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

• albega
In summary, Kelvin's circulation theorem can be applied to prove the conservation of circulation around any part of a vortex tube, as long as the curve c(t) stays within the boundaries of the vortex tube.
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 :)

Last edited:
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.

## 1. What is Kelvin's Circulation Theorem?

Kelvin's Circulation Theorem is a fundamental principle in fluid dynamics that explains the relationship between the circulation of a fluid and its vorticity. It states that in a steady, inviscid (frictionless) flow, the circulation around any closed loop in the fluid is constant.

## 2. Who discovered Kelvin's Circulation Theorem?

The theorem is named after William Thomson, also known as Lord Kelvin, a Scottish physicist and mathematician who first published it in 1867.

## 3. What are some real-world applications of Kelvin's Circulation Theorem?

Kelvin's Circulation Theorem has many practical applications, including in airplane wing design, weather forecasting, ocean currents, and even blood flow in the human body. It is also used in the study of tornadoes and hurricanes.

## 4. How is Kelvin's Circulation Theorem related to Bernoulli's principle?

Kelvin's Circulation Theorem is closely related to Bernoulli's principle, which states that in an inviscid flow, the sum of pressure and kinetic energy per unit volume is constant. This is because the circulation theorem can be derived from Bernoulli's principle.

## 5. Are there any limitations to Kelvin's Circulation Theorem?

Yes, there are some limitations to the theorem. It only applies to steady, inviscid flows and does not take into account the effects of viscosity or turbulence. It also assumes that the fluid is incompressible and has a constant density. In practical applications, these assumptions may not always hold true, and therefore the theorem may not accurately predict the behavior of the fluid.

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