# Show incompressibility and irrotationality.

For a source flow, show that it is a physically possible incompressible flow except at origin.Also, show that it is irrotational.

Asked on 31st December 2018 in
For a source flow, velocity $$\mathop V\limits^ \to = {V_r}\mathop {{e_r}}\limits^ \to = \frac{\Lambda }{{2\pi r}}\mathop {{e_r}}\limits^ \to$$In polar co-ordinates :, the curl of velocity field will be $\nabla \cdot \mathop V\limits^ \to = \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{V_r}} \right) + \frac{1}{r}\frac{{\partial {V_\theta }}}{{\partial \theta }}$$\nabla \cdot \mathop V\limits^ \to = \frac{1}{r}\frac{\partial }{{\partial r}}\left[ {r\frac{\Lambda }{{2\pi r}}} \right] + \frac{1}{r}\frac{{\partial \left( 0 \right)}}{{\partial \theta }}$$\nabla \cdot \mathop V\limits^ \to = \frac{1}{r}\frac{\partial }{{\partial r}}\left( {\frac{\Lambda }{{2\pi }}} \right) + 0 = 0$Therefore, the flow is a physically possible incompressible flow, except at origin where r =0.
Also,$\nabla \times \mathop V\limits^ \to = \frac{1}{r}\left| {\begin{array}{*{20}{c}}{\mathop {{e_r}}\limits^ \to }&{r\mathop {{e_\theta }}\limits^ \to }&{\mathop {{e_z}}\limits^ \to }\\{\frac{\partial }{{\partial r}}}&{\frac{\partial }{{\partial \theta }}}&{\frac{\partial }{{\partial z}}}\\{{V_r}}&{r{V_\theta }}&{{V_z}}\end{array}} \right| = \frac{1}{r}\left| {\begin{array}{*{20}{c}}{\mathop {{e_r}}\limits^ \to }&{r\mathop {{e_\theta }}\limits^ \to }&{\mathop {{e_z}}\limits^ \to }\\{\frac{\partial }{{\partial r}}}&{\frac{\partial }{{\partial \theta }}}&{\frac{\partial }{{\partial z}}}\\{\frac{\Lambda }{{2\pi r}}}&0&0\end{array}} \right|$$\nabla \times \mathop V\limits^ \to = – r\mathop {{e_\theta }}\limits^ \to \left( {\frac{{\partial 0}}{{\partial r}} – \frac{{\partial \Lambda /2\pi r}}{{\partial z}}} \right) + \mathop {{e_z}}\limits^ \to \left( {\frac{{\partial 0}}{{\partial r}} – \frac{{\partial \Lambda /2\pi r}}{{\partial \theta }}} \right) = 0$Therefore, $$\nabla \times \mathop V\limits^ \to = 0$$ (everywhere in the flow).