 Solution

The general continuity equation is ${\rho }_t+\nabla \cdot (\rho u)=0$, or $D\rho /Dt+\rho \nabla \cdot u=0$. An incompressible flow has $D\rho /Dt=0$, so for positive density $\boxed{\nabla \cdot u=0}$.

For the displayed velocity field, on $x^2+y^2>0$,The sum is zero. A stream function with the required sign convention isIts derivatives give exactly $u=({\psi }_y,-{\psi }_x)$. The origin is excluded because the prescribed velocity is singular there.