A dissipative anisotropic fluid model for non-colloidal particle dispersions

Abstract

This article examines a reduced form of the ‘purely dissipative’ model proposed several years ago as a general continuum model for the rheology of non-colloidal particle dispersions, ranging from Stokesian suspensions to non-cohesive granular media. Essential to the model is a positive-definite viscosity tensor $\boldsymbol{\eta}$, depending on the history of deformation and providing a crucial restriction on related models for anisotropic fluids and suspensions. In the present treatment, $\boldsymbol{\eta}$ is assumed to be as an isotropic function of a history-dependent second-rank ‘texture’ or ‘fabric’ tensor ${\textsfbi A}$. A formula for $\boldsymbol{\eta}({\textsfbi A})$ borrowed from the analogous theory of linear elasticity, and its subsequent expansion for weak anisotropy provides an explicit expression for the stress tensor in terms of fabric, strain-rate and eight material constants.

Detailed consideration is given to the special case of Stokesian suspensions, which represent an intriguing subset of memory materials without characteristic time. For this idealized fluid one finds linear dependence of all stresses, including viscometric normal stress, on present deformation rate, with the provision for an arbitrary fabric evolution (‘thixotropy’) in unsteady deformations. As a concrete example, a co-rotational memory integral is adopted for ${\textsfbi A}$ in terms of strain-rate history, and a memory kernel with two-mode exponential relaxation gives close agreement with the rather sparse experimental data on transient shear experiments. In the proposed model, an extremely rapid mode of relaxation is required to mimic the incomplete reversal of stress observed in experiments involving abrupt reversal of steady shearing, supporting the conclusion of others that non-hydrodynamic effects, with breaking of Stokesian symmetry, may be implicated in such experiments.

Qualitative comparisons are made to a closely related model, derived from a micro-mechanical analysis of Stokesian suspensions, but also involving non-Stokesian effects.

The present analysis may point the way to improved micro-mechanical analysis and to further experiments. Possible extensions of the model to the viscoplasticity of dry and liquid-saturated granular media also are discussed briefly.