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Both circulation and vorticity have to do with rotation of a fluid element.How are vorticity and circulation are related.

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  • $\begingroup$ In formal-ish settings, circulation is the spacial integral of vorticity. Vorticity is usually a point property, or a property of a very small region (such as a FEA cell). We can derive the vorticity distribution function over a lifting surface (and wake surfaces) that will result in flow parallel to the surfaces at the surface, and integrating the resulting velocity vector perturbations over the domain yields circulation. $\endgroup$
    – Phil Sweet
    Dec 17 '19 at 12:33
  • $\begingroup$ Also, be very careful how you use the word "rotation" in vector field contexts. Both the vortex sources and the resulting circulation are irrotational in the formal sense. $\endgroup$
    – Phil Sweet
    Dec 17 '19 at 22:29
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No. If no vorticity then, no circulation.

Mathematically, The flux of vorticity is circulation.

$ \Gamma = \int\int \omega~ ds$

So, $\Gamma \rightarrow $ 0 when $\omega \rightarrow $ 0.

Physically, vorticity is just not a rotation of a fluid element. But it is the rotation of that element about its own axis (spin). (Please know about free/forced vortex for a kind of imagination).

So, circulation is non-zero only when there is a flux of vorticity.

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  • $\begingroup$ It is very interesting and can further go on. The kutta-Jowkoski theorem relates lift with circulation ($L = \rho u \Gamma$) and Blasius theorem says momentum imparted on a body by the inviscid flow is zero. So, in nature, viscosity creates friction and hence vorticity $\rightarrow$ circulation and $\rightarrow$ lift! $\endgroup$
    – mustang
    Dec 27 '19 at 10:53
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The Biot-Savart Law can be interpreted as the relationship between the velocity induced by a vortex tube and the strength of the vortex tube, i.e. its circulation.

See: https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law

Note: you need to be a little careful in ascribing causes and effects due to the unfortunate adjective "induced" in induced velocity. In classical fluid mechanics there is no action-at-a-distance, so vorticity cannot cause changes to fluid velocities at remote locations.

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