Hydrodynamic interaction

Bodies immersed in one fluid affect each other through the flow that each produces. At large separation, a force-driven sphere generates a Stokeslet that modifies the other sphere's velocity. These interactions produce off-diagonal entries in the hydrodynamic mobility matrix.

Two spheres whose prescribed translations have a phase lag can exert a nonzero mean force on the fluid while each returns to its starting position. Their changing separation correlates the interaction strength with the other sphere's velocity. This pump is externally driven, so it is distinct from force-free swimming governed by the scallop theorem.

The centre separation for equal-amplitude oscillations has minimum ${\ell }_0-2\delta |\sin (\varphi /2)|$. A bound ${\ell }_0\ge \delta$ alone cannot guarantee nonoverlap. An asymptotic mobility calculation needs this minimum much larger than the sphere radii.

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.

For well-separated spheres moving along their line of centres, the leading mobility matrix has diagonal entries $m_i=(6\pi \mu a_i{)}^{-1}$ and off-diagonal entries $h=(4\pi \mu \ell {)}^{-1}$. This is the axial Stokeslet interaction. Finite-size and repeated-reflection corrections enter at higher order.