A Question about Kelvin’s circulation theorem

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Kelvin's circulation theorem states that in inviscid flow with constant density and conservative body forces, the circulation around a contour is conserved for fluid particles. This raises the question of whether vorticity is also preserved for each fluid particle under these conditions. Analyzing the vorticity equation, it can be shown that under the same assumptions as Kelvin's theorem, the equation reduces to \frac{D\omega}{Dt} = 0, indicating that vorticity is indeed conserved. Therefore, Kelvin's theorem not only preserves circulation but also ensures the preservation of vorticity for fluid particles. This relationship highlights the fundamental principles of fluid dynamics in inviscid flows.
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In Currie’s fluid mechanics textbook, there is a statement “the vorticity of each fluid particle will be preserved.” as the result of Kelvin’s circulation theorem.
Kelvin’s circulation theorem claims that
For inviscid flow, constant density or barotropic fluid, conservative body force,
the circulation around an arbitrary contour is conserved following same fluid particle.
Does Kelvin’s theorem also guarantee the vorticity on each fluid particles? Why?
 
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Take a look at the vorticity equation under the same conditions as Kelvin's circulation theorem. Does it reduce to \frac{D\omega}{Dt} = 0?
 

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