Vorticity describes the tendency for a ﬂuid to rotate locally, deﬁned as
Vorticity is diﬃcult to picture under shear because the deformation masks the local rotation. By separating the deformation, as in the ﬁgure in Sec. 2.10 , the local rotation is revealed.
Vorticity is often demonstrated by a vortex ring produced by an air “cannon” with smoke to visualise the ﬂow. It reveals ﬂow circulation around sections of the torus, from front to back. The vorticity vectors are normal to the planes of circulation, along the axis of the torus.
Stokes’s theorem8 relates circulation — the integral of around a closed curved line, — to the integral of vorticity over a section of surface bounded by the curve, according to:
Vorticity is related to the spin tensor in Eq. (2.33 ) by , where is the Hodge dual operator which extracts components of a vector from a tensor as shown below: