Mutual drag reduction of two distant translating spheres ($D_{\perp }/D_0=1-3a/(4b),D_{\parallel }/D_0=1-3a/(2b)$)

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.