 Leading interaction of two sedimenting spheres

For two well-separated spheres with isolated downward speeds $V_i$ and radii $a_i$, let $\mathbf{r}={\mathbf{x}}_1-{\mathbf{x}}_2$, $r=|\mathbf{r}|$, $\mathbf{n}=\mathbf{r}/r$, and let $\hat{\mathbf{z}}$ point upwards. Their leading Stokeslet interactions giveEach sphere's Stokes drag law supplies force $-6\pi \mu a_i{V}_i\hat{\mathbf{z}}$ on the fluid. Evaluate the other's Stokeslet at its centre and subtract the two velocities to obtain the equation. Corrections from finite size and potential dipoles are higher order in radius divided by separation. The approximation must remain well separated throughout the segment of trajectory to which it is applied.