Convective acceleration ($(\mathbf{u}\cdot \nabla )\mathbf{u}$)

Convective acceleration is the spatial part $(\mathbf{u}\cdot \nabla )\mathbf{u}$ of the material derivative of fluid velocity. It can remain nonzero in steady flow, because a moving particle passes through positions with different velocities. The identity $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (|\mathbf{u}{|}^2/2)-\mathbf{u}\times (\nabla \times \mathbf{u})$ relates it to vorticity.

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 $\theta$.