Relationship between angular velocity and curl of a vector
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Denote by v the velocity field of a planar fluid rotating with angular velocity ω around the origin 0 of the z-plane. At distance r from 0 we then have |v|=rω. Two rays from 0 at arguments ϕ and ϕ+dϕ, together with two concentric circles of radii r and r+dr determine an infinitesimal quadrangle Q with center z0. Let us compute the circulation of v around ∂Q, i.e., the line integral ∫∂Qv⋅dz.
Along the (counterclockwise) outer arc we obtain the contribution ω(r+dr)⋅(r+dr)dϕ, along the (clockwise) inner arc the contribution −ωr⋅rdϕ, and the two radial segments give no contribution since they are orthogonal to v. It follows that, up to first order in dr we obtain
∫∂Qv⋅dz≐2ωrdrdϕ .(1)
On the other hand, by Green's (or Stokes') theorem we have
∫∂Qv⋅dz=∫Qcurl(v(z))d(z)≐curl(z0)area(Q)≐curl(z0)drrdϕ .(2)
Comparing (1) and (2) we can see that necessarily curl(z0)=2ω, independently of z0. Looking back we recognize the reason for this factor of 2: Going outwards not only the speed |v| increases, but also the length of the arcs corresponding to a given dϕ.
A student confronted with the above field v or curl the first time is inclined to think that curl(v) is somehow concentrated at the origin 0. But this is not the case: curl(v)=2ω can be felt everywhere!