Stress in a dilute suspension of spheres in a dilute polymer solution subject to simple shear flow at finite Deborah numbers

Abstract

The influence of particle-polymer interactions on the ensemble average stress is derived as a function of the Deborah number for a dilute suspension of spheres in an Oldroyd-B fluid in the limit of small polymer concentrations. The slow rate of decay of the particle-induced polymer stress with separation from a particle presents a challenge to the derivation of the average stress, which can be overcome by removing the linearized polymer stress disturbance before computing the bulk average stress from the particle-induced disturbance. The linearized stress can be shown to have zero ensemble average. The polymer influence on the particle's stresslet is computed with the aid of a generalized reciprocal theorem based on a regular perturbation from Newtonian flow for small polymer concentration. The analysis shows that the particle-polymer contributions to the shear stress and first normal stress difference shear thicken as has been observed in the experiments of Scirocco et al.[Shear thickening in filled Boger fluids, J. Rheol. 49, 551 (2005)]. The particle-polymer contribution to the second normal stress difference is positive at small Deborah numbers but changes sign at a Deborah number of about 2.3.

Einstein viscosity with fluid elasticity

“…in a homogenous suspension [21,22] (see also Appendix B.) This implies that also the stress field settles down to its mean value as |r| ∼ φ −1/3 .…”

“…If one chooses to approximate the microscopic problem with an unbounded flow without regularization, it appears that the volume averaged stress is not equal to the ensemble averaged stress, even in a statistically homogenous suspension [11,21,22]. This apparent ergodicity breaking has led to confusion regarding the correct averaging procedure, where some terms required ensemble averaging, whereas others could be volume averaged [11,17,21,22]. We remove this ambiguity by correctly imposing the mean field conditions.…”

“…Yang et al [5] and Koch et al [21] independently studied suspension stress in shear flow by numerical simulation and a semi-analytical theory, respectively. They both found that as Wi is increased from vanishingly small values, the stresslet contribution to the dilute suspension shear viscosity decreases, but the particle induced fluid stress increases.…”

Force moments of an active particle in a complex fluid

“…The zeroth moment and the antisymmetric first moment lead to a force and torque absorbed into the first two terms of (13). Introducing a more compact notation, where…”

“…This approach was extended to find the motion of active particles in otherwise quiescent fluids [6,7] and background flows [8] and may also be used to describe the motion of multiple particles in complex fluids [9] (and Newtonian fluids [10]). Recently, the reciprocal theorem has also been used to find the first force moment on a passive sphere in a viscoelastic linear flow [11][12][13][14].…”