Past exam of the advanced fluid dynamics course of the University of Oxford / 2019 / 2 / b / Solution

Neglect the inertia of the sphere and balance the applied force against Stokes drag. The sphere translates with $\mathbf{U}={\mathbf{F}}_1/\zeta$, where $\zeta =6\pi \mu a$. The applied force is also the net force transmitted to the fluid; the fluid's drag on the sphere has the opposite sign.

Translate the origin to ${\mathbf{r}}_1$. In the solution of part a, retain the terms of order $a/r$ and discard those of order $a^3/r^3$. With $\mathbf{n}=(\mathbf{x}-{\mathbf{r}}_1)/r$,This is the Stokeslet, or equivalently the Oseen tensor $\mathsf{G}(\mathbf{r})=(I+\mathbf{n}{\mathbf{n}}^T)/(8\pi \mu r)$ acting on ${\mathbf{F}}_1$. The omitted finite-size velocity is of order $a^2|{\mathbf{F}}_1|/(\mu r^3)$, so the displayed term is the leading far field for $r\gg a$. The singular Stokeslet must not be evaluated at its own centre to obtain a particle's self mobility.