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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.

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  • $\begingroup$ Sorry for this question: by interface you mean phase surface separation? It is synonymous to surface of the fluid? $\endgroup$
    – kakaz
    Sep 26 '19 at 16:29
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    $\begingroup$ 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. $\endgroup$
    – energy
    Sep 26 '19 at 16:31

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