= Sphere theorem for a Stokes streamfunction
{title2=$\widetilde\Psi(r,\theta)=\Psi(r,\theta)-(r/a)\Psi(a^2/r,\theta)$}
Suppose the incident <axisymmetric flow> is an <incompressible flow> and an <irrotational flow>, with <Stokes streamfunction> $\Psi$ regular inside $r<a$. The displayed transformation adds an impermeable <sphere> at $r=a$ without new finite exterior singularities. Writing $s=a^2/r$ and $G=(r/a)\Psi(s,\theta)$, direct differentiation shows
$$
\left(r^2\partial_r^2+\partial_\theta^2-\cot\theta\,\partial_\theta\right)G
=\frac r a\left(s^2\Psi_{ss}+\Psi_{\theta\theta}-\cot\theta\,\Psi_\theta\right)=0.
$$
Hence the <Stokes streamfunction equation for irrotational flow> is preserved. At $r=a$, $\widetilde\Psi=0$, making the radial <velocity> zero. Exterior points sample only the regular interior under inversion. If $\Psi$ vanishes at the origin and is $O(r)$ there with smooth regular derivatives, the induced <velocity> decays at infinity. This is an inversion construction for an axisymmetric streamfunction, distinct from a scalar <Kelvin transform>.
Back to article page
