a system of two fluids, fluid $1$, and fluid $2$, separated by an interface, and define a discontinuous function, $c {\left( \vec{x} \right)}$, to distinguish between the two phases: $$c {\left( \vec{x} \right)}
=
  \left\{
    \begin{array}{ll}
      c_{1}                           & {\text{in liquid 1}}  \\
      {\left<{c}\right>} = {\left( c_{1} + c_{2} \right)}/2 & {\text{on interface}} \\
      c_{2}                           & {\text{in liquid 2}}
    \end{array}
  \right.$$ An example of such a function would be the density of two different incompressible liquids, $\rho_{1}$, and $\rho_{2}$. In this case the position of the contact line can be found with: $$\rho {\left( \vec{x}_{L} \right)}
=
  {\left<{\rho}\right>}$$ In order to change this problem from a boundary value problem on the contact line to an approximate continuous model, one can define a continuous function $\tilde{c}$, which varies smoothly over thickness $h$ going from $c_{1}$ to $c_{2}$ , $c_{1} \le \tilde{c} \le c_{2}$. $h$ is a length scale of the order of the grid size $\Delta x$, and defines the width of the transition region from $c_{1}$ to $c_{2}$. The two functions $c$ and $\tilde{c}$ are related through the interpolation function $\mathcal{P} {\left( \vec{x} \right)}$: $$\tilde{c} {\left( \vec{x} \right)}
=
  \frac{1}{h^{2}} \int_{A} c {\left( \vec{x}' \right)} \mathcal{P} {\left( \vec{x}'-\vec{x} \right)} d^{2} \vec{x}$$ 