\left( \alpha \right)}
  \frac{\Delta A_{w}}{\Delta V}$$ and $\nabla_{2D} \alpha$ is implemented 