Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 75 / 1 / a

For two incompressible Stokes flows $(\mathbf{u},\mathbf{\sigma })$ and $(\hat{\mathbf{u}},\hat{\mathbf{\sigma }})$ with the same viscosity and no body forces, the Lorentz reciprocal theorem for Stokes flow statesIndeed the divergence of their difference is zero: stress divergences vanish, pressure contracts with zero velocity divergence, and the two symmetric strain-rate contractions cancel.

Apply it to the decaying disturbance ${\mathbf{u}}^{\prime }=\mathbf{u}-{\mathbf{u}}_{\infty }$ around the sphere and an auxiliary translating sphere of velocity $\mathbf{V}$. On the body surface, ${\mathbf{u}}^{\prime }=\mathbf{U}+{\mathbf{\Omega }}_s\times \mathbf{x}-{\mathbf{u}}_{\infty }$. Using normals from the body into the fluid, the given auxiliary traction is $-3\mu \mathbf{V}/(2a)$; reversing both normal conventions does not change the reciprocal equality. The background traction integrates to zero because it is regular inside the sphere. The disturbance's hydrodynamic force is consequently ${\mathbf{F}}_H=-\mathbf{F}$, where $\mathbf{F}$ is the externally applied force. The rotational term integrates to zero. Thus for arbitrary $\mathbf{V}$,Therefore $\mathbf{U}=\mathbf{F}/(6\pi \mu a)+⟨{\mathbf{u}}_{\infty }{⟩}_{S_a}$.

The background Stokes equations imply ${\nabla }^2{p}_{\infty }=0$ and ${\nabla }^4{\mathbf{u}}_{\infty }=0$. The spherical mean of a biharmonic function gives exactly $⟨{\mathbf{u}}_{\infty }⟩={\mathbf{u}}_{\infty }+(a^2/6){\nabla }^2{\mathbf{u}}_{\infty }$, evaluated at the sphere center. One can obtain it from Taylor expansion and $⟨x_i{x}_j⟩=a^2{\delta }_{ij}/3$; every subsequent spherical average contains ${\nabla }^4$ and vanishes. This requires the background to be regular throughout the enclosed ball. Hence the Faxén relation isThe printed denominator $6\pi \mu$ omits $a$. The given traction integrates to $-6\pi \mu a\mathbf{V}$, fixing that factor; the printed force term also fails dimensional consistency unless its force were redefined, which the question does not do.