# About parametrization of the interface of a fluid

In Navier Stokes Equation, more precisly in the evolution of a fluid interface I have an extension of the function $$u$$ such that:

$$u(x,t)=\frac{1}{2\pi} \operatorname{P.V.}\int_\Omega\frac{(x-z(\beta,t))^{\perp}}{|x-z(\beta,t)|^2}\omega(\beta,t)d\beta.$$

And then I take the limit as $$\lim_{x\to z(\alpha,t)}u(x,t)$$ then I have this:

$$u(z(\alpha,t),t)=BR(z,w)(\alpha,t)+\frac{1}{2}\omega(\alpha,t)\frac{\partial_{\alpha}z}{|\partial_{\alpha}z|^2}.$$

I understand the Birkhoff-Rott integral step but I do not know why the other term?. Is it because it has somethign related with principal value and putting the limit inside?

Greetings.

• Sorry for this question: by interface you mean phase surface separation? It is synonymous to surface of the fluid? – kakaz Sep 26 '19 at 16:29
• Yes, it is the parametrization of the interface which is the separation between 2 fluids of different density, in my particular case, one is the vacum so its density is 0 and the other one is the water. – energy Sep 26 '19 at 16:31