For an incompressible ﬂuid (with other simplifying assumptions), vorticity obeys the transport equation , Eq. (6.2 ). This is a typical advection-diﬀusion equation, similar to Eq. (2.65 ) for heat, which is expressed in terms of the local time derivative and advection in conservative form by
Instead, like heat, vorticity evolves locally only, with a range of inﬂuence limited by advective and diﬀusive transport, as discussed on page 126 .
Advection of vorticity is clearly illustrated by the smoke ring shown in Sec. 2.11 . Diﬀusion occurs by viscous torques transferring angular momentum between ﬂuid elements.
The source of vorticity, is due to vortices changing shape under the inﬂuence of a velocity gradient . If a vortex is stretched, e.g. under shear as shown above, its radius decreases, so angular velocity, and thus , increases. Similarly, decreases if the vortex is compressed.
For incompressible ﬂow with constant (and zero, or constant, body force), vorticity obeys the following transport equation: