2.11 Vorticity
Vorticity describes the tendency for a fluid to rotate locally, defined as
![]() |
(2.37) |


![]() |
(2.38) |
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:
![]() |
(2.39) |






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:
![]() |
(2.40) |
