Vortex motions are vital elements of fluid flows, most dynamical aspects of fluid motions are featured by vortex motions.
Analytical study of vortex motions is mostly based on vorticity which is defined as twice the angular vorticity of local rotation.
Here the fluid density \(\rho\) is assumed to be constant, so that the flow is barotropic.
Equation of the vorticity \(\boldsymbol{\omega} = \nabla \times \mathbf{u}\) was given for compressible flows in an inviscid fluid \((\nu = 0)\) as
\[
\frac{\partial \boldsymbol{\omega}}{\partial t} + \nabla \times (\boldsymbol{\omega} \times \mathbf{u}) = 0
\]
For viscous incompressible flows
\[
\frac{\partial \boldsymbol{\omega}}{\partial t}
+ \nabla \times (\boldsymbol{\omega} \times \mathbf{u})
= \underbrace{
\nu \nabla^2 \boldsymbol{\omega}
}_{\text{rate of change of } \omega \text{ caused by diffusion of vorticity} \\ \text{in the same way that } \nu \nabla^2 \mathbf{u} \\ \text{represents acceleration caused by diffusion of momentum}}
\]
\[
\underbrace{
\frac{\partial \boldsymbol{\omega}}{\partial t}
+ \underbrace{
(\mathbf{u} \cdot \nabla) \boldsymbol{\omega}
}_{\text{rate of change of vorticity}}
}_{\displaystyle \frac{D \boldsymbol{\omega}}{D t}}
=
\underbrace{
(\boldsymbol{\omega} \cdot \nabla) \mathbf{u}
}_{\text{rate of change of vorticity} \\ \text{caused by the stretching} \\ \text{and tilting of vortex lines}}
+
\underbrace{
\nu \nabla^2 \boldsymbol{\omega}
}_{\text{rate of change of } \omega \text{ caused by diffusion of vorticity} }
\]