 Solution

For constant viscosity, differentiating $\sigma =-pI+\mu (\nabla u+\nabla u^T)$ gives $\nabla \cdot \sigma =-\nabla p+\mu {\nabla }^{2}u$, since $\nabla \cdot u=0$. This is zero by the body-force-free Stokes equations.

Apply the divergence theorem to the annular fluid. Its outer normal is $+n_R$, but its inner normal is minus the sphere's outward normal. Therefore $0={\int }_{S_R}\sigma n_{R}dS-{\int }_{S_a}\sigma n_{a}dS$. The second integral is the hydrodynamic force on the inner sphere, $-F$, because that sphere exerts $F$ on the fluid. Thus again $G=-F$, without differentiating the far field. Local momentum conservation carries the same total force through every enclosing surface.