Is Kelvin's Circulation Theorem Applicable to Vortex Tube Conservation?

[albega]

(Edited to make an answer more likely)

So first let's quickly summarize what this is. If you have some closed curve c(t) around a set of fluid elements, Kelvin's circulation theorem says that the circulation around this curve is constant as the curve and its corresponding fluid elements move around the fluid. And of course because this circulation is equivalently a surface integral of the vorticity we can say a few things about vorticity.

The following page proves that the circulation around any part of a vortex tube is conserved:
http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node57.html
However this proof does not use Kelvin's circulation theorem. My question is could we actually apply the circulation theorem in this case to obtain the same conclusion? I feel as though really we shouldn't because we can't assume we are able to get a curve c(t) that will move along a given vortex tube, but my notes seem to suggest it can be applied in this way...

Can anyone help with this, thanks :)

 

[gtoguy97]

Yes, you can use Kelvin's circulation theorem to prove the conservation of circulation around any part of a vortex tube. The proof is similar to the one provided in the link you shared, but with the added step of showing that the curve c(t) moves along the vortex tube as it changes shape and direction. To do this, you will need to make sure that the curve c(t) stays within the boundaries of the vortex tube as it changes shape and direction. You can then use the circulation theorem to show that the circulation around the curve c(t) remains constant.