= Centripetal acceleration of an azimuthal flow
{title2=$a_r=-U^2/r,\quad a_\theta=-U^2\cot\theta/r,\quad a_\phi=0$}
For a <velocity field> $\mathbf u=U(r)e_\phi$ 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 $U'(r)$ cancel because a particle on such a streamline stays at fixed $r$ and $\theta$.
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