= Mutual drag reduction of two distant translating spheres
{title2=$D_\perp/D_0=1-3a/(4b),\quad D_\parallel/D_0=1-3a/(2b)$}
Two identical radius-$a$ spheres moving together through a viscous fluid at separation $b\gg a$ entrain each other. The leading far velocity of one sphere at the other is $3aU/(4b)$ for separation perpendicular to the velocity, and $3aU/(2b)$ for parallel separation. Replacing $U$ by its velocity relative to this induced flow in <Stokes law> gives the displayed drag reductions relative to $D_0=6\pi\mu aU$, with errors of order $(a/b)^2$. Reflection symmetry and linearity imply equal drag on the two spheres in either geometry.
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