A steady velocity field $\mathbf{u}=\left(u_{r}, u_{\theta}, u_{z}\right)$ is given in cylindrical polar coordinates $(r, \theta, z)$ by

$$u_{r}=-\alpha r, \quad u_{\theta}=\frac{\gamma}{r}\left(1-e^{-\beta r^{2}}\right), \quad u_{z}=2 \alpha z$$

where $\alpha, \beta, \gamma$ are positive constants.

Show that this represents a possible flow of an incompressible fluid, and find the vorticity $\omega$.

Show further that

$$\operatorname{curl}(\mathbf{u} \wedge \boldsymbol{\omega})=-\nu \nabla^{2} \boldsymbol{\omega}$$

for a constant $\nu$ which should be calculated.

[The divergence and curl operators in cylindrical polars are given by

$$\begin{aligned} \operatorname{div} \mathbf{u} &=\frac{1}{r} \frac{\partial}{\partial r}\left(r u_{r}\right)+\frac{1}{r} \frac{\partial u_{\theta}}{\partial \theta}+\frac{\partial u_{z}}{\partial z} \\ \operatorname{curl} \mathbf{u} &=\left(\frac{1}{r} \frac{\partial u_{z}}{\partial \theta}-\frac{\partial u_{\theta}}{\partial z}, \frac{\partial u_{r}}{\partial z}-\frac{\partial u_{z}}{\partial r}, \frac{1}{r} \frac{\partial}{\partial r}\left(r u_{\theta}\right)-\frac{1}{r} \frac{\partial u_{r}}{\partial \theta}\right) \\ \text { and ,when } \boldsymbol{\omega} &=[0,0, \omega(r, z)], \\ \nabla^{2} \boldsymbol{\omega} &\left.=\left[0,0, \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \omega}{\partial r}\right)+\frac{\partial^{2} \omega}{\partial z^{2}}\right] .\right] \end{aligned}$$