Hydrodynamic diffusion in the absence of Brownian motion is studied via active microrheology in the ‘pure-hydrodynamic’ limit, with a view towards elucidating the transition from colloidal microrheology to the non-colloidal limit, falling-ball rheometry. The phenomenon of non-Brownian force-induced diffusion in falling-ball rheometry is strictly hydrodynamic in nature; in contrast, analogous force-induced diffusion in colloids is deeply connected to the presence of a diffusive boundary layer even when Brownian motion is very weak compared with the external force driving the ‘probe’ particle. To connect these two limits, we derive an expression for the force-induced diffusion in active microrheology of hydrodynamically interacting particles via the Smoluchowski equation, where thermal fluctuations play no role. While it is well known that the microstructure is spherically symmetric about the probe in this limit, fluctuations in the microstructure need not be – and indeed lead to a diffusive spread of the probe trajectory. The force-induced diffusion is anisotropic, with components along and transverse to the line of external force. The latter is identically zero owing to the fore–aft symmetry of pair trajectories in Stokes flow. In a naïve first approach, the vanishing relative hydrodynamic mobility at contact between the probe and an interacting bath particle was assumed to eliminate all physical contribution from interparticle forces, whereby advection alone drove structural evolution in pair density and microstructural fluctuations. With such an approach, longitudinal force-induced diffusion vanishes in the absence of Brownian motion, a result that contradicts well-known experimental measurements of such diffusion in falling-ball rheometry. To resolve this contradiction, the probe–bath-particle interaction at contact was carefully modelled via an excluded annulus. We find that interparticle forces play a crucial role in encounters between particles in the hydrodynamic limit – as they must, to balance the advective flux. Accounting for this force results in a longitudinal force-induced diffusion $D_{\Vert }=1.26aU_{S}{\it\phi}$, where $a$ is the probe size, $U_{S}$ is the Stokes velocity and ${\it\phi}$ is the volume fraction of bath particles, in excellent qualitative and quantitative agreement with experimental measurements in, and theoretical predictions for, macroscopic falling-ball rheometry. This new model thus provides a continuous connection between micro- and macroscale rheology, as well as providing important insight into the role of interparticle forces for diffusion and rheology even in the limit of pure hydrodynamics: interparticle forces give rise to non-Newtonian rheology in strongly forced suspensions. A connection is made between the flow-induced diffusivity and the intrinsic hydrodynamic microviscosity which recovers a precise balance between fluctuation and dissipation in far from equilibrium suspensions; that is, diffusion and drag arise from a common microstructural origin even far from equilibrium.
Brownian motion in fluctuating periodic potentials
