= Leading interaction of two sedimenting spheres
{title2=$\dot{\mathbf r}=-\delta\widehat{\mathbf z}-3D(I+\mathbf n\mathbf n)\widehat{\mathbf z}/(4r)$}
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 $\widehat{\mathbf z}$ point upwards. Their leading <Stokeslet> interactions give
$$
\dot{\mathbf r}=-\delta\widehat{\mathbf z}-\frac{3D}{4r}(I+\mathbf n\mathbf n)\widehat{\mathbf z},\qquad\delta=V_1-V_2,\quad D=a_2V_2-a_1V_1.
$$
Each <sphere>'s <Stokes drag law> supplies <force> $-6\pi\mu a_iV_i\widehat{\mathbf z}$ 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.
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