Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 77 / 3 / a

A Stokeslet is the flow produced by a point force. With $C=1/(8\pi \mu )$, its tensor is $G_{ij}=C({\delta }_{ij}/r+r_i{r}_j/r^3)$. A separated pair of equal and opposite forces has no force monopole. Expanding their difference in the separation gives a directional derivative of this tensor. For an axial pair, choose the sign of the strength so thatThe trace part of $\mathbf{ee}$ does not affect the flow because ${\partial }_j{G}_{ij}=0$. The remaining symmetric force-dipole contribution is the stresslet, decaying as $r^{-2}$ instead of the Stokeslet's $r^{-1}$. A force-free swimming cell has thrust balanced by drag, so this is typically its leading far field. With $S>0$ it is a pusher microswimmer: axial outflow and transverse inflow. It is distinct from the antisymmetric torque singularity, the rotlet.