Starting from the Euler equations for incompressible, inviscid flow

$$\rho \frac{D \mathbf{u}}{D t}=-\nabla p, \quad \nabla \cdot \mathbf{u}=0$$

derive the vorticity equation governing the evolution of the vorticity $\boldsymbol{\omega}=\nabla \times \mathbf{u}$.

Consider the flow

$$\mathbf{u}=\beta(-x,-y, 2 z)+\Omega(t)(-y, x, 0)$$

in Cartesian coordinates $(x, y, z)$, where $t$ is time and $\beta$ is a constant. Compute the vorticity and show that it evolves in time according to

$$\boldsymbol{\omega}=\omega_{0} \mathrm{e}^{2 \beta t} \mathbf{k}$$

where $\omega_{0}$ is the initial magnitude of the vorticity and $\mathbf{k}$ is a unit vector in the $z$-direction.

Show that the material curve $C(t)$ that takes the form

$$x^{2}+y^{2}=1 \quad \text { and } \quad z=1$$

at $t=0$ is given later by

$$x^{2}+y^{2}=a^{2}(t) \quad \text { and } \quad z=\frac{1}{a^{2}(t)},$$

where the function $a(t)$ is to be determined.

Calculate the circulation of $\mathbf{u}$ around $C$ and state how this illustrates Kelvin's circulation theorem.