A weakly nonlocal anisotropic fluid model for inhomogeneous Stokesian suspensions

Abstract

A continuum model is proposed for a weakly inhomogeneous Stokesian suspensions, as an extension with minor amendments of a previous work on homogeneous suspensions [J. D. Goddard, J. Fluid Mech. 568, 1 (2006)]. In the present model, stress and particle flux are given as invariant tensor functions of particle volume fraction ϕ, deformation rate E, and second-rank anisotropy tensor A, in a form that is also linear in E and the gradients of ϕ, E, and A. In contrast to models without history dependence, all nonlinear dependence of particle flux on E arises from the evolution of A. Detailed attention is paid to unsteady viscometric flow, where a contribution of streamline curvature to particle migration emerges as a natural consequence of tensorial gradients. The model predicts equal curvature-induced fluxes in gradient and vorticity directions but there is an unexplained disagreement with recent experiments on Couette and torsional flows. A previously proposed corotational evolution equation for A, with a two-mode exponential relaxation, is employed to investigate the transient response following the reversal of shearing in sinusoidal and in steady shear. The model predicts roughly equal response for the two flows if sinusoidal strains are of order unity, which is consistent with some but not all experiments. The model for particle flux admits an asymmetric diffusion tensor which, owing to Stokesian reversibility, can become nonpositive upon abrupt reversal of shearing. This effect is diminished by non-Stokesian response on short strain scales, which, although poorly understood, appears essential to elementary models without dependence on shear history. A synthesis is given of multipolar Stokesian resistance and the associated Stokesian dynamics, showing how these follow from a single grand resistance kernel. In addition to unifying and extending large literature on Stokesian resistance formulae, this provides some justification for the proposed continuum model and possible multipolar extensions.