V}
=
  \vec{F}_{\tau A} \frac{\Delta A_{w}}{\Delta V} g {\left( \tilde{c} \right)}
\label{eqn:volumeForceCor}$$ where the same definition is used for $\vec{F}_{\tau A}$ as in equation \[eqn:volumeForce\].

Numerical implementation
------------------------

This section focuses on the numerical implementation of the above derived equation in the Volume Of Fluid (VOF) solver that comes with OpenFOAM [@jasak1996; @openfoam]. This involves implementing the uncompensated Young’s stress and the Navier slip boundary condition. In this code the general phase parameter $\tilde{c} {\left( \vec{x} \right)}$ is called $\alpha$, and has the following properties: $$\alpha
=
  \left\{
  \begin{array}{ll}
  0      & {\text{in phase 1}}    \\
  (0, 1) & {\text{on interface}}    \\
  1      & {\text{in phase 2}}
  \end{array}
  \right.$$ Phase parameter $\alpha$ is stored as a separate field, just like velocity and pressure, and its evolution is calculated using the following transport equation: $$\frac{\partial \alpha}{\partial t}
+ \nabla \cdot {\left( \alpha \vec{v} \right)}
+ \nabla \cdot {\left( \alpha {\left( 1 - \alpha \right)} \vec{v}_{lg} \right)}
=
  0$$ where $\vec{v} = \alpha \vec{v}_{l} + {\left( 1 - \alpha \right)} \vec{v}_{g}$ is the phase averaged velocity, and $\vec{v}_{lg} = \vec{v}_{l} - \vec{v}_{g}$ is the velocity difference between the liquid and the gas phase. This equation is equivalent to a material derivative, but rewritten 