Consider a velocity field where the radial and tangential components of velocity are Vr = 0 and Vθ = cr, respectively, where c is a constant. Is the flow field irrotational? Prove your answer.
Consider a velocity field where the radial and tangential components of velocity are \(V_{r} = 0\) and \(V_{θ} = cr\), respectively, where \(c\) is a constant. Is the flow field irrotational? Prove your answer.
Here \(V_{r} =0\) and \(V_{\theta} = cr\)
\[\nabla \times \vec{V}=\vec{e_{z}}\left [ \frac{\partial }{\partial r}\left ( \frac{c}{r} \right )+\frac{cr}{r} -\frac{1}{r}\left ( \frac{\partial (0)}{\partial \theta} \right )\right ]
\\=\vec{e_{z}}\left ( c+c-0 \right )=2c\vec{e_{z}}\]
Here vorticity is finite.Therefore the flow is rotational.
