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.

In the free-draining approximation, each separated component of a body experiences its isolated viscous drag, and hydrodynamic interactions between components are neglected. For widely separated spheres of radii $a_i$ in an unbounded Newtonian fluid of dynamic viscosity $\mu$, Stokes law gives ${\mathbf{f}}_i=-{\zeta }_i{\mathbf{v}}_i$, with ${\zeta }_i=6\pi \mu a_i$. This is a leading approximation in sphere size divided by separation. Any connecting rod must also have negligible drag; a small diameter alone does not guarantee this for an arbitrarily long rod.

Consider two widely separated rigid spheres of equal radius $a$, with separation $R\mathbf{n}$, in Stokes flow. Apply a force $6\pi \mu a{\mathbf{U}}_0$ to the first sphere and let the second be force-free and torque-free. The second sphere samples the rate-of-strain tensor of the first sphere's Stokeslet, $\mathbf{E}=3a({\mathbf{U}}_0\cdot \mathbf{n})(\mathbf{I}-3\mathbf{nn})/(4R^2)$. Its leading disturbance is the stresslet of a sphere in a uniform straining Stokes flow. Evaluating that disturbance back at the first sphere givesThe hydrodynamic mobility matrix decreases at fixed force, consistently with the increase of resistance caused by an additional rigid inclusion.

If the otherwise force-free second sphere is held against rotation, its angular velocity relative to the incident fluid is $-3a({\mathbf{U}}_0\times \mathbf{n})/(4R^2)$. The rotlet reflected back to the first sphere addsThis follows by evaluating the rotating sphere in Stokes flow at displacement $-R\mathbf{n}$ and using the vector triple-product identity. The holding couple does no work because the held sphere's angular velocity is zero.

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.