Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 329 / 1 / c / Solution

Apply the Lorentz reciprocal theorem for Stokes flow to ${\mathbf{u}}_1$ and the test flow around a sphere rotating with arbitrary $\hat{\mathbf{\Omega }}$. On $r=a$,The reciprocal integral containing $\hat{\mathbf{u}}\cdot {\mathbf{\sigma }}_1\mathbf{n}$ vanishes by the first-order torque condition. Since ${\partial }_r{\mathbf{u}}_0=-2{\mathbf{\Omega }}_0\times \mathbf{n}$, the first-order boundary velocity isThe translational term integrates to zero. Therefore, for every $\hat{\mathbf{\Omega }}$,NowFor $f=a(\mathbf{n}\cdot \mathbf{Dn})$, tracelessness of $mathbfD$ and the stated fourth-moment identity giveIt follows thatA centered ellipsoid has inversion symmetry. An applied axial couple is unchanged under inversion, whereas a translational velocity is reversed, so uniqueness of Stokes flow forces