is the equilibrium Young’s contact angle, $\hat{t}_{w}$ is the vector tangent to the wall and normal to the contact line, and the distribution $\delta_{{\text{cl}}}$ is defined as [@gerbeau2009]: $${\left<{\delta_{{\text{cl}}}, \psi}\right>}
=
  \int_{{\text{cl}}} \psi d \lambda_{{\text{cl}}}$$ where $\psi$ is any smooth function, and $\lambda_{{\text{cl}}}$ denotes the Lebesgue measure (i.e. the length measure) on the contact line. The slip length in equation \[eqn:gnbc\] is equal to $l_{s} = \mu / \beta$.

Analogous to the methods of @brackbill1992 a Volume Of Fluid expression is derived for the above equation for the Generalized Navier Boundary Condition. However, instead of relating the uncompensated Young’s stress to a velocity using a slip coefficient, the uncompensated Young’s stress is modeled as an extra body force acting at the contact line in the Navier-Stokes equation. The reason for this approach is that the intrinsic contact line slip in a VOF code is large enough that converting the contact line tension directly to a velocity on the wall does not move the contact line. The Navier-slip boundary condition component of the above equation is left unchanged in our implementation of the Generalized Navier Boundary Condition.

The goal of this derivation is to find an expression that rewrites the contact 