 Solution

For the stated incompressible planar flow, use the stream function convention $u={\psi }_y$, $v=-{\psi }_x$. It gives $u_x+v_y=0$ and the vorticity isThus $\boxed{\mathbf{\omega }=-{\nabla }^2\psi \hat{\mathbf{z}}}$, where the Laplacian is the two-dimensional one.