Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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 $\vec{v}(\vec{r},t)$?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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: $$\kappa=\nabla \cdot \mathbf{\frac{n}{|n|}} $$

Apart from the source term you indeed also have boundary conditions on the interface which are basically the standard free slip condition and a jump for the normal stress coming again from the Laplace pressure. There is a good explanation of these in the first part of the seminal work on fluid-fluid CFD by Brackbill.

If you are interested in the curvature itself, I think Slides 22-28 of this course on wetting are probably also a good source to take a look at for more background.

share|improve this answer
    
thanks for the links! seems to be what I asked for –  xaxa May 8 '13 at 8:15

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.