For vortex flow, calculate: (a). The time rate of change of the volume of a fluid element per unit volume. (b). The vorticity.

a) Let for a vortex flow $u=\frac{cy}{x^{2}+y^{2}}$ and $v=\frac{-cx}{x^{2}+y^{2}}$.In polar coordinates the time rate of change of a fluid element per unit volume is given as $$\nabla\cdot \vec{V}=\frac{1}{r}\frac{\partial }{\partial r}\left ( rV_{r} \right )+\frac{1}{r}\frac{\partial V_{\theta}}{\partial \theta}$$

The velocity component  is transformed by using $x=rcos\theta$ and $y=rsin\theta$.

$$V_{r}=ucos\theta+vsin\theta \\V_{\theta}=-usin\theta+vcos\theta$$

$$u=\frac{cy}{x^{2}+y^{2}}=\frac{crsin\theta}{r^{2}}=\frac{c sin\theta}{r} \\v=\frac{-cx}{x^{2}+y^{2}}=\frac{crcos\theta}{r^{2}}=\frac{-c cos\theta}{r} \\V_{r}=\frac{c}{r}cos\theta sin\theta-\frac{c}{r}cos\theta sin\theta=0 \\V_{\theta}=\frac{-c}{r}sin^{2}\theta-\frac{c}{r}cos^{2}\theta=\frac{-c}{r}$$

$$\nabla\cdot \vec{V}=\frac{1}{r}\frac{\partial }{\partial r}\left ( 0 \right )+\frac{1}{r}\frac{\partial }{\partial \theta}\left ( \frac{-c}{r} \right )=0+0=0$$

b) The vorticity

Vorticity is given as $$\nabla \times \vec{V}=\vec{e_{z}}\left [ \frac{\partial }{\partial r}\left ( \frac{-c}{r} \right )-\frac{c}{r^{2}}-\frac{1}{r}\frac{\partial (0)}{\partial \theta} \right ] \\=\vec{e_{z}}\left [ \frac{c}{r^{2}}-\frac{c}{r^{2}} -0\right ] \\=0$$

Here $\nabla \times \vec{V}=0$ except at origin, where r=0.The flow field is singular at the origin.