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albega

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(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 :)

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 :)

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