 Solution

For an axisymmetric flow without azimuthal velocity, define the Stokes streamfunction byThese formulas enforce incompressibility. The only nonzero curl component is $R^{-1}[{\partial }_R(R{u}_{\theta })-{\partial }_{\theta }{u}_R]$. Substitution givesCurling the Stokes flow equations eliminates pressure and gives ${\nabla }^2(\nabla \times u)=0$. For an azimuthal field, its vector Laplacian is the scalar Laplacian minus $1/(R^2{\sin }^2\theta )$; applying that operator to $-D^2\Psi /(R\sin \theta )$ gives $-D^2(D^2\Psi )/(R\sin \theta )$. Hence the streamfunction equation is $\boxed{D^2{D}^2\Psi =0}$.

For the proposed sphere solution put $f(R)=R^2-3aR/2+a^3/(2R)$, so $\Psi =(U/2)f(R){\sin }^2\theta$. Its velocity isBoth components vanish at $R=a$ and tend to the components of $U{e}_z$ at infinity. Also $D^2\Psi =(U/2)(f^{\prime \prime }-2f/R^2){\sin }^2\theta =(3Ua/(2R)){\sin }^2\theta$, and applying $D^2$ once more gives zero. This verifies the differential equation and both physical boundary conditions.

For a sphere translating upward through otherwise stationary fluid, subtract the uniform background in the sphere frame and reverse its far-flow direction. Its leading disturbance is the upward StokesletPlace an image at $z=+d$ with the opposite, downward velocity $-U$, while the real sphere is at $z=-d$. The sum has $u_z$ odd under $z\mapsto -z$ and $u_r$ even. Thus at $z=0$, $u_z=0$ and ${\partial }_z{u}_r=0$; moreover ${\partial }_r{u}_z=0$. The tangential stress ${\sigma }_{rz}=\mu ({\partial }_z{u}_r+{\partial }_r{u}_z)$ is therefore zero, as required by the stress-free boundary condition.

At a point on the plane with radial coordinate $r$, the real Stokeslet gives $u_r=3aUrd/[4(r^2+d^2{)}^{3/2}]$. The image gives the same radial contribution, soThe surface flow is outward. Although the isolated-sphere dipole term is only $O(U(a/d{)}^3)$, the next correction is $O(U(a/d{)}^2)$: the image's velocity at the real sphere is downward and of size $3aU/(4d)$, so satisfying the real sphere's prescribed velocity requires a relative $O(a/d)$ correction to its Stokeslet strength. Since the leading surface flow is $O(Ua/d)$, the first reflection already changes it at the displayed second order. Ignoring this boundary-interaction correction would give the wrong next power.