Show that a source flow is a physically possible incompressible flow everywhere except at the origin. Also show that it is irrotational everywhere.
Show that a source flow is a physically possible incompressible flow everywhere except at the origin. Also show that it is irrotational everywhere.
For a source flow ;
\[\overrightarrow{V}=V_{r}\overrightarrow{e}_{r}=\frac{\Lambda }{2\pi r}\overrightarrow{e_{r}}\]
In polar coordinates :
\[\nabla \cdot \overrightarrow{V}=\frac{1}{r}\frac{\partial }{\partial r}\left ( rV_{r} \right )+\frac{1}{r}\frac{\partial V_{\theta}}{\partial \theta}
\\\Rightarrow\nabla \cdot \overrightarrow{V}=\frac{1}{r}\frac{\partial }{\partial r}\left [ r\;\frac{\Lambda }{2\pi r} \right ]+\frac{1}{r}\frac{\partial(0)}{\partial \theta}
\\\Rightarrow\nabla \cdot \overrightarrow{V}=0+0=0\]
This shows that the flow is incompressible.For the flow to be irrotational
\[\nabla\times \overrightarrow{V}=0\]
\[\nabla \times \overrightarrow{V}=\frac{1}{r}\begin{vmatrix}
\overrightarrow{e}_{r} &r\overrightarrow{e}_{\theta} & \overrightarrow{e}_{z}\\
\frac{\partial }{\partial r}& \frac{\partial }{\partial \theta} & \frac{\partial }{\partial z}\\
V_{r}& rV_{\theta} & V_{z}
\end{vmatrix}=\frac{1}{r}\begin{vmatrix}
\overrightarrow{e}_{r} &r\overrightarrow{e}_{\theta} & \overrightarrow{e}_{z}\\
\frac{\partial }{\partial r}& \frac{\partial }{\partial \theta} & \frac{\partial }{\partial z}\\
\frac{\Lambda }{2\pi r}&0 & 0
\end{vmatrix}\]
\[\Rightarrow\nabla\times \overrightarrow{V}=-r\overrightarrow{e_{\theta}}\left ( \frac{\partial (0)}{\partial r} -\frac{\partial }{\partial z}\left ( \frac{ \Lambda }{2\pi r} \right )\right )+\overrightarrow{e}_{z}\left ( \frac{\partial (0)}{\partial r} -\frac{\partial }{\partial \theta}\left ( \frac{ \Lambda }{2\pi r} \right ) \right )\]
\[\Rightarrow\nabla\times \overrightarrow{V}=-r\overrightarrow{e_{\theta}}\left (0-0\right )+\overrightarrow{e}_{z}\left ( 0-0 \right )=0\]
This shows that it is irrotational everywhere.
