= Solution
A moving or externally forced colloid disturbs the surrounding fluid, changing the <velocity> of other colloids even without direct contact. These <hydrodynamic interactions> are long-ranged at small <Reynolds number>. In an unbounded incompressible Newtonian liquid, the <Stokes flow> response to a point <force> is the <Stokeslet>:
$$
-\nabla p+\eta\nabla^2\mathbf u+\mathbf F\,\delta(\mathbf r)=0,\qquad\nabla\cdot\mathbf u=0,
\qquad u_i(\mathbf r)=\mathsf G_{ij}(\mathbf r)F_j.
$$
Its <Oseen tensor> is
$$
\boxed{\mathsf G_{ij}(\mathbf r)=\frac1{8\pi\eta r}
\left(\delta_{ij}+\frac{r_ir_j}{r^2}\right).}
$$
This is a mobility tensor, with units <velocity> per <force>, rather than a <stress> tensor. At separation $r$ the response parallel to the line of centres is $1/(4\pi\eta r)$; the perpendicular response is $1/(8\pi\eta r)$. Flow is not isotropic even though the solvent is. The disturbance decays as $1/r$, so treating all drag <forces> as independent can miss correlated motion.
For well-separated equal spheres of <radius> $a$, the leading <hydrodynamic mobility matrix> gives
$$
\mathbf V_i=\frac{\mathbf F_i}{6\pi\eta a}
+\sum_{j\ne i}\mathsf G(\mathbf r_i-\mathbf r_j)\mathbf F_j.
$$
The first term is the isolated-sphere self mobility; the remaining terms couple distinct particles. At thermal equilibrium the <fluctuation-dissipation theorem> also predicts correlated short-time increments, $\langle\Delta r_{i\alpha}\Delta r_{j\beta}\rangle\simeq2k_BT\mu_{i\alpha,j\beta}\Delta t$ with the configuration locally fixed. Thus correlated Brownian <displacements> can reveal hydrodynamic coupling without an applied <force>.
The point-force approximation requires $r\gg a$ and negligible inertia. It diverges at the origin and cannot replace the finite self mobility. Near-contact lubrication, walls, many-body corrections, and a viscoelastic solvent require more complete mobility laws. A force-free particle can instead have a faster-decaying leading disturbance, so the $1/r$ claim specifically concerns a net-force-driven object.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2012/paper-1-oseen-response.png]
{title=Oseen mobility directions}
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