Past exam of the advanced fluid dynamics course of the University of Oxford / 2018 / 2 / c

Let $S_a$ be the sphere surface, ${\mathbf{n}}_s$ point from the sphere into the fluid, and writeBecause the prescribed exterior body forces are unchanged, they cancel on subtraction: the disturbance is a homogeneous Stokes flow in the exterior domain. On the sphere the no-slip boundary condition isallowing an arbitrary sphere rotation ${\mathbf{\omega }}_s$.

Use as auxiliary flow the translating-sphere solution of part b, with arbitrary test velocity $\mathbf{U}$. Its surface velocity is $\mathbf{U}$ and its surface traction is $-3\mu \mathbf{U}/(2a)$. Applying the Lorentz reciprocal theorem on the exterior domain gives, after cancelling the common inner-normal minus sign,The outer surface terms vanish for decaying disturbances: velocities are at most $O(r^{-1})$, stresses $O(r^{-2})$, and the integrated cross-work is $O(r^{-1})$.

There are no body forces inside the sphere in the original ambient problem. Thus the smooth ambient stress tensor has zero divergence there, andAlso ${\int }_{S_a}\mathbf{x}dS=0$, so the rotation term drops out of the left side. Consequently the reciprocal identity reduces toSince $\mathbf{U}$ is arbitrary, the result before imposing mechanical balance isFor a freely suspended sphere with negligible inertia and no applied external force, ${\mathbf{F}}_H=0$. Its sphere velocity as a surface average of ambient Stokes flow is thereforeThe implicit force-free condition is essential. More generally, an external force ${\mathbf{F}}_{\mathrm{ext}}$ gives $\mathbf{V}=⟨{\mathbf{u}}_{\infty }{⟩}_{S_a}+{\mathbf{F}}_{\mathrm{ext}}/(6\pi \mu a)$. For example, an externally pulled sphere in the zero ambient flow has nonzero velocity, contrary to an unrestricted surface-average-only claim. No torque-free assumption is needed for this translational identity, because the auxiliary surface traction is constant.