Past exam of the advanced fluid dynamics course of the University of Oxford / 2023 / 2 / b / Solution

Use the force flux at infinity in Stokes flow. With an outward normal from a large sphere, the force on the body is $\mathbf{F}={\int }_{S_R}\sigma \mathbf{n}dS$. Let $C=3a/4$. The leading far-field velocity and pressure areTheir rate-of-strain tensor isIn the Newtonian fluid stress tensor ${\sigma }_{ij}=-p{\delta }_{ij}+2\mu e_{ij}$, the isotropic pieces cancel. Consequently,The second spherical moment integral $\int n_i{n}_{j}d\Omega =(4\pi /3){\delta }_{ij}$ follows from rotational symmetry and ${\sum }_i{n}_i^2=1$. Since $dS=R^{2}d\Omega$, taking $R\to \infty$ gives the Stokes drag law:The omitted dipole contributes a surface integral of order $R^{-2}$ and vanishes in this limit. The sign refers to the fluid's drag on the moving sphere.