= Solution
The steady <Stokes equations> are $-\nabla p+\mu\nabla^2\mathbf u=0$, $\nabla\cdot\mathbf u=0$. Taking the curl gives $\nabla^2\boldsymbol\omega=0$. For an axisymmetric azimuthal component $\omega\mathbf e_\phi$, the vector Laplacian's azimuthal component is $(\nabla^2-1/(R^2\sin^2\theta))\omega$. Expanding the derivatives shows
$$
\left(\nabla^2-\frac1{R^2\sin^2\theta}\right)\frac{F}{R\sin\theta}
=\frac1{R\sin\theta}D^2F.
$$
With $\omega=-D^2\Psi/(R\sin\theta)$, the vorticity equation consequently gives \b[$D^4\Psi=0$]. This derives the streamfunction equation from momentum rather than treating the scalar vorticity component as an ordinary scalar <harmonic function>.
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