(373d) Single Particle Motion in a Sheared Colloidal Dispersion

In conventional (macro-)rheometry, rheological properties are measured by subjecting a material to a bulk shearing motion, actuated by oppositely moving plates, for example. In contrast, (active) micro-rheological techniques utilize a colloidal "probe" driven through an otherwise quiescent fluid. Here, we illustrate effects arising from combination of micro- and macro- forcings: specifically, we consider a colloidal probe driven at fixed force through a dilute suspension of hard-sphere "bath" particles undergoing simple shear flow.  The distortion to the equilibrium suspension microstructure caused by the probe is characterized by a (micro) Peclet number Pe_f (a dimensionless pulling force),  and the distortion due to the ambient shear is represented by a (macro) Peclet number Pe_s (a dimensionless shear rate).  Using asymptotic expansions at small Peclet numbers (more precisely, Pe_s^3/2« Pe_f « Pe_s^1/2« 1), we demonstrate that a probe forced along the velocity axis of the shear experiences a cross-streamline drift of O(Pe_sΦU_S), resulting from a nonlinear combination of micro- and macro- microstructural deformations (Φ is the bath particle volume fraction and U_S is Stokes' settling velocity).  The magnitude of the drift velocity is sensitive to the degree of hydrodynamic interactions between the probe and bath particles. Next, we consider a probe forced orthogonal to the imposed shear. Here, the probe experiences a shear-driven enhancement in rectilinear velocity of O(Pe_s^3/2ΦU_S): this non-analytic contribution originates from the microstructural deformation in the shear dominated (outer) region far from the probe. The connection of this result to recent work on particle sedimentation in orthogonal linear flows of viscoelastic fluids is discussed.