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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.

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