Past exam of the soft condensed matter course of the University of Cambridge / 2012 / Paper 1 / 2 / c

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:Its Oseen tensor isThis 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 givesThe 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, $⟨\Delta r_{i\alpha }\Delta r_{j\beta }⟩\simeq 2k_{B}T{\mu }_{i\alpha ,j\beta }\Delta t$ 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.