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Obviously there are two different kinds of flow discussed, namely

(a) rigid rotation

The velocity field then is

##\vec{v}(\vec{r})=\vec{\omega} \times \vec{r}##

with ##\vec{\omega}=\text{const}##.

(b) "potential vortex"

##\vec{v}(\vec{r})=\frac{C}{2 \pi \rho^2} \begin{pmatrix}-y \\ x \\ 0 \end{pmatrix},##

where ##\rho=\sqrt{x^2+y^2}##.

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