\right)}$. Integral \[gauss\] can thus be approximated as: $$\frac{1}{h^{2}} \int_{L} \hat{n}_{2D} {\left( \vec{x}_{L} \right)} \mathcal{P} {\left( \vec{x}-\vec{x}_{L} \right)} d L
\approx
  \frac{1}{h^{2}} \hat{n}_{2D} {\left( \vec{x}_{L0} \right)} \int_{L} \mathcal{P} {\left( \vec{x}-\vec{x}_{L} \right)} d L
+
  \mathcal{O} {\left( \frac{h}{R} \right)}
\label{approximate}$$ where $\vec{x}_{L0}$ is the point on $L$ closest to $x$, and $R$ is the radius of the contact line.

The integral in equation \[approximate\] can be bounded by: $$\frac{1}{h} \int_{L} \mathcal{P} {\left( \vec{x}-\vec{x}_{L} \right)} d L
\le
  \mathcal{P} {\left( \vec{x}-\vec{x}_{L0} \right)}$$ where in the limit $h \to 0$, $\mathcal{P} {\left( \vec{x}-\vec{x}_{L0} \right)}$ is zero everywhere except for $\vec{x} = \vec{x}_{L0}$. Taking the corresponding limit of $\nabla_{2D} \tilde{c} {\left( \vec{x} \right)}$ over the interface gives: $$\lim_{h \to 0} \int \hat{n}_{2D} {\left( \vec{x}_{L0} \right)} \cdot \nabla_{2D} \tilde{c} {\left( \vec{x} \right)} dx
= 
  {\left[ c \right]}$$ As $h$ goes to $0$, $\nabla_{2D} \tilde{c} {\left( \vec{x} \right)}$ is thus equivalent to: $$\lim_{h \to 0} \nabla_{2D} \tilde{c} {\left( \vec{x} \right)}
=
  \hat{n}_{2D} {\left[ c \right]} \delta_{2D} {\left[ \hat{n}_{2D} \cdot {\left( \vec{x} - \vec{x}_{L} \right)} \right]} 
=
  \nabla_{2D} c {\left( \vec{x} \right)}
\label{delta}$$

Because the Brackbill surface tension already takes care of the $\cos(\theta)$ component of the reduced Young’s stress, only the $\cos{\theta_{0}}$ component needs to be 