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Both candidate fields are divergence-free. For solid-body rotation,each component is linear in $\mathbf{x}$, so ${\nabla }^2\mathbf{u}=0$. Also,is harmonic for $r>0$, because ${\nabla }^2(1/r)=0$ there and constant-coefficient derivatives commute with the Laplacian. Thus both solve the Stokes equations with $p=0$.

The rotationally symmetric solution is consequentlyThe no-slip boundary condition givesThereforeandThis is the Rotational Stokes flow between concentric spheres.