Irrotational flow

A flow is called rotational if $\nabla  \times V \ne 0$ at every point in a flow. The fluid element has a finite angular velocity. A flow field is called irrotational if $\nabla  \times V = 0$ at every point in a flow. The fluid elements have no angular velocity and the motion is translation.

Example: A velocity field has a radial component of velocity ${V_r} = 0$ and tangential components of velocity ${V_\theta } = 4r$, respectively. Is this flow rotational or irrotational? 

Solution: Here ${V_r} = 0$ and ${V_\theta } = 4r$ $$\begin{array}{*{20}{l}}{\nabla  \times \mathop V\limits^ \to   = {e_z}\left[ {\frac{{\partial {V_\theta }}}{{\partial r}} + \frac{{{V_\theta }}}{r} - \frac{1}{r}\frac{{\partial \left( {{V_r}} \right)}}{{\partial \theta }}} \right]}\\{\nabla  \times \mathop V\limits^ \to   = {e_z}\left[ {\frac{{\partial \left( {4r} \right)}}{{\partial r}} + \frac{{4r}}{r} - \frac{1}{r}\frac{{\partial \left( 0 \right)}}{{\partial \theta }}} \right]}\\{\nabla  \times \mathop V\limits^ \to   = {e_z}\left[ {4 + 4 - 0} \right] = 8{e_z}}\end{array}$$ therefore, ${\nabla  \times \mathop V\limits^ \to  }$= finite.The flow is rotational.