# boundary conditions for liquid with surface tension

so one uses equations of motion to describe liquids (e.g. Navier–Stokes equations). These are equations for $\vec{v}(\vec{r},t)$ with boundary conditions on the surface $S$ of the liquid (e.g. $\vec{v}(\vec{r}\in S,t) = \vec{0}$).

How should one incorporate surface tension $\sigma$ in these equations/boundary conditions? It seems, only boundary conditions must change, and $\Delta p = \sigma (1/R_1 + 1/R_2)$ is the first thing that comes to mind, but how to get $1/R$ from $\vec{v}(\vec{r},t)$?

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You don't want $1/R$ (although technically it means the same) but rather the full curvature term: $\Delta p=\sigma \kappa$. In fact you will get a source term in the Navier-Stokes equations that looks like this: $$\sigma \kappa \delta(n) \mathbf{n}$$ where $\delta(n)$ is the Dirac Delta function that only has a value at the interface and $\mathbf{n}$ is the interface normal. The curvature $\kappa$ can be written as the divergence of the unit interface normal: $$\kappa=\nabla \cdot \mathbf{\frac{n}{|n|}}$$