Sphere theorem for a Stokes streamfunction ($\tilde{\Psi }(r,\theta )=\Psi (r,\theta )-(r/a)\Psi (a^2/r,\theta )$)

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 showsHence the Stokes streamfunction equation for irrotational flow is preserved. At $r=a$, $\tilde{\Psi }=0$, 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.