= Stokes streamfunction equation for irrotational flow
{c}
{title2=$r^2\Psi_{rr}+\Psi_{\theta\theta}-\cot\theta\,\Psi_\theta=0$}
For an <axisymmetric flow> without azimuthal <velocity>, the spherical <Stokes streamfunction> gives $u_r=\Psi_\theta/(r^2\sin\theta)$ and $u_\theta=-\Psi_r/(r\sin\theta)$. These formulas enforce <incompressibility>. The only possibly nonzero component of <vorticity> is
$$
(\nabla\times\mathbf u)_\varphi=-\frac{r^2\Psi_{rr}+\Psi_{\theta\theta}-\cot\theta\,\Psi_\theta}{r^3\sin\theta}.
$$
Thus the displayed equation is precisely the condition for <irrotational flow>, away from coordinate poles, with limits interpreted through regularity of the physical <velocity field>. It differs from the scalar spherical <Laplace equation>: $\Psi$ is a <Stokes streamfunction>, not a <velocity potential>.
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