Phase-dependent mean force of an externally driven sphere pair ($⟨F_1+F_2{⟩}_t\sim 9\pi \mu a_1{a}_2{\delta }^2\omega {\ell }_0^{-2}\sin \varphi$)

For equal oscillation amplitudes $\delta$, frequency $\omega$, mean separation ${\ell }_0$ and phase lag $\varphi$, the leading mean total force on the fluid is $9\pi \mu a_1{a}_2{\delta }^2\omega \sin \varphi /{\ell }_0^2$. Each sphere supplies half. The large-separation calculation expands the Stokeslet interaction; reciprocal in-phase and antiphase motions have zero mean force.