Consider a 2D plane flow that is now represented in terms of the polar coordinates. The flow has then only \(v_{r}\) and \(v_{\theta}\) components and no \(v_{z}\) component. How is the streamfunction defined here? Show that the continuity equation (in polar coordinates) is automatically satisfied by this function.

Also show that the vorticity of flow can be represented as

\[-\omega=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \psi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \psi}{\partial \theta^{2}}\]

or in other words

\[\omega=-abla^{2} \psi\]

where \(abla^{2}\) denotes the Laplacian polar coordinates.