Centripetal acceleration of an azimuthal flow ($a_r=-U^2/r,a_{\theta }=-U^2\cot \theta /r,a_{\varphi }=0$)

For a velocity field $\mathbf{u}=U(r)e_{\varphi }$ in spherical coordinates, the convective acceleration is $(\mathbf{u}\cdot \nabla )\mathbf{u}=-U(r{)}^2(e_r+\cot \theta e_{\theta })/r$, away from the polar axis. Each streamline is a circle of cylindrical radius $r\sin \theta$, so this is centripetal acceleration of magnitude $U^2/(r\sin \theta )$ directed toward the polar axis. The terms containing $U^{\prime }(r)$ cancel because a particle on such a streamline stays at fixed $r$ and $\theta$.