Effects of Time-Periodic Fields on the Rheology of Suspensions of Brownian Dipolar Spheres

Abstract

The present contribution studies the rotary motion of a spherical dipolar particle suspended in homogeneous shear in the presence of a time-periodic external field, with the goal of describing the rheology (i.e. the macroscopic stress) of a dilute suspension of such particles in the limit of weak Brownian rotary diffusion. In this singular limit, the macroscopic behaviour of the suspension is largely dependent upon the deterministic rotary motion of the particles. This motion is governed by a nonlinear and non-autonomous system. The analysis reveals two modes of motion: convergence of all particles to a global time-periodic attractor (TPA), and quasi-periodic (QP) motion. The former mode, which is characterized by both frequency and phase locking is shown to result from an appropriate resonance interaction of the respective effects of the fluid shear and external field. The distinction between the two modes of motion is essential in the calculation of the particle contribution to the effective stress. Thus, when TPAs occur, diffusive effects are confined to a narrow domain about the attractor. If, on the other hand, the rotary motion is QP, the (weak) diffusion has a global effect throughout the entire orientation space. A sufficient condition for the occurrence of a global TPA is here established for the particular square-wave oscillation of the external field (and is elsewhere extended to cover more general modes of time variation). Explicit results for the bulk stress are presented for the case of a TPA rotary motion. These results indicate that the particle contribution to the bulk stress may in some cases be negative (i.e. reduce the suspension effective viscosity). These trends are rationalized in terms of the particle deterministic rotary motion.