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Put $s=a^2/r$ and define the image term $G(r,\theta )=(r/a)\Psi (s,\theta )$. The key step is to verify that inversion preserves the Stokes streamfunction equation for irrotational flow. Direct differentiation giveswhere the quantities on the right are evaluated at $(s,\theta )$. HenceBy linearity, $\tilde{\Psi }=\Psi -G$ obeys the same equation and describes an incompressible flow and an irrotational flow wherever the original exterior flow is regular.

On the sphere, $s=a$ and the two terms cancel identically:Thus the sphere is a stream surface and satisfies the no-penetration boundary condition. Tangential slip is allowed in this potential flow model. For every finite $r>a$, the image argument satisfies $0<s<a$, where $\Psi$ has no singularities. The prefactor $r/a$ is regular there. Therefore the construction creates no new finite exterior singularities; any original exterior singularities remain those of the prescribed incident flow. This is the sphere theorem for a Stokes streamfunction.

For the far-field claim, the stated bound givesOne must compare velocities, rather than require the correction to the Stokes streamfunction to tend to zero. Smoothness at the original origin, together with $\Psi (0,\theta )=0$, gives ${\Psi }_s=O(1)$ and ${\Psi }_{\theta }=O(s)$ there. Thus $G_r=O(r^{-1})$ and $G_{\theta }=O(1)$, and the correction hasThese estimates hold away from the coordinate poles, with the regular-axis limits supplied by the original smooth physical flow. ConsequentlyThe far-field velocity field is unchanged. For a fully smooth incident velocity field at the origin, the usual regular expansion is stronger and makes the sphere disturbance decay as $r^{-3}$.

The incident uniform flow $U{\mathbf{e}}_z$ has spherical components $u_r=U\cos \theta$, $u_{\theta }=-U\sin \theta$. Integrating the Stokes streamfunction formulas, with the additive constant chosen to vanish on the axis, givesIt is regular at the origin. Applying the sphere theorem for a Stokes streamfunction yieldsThe resulting velocity components areTo find the velocity potential, integrate ${\varphi }_r={\tilde{u}}_r$ and verify $r^{-1}{\varphi }_{\theta }={\tilde{u}}_{\theta }$. This givesThe first term is the incident uniform flow, and the second is a potential dipole located inside the excluded sphere. At $r=a$, the radial velocity vanishes, while the far-field velocity tends to $U{\mathbf{e}}_z$. The poles are stagnation points; the surface tangential speed reaches $3U/2$ at the equator. These formulas describe uniform potential flow around a fixed sphere.

For a fixed impermeable sphere of radius $a$ and incident uniform flow $U{\mathbf{e}}_z$, the velocity potential is the displayed expression and the Stokes streamfunction is $\Psi =(U/2)(r^2-a^3/r){\sin }^2\theta$. Differentiation gives $u_r=U(1-a^3/r^3)\cos \theta$ and $u_{\theta }=-U(1+a^3/(2r^3))\sin \theta$. Thus the no-penetration boundary condition holds, the poles are stagnation points, and the equatorial surface speed is $3U/2$. The disturbance is a potential dipole decaying as $r^{-3}$ in velocity. The sphere theorem for a Stokes streamfunction constructs this solution from ${\Psi }_0=U{r}^2{\sin }^2\theta /2$.