Helmholtz’s Third Vortex Theorem

The third vortex theorem of Helmholtz’s states that:

“the circulation of a vortex tube remains constant in time."

Using Helmholtz’s second theorem and Kelvin’s circulation theorem, the above statement can be inter­preted as “a closed line generating the vortex tube is a material line whose circulation remains constant.” Helmholtz’s second and third theorems hold only for inviscid and barotropic fluids.

5.9.2 Helmholtz’s Fourth Vortex Theorem

The fourth theorem states that:

“the strength of a vortex remains constant in time."

This is similar to the fact that the mass flow rate through a streamtube is invariant as the tube moves in the flow field. In other words, the circulation distribution gets adjusted with the area of the vortex tube. That is, the circulation per unit area (that is, vorticity) increases with decrease in the cross-sectional area of the vortex tube and vice versa.