# Is it possible to have circulation without vorticity?

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

• 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. Dec 17 '19 at 12:33
• 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. Dec 17 '19 at 22:29

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.

• 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! Dec 27 '19 at 10:53

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.

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.