Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 329 / 1 / c / i

The Linearity of Stokes flow makes every velocity linear in ${\mathbf{G}}_1$. The only available isotropic polar vector built from the axial vector ${\mathbf{G}}_1$ and separation vector $\mathbf{R}$ is ${\mathbf{G}}_1\times \mathbf{R}$. Dimensional analysis then givesAn angular velocity is axial, so the two independent isotropic possibilities are ${\mathbf{G}}_1$ and $({\mathbf{G}}_1\cdot \mathbf{R})\mathbf{R}$. Hencefor dimensionless scalar functions ${\beta }_i,{\gamma }_i$.