Use Euler's equations to derive the vorticity equation

$$\frac{D \boldsymbol{\omega}}{D t}=\boldsymbol{\omega} \cdot \nabla \mathbf{u}$$

where $\mathbf{u}$ is the fluid velocity and $\boldsymbol{\omega}$ is the vorticity.

Consider axisymmetric, incompressible, inviscid flow between two rigid plates at $z=h(t)$ and $z=-h(t)$ in cylindrical polar coordinates $(r, \theta, z)$, where $t$ is time. Using mass conservation, or otherwise, find the complete flow field whose radial component is independent of $z$.

Now suppose that the flow has angular velocity $\boldsymbol{\Omega}=\Omega(t) \mathbf{e}_{z}$ and that $\Omega=\Omega_{0}$ when $h=h_{0}$. Use the vorticity equation to determine the angular velocity for subsequent times as a function of $h$. What physical principle does your result illustrate?