Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 38C / d / Solution

The Lorentz reciprocal theorem for Stokes flow states that two Stokes solutions $({\mathbf{u}}^{(1)},{\mathbf{\sigma }}^{(1)})$ and $({\mathbf{u}}^{(2)},{\mathbf{\sigma }}^{(2)})$ in the same fluid domain satisfyApply it to the same rod translating with arbitrary velocities ${\mathbf{U}}_1$ and ${\mathbf{U}}_2$. The integral at infinity vanishes, and the rod-surface integrals giveSince ${\mathbf{F}}_i=-\mathbf{R}{\mathbf{U}}_i$,for every pair of vectors. Therefore