An anisotropic-fluid model for inhomogeneous Stokesian suspensions

A constitutive model is proposed for a suspension of rigid spheres with spatially non-uniform strain rate \textbf{E} and particle concentration $\phi $. As in [1], the model involves a 4$^{th}$ rank viscosity tensor depending on $\phi $ and a 2$^{nd}$ rank structure tensor \textbf{A} determined by a kinematic evolution equations. The particle flux \textbf{j }is a linear function of the spatial gradients in $\phi $, \textbf{E}, {\&} \textbf{A }. In contrast to existing models [2,3], the constitutive equations exhibits Stokesian linearity in \textbf{E}, and all nonlinear suspension-dynamics effects are represented by \textbf{A} and its evolution. An expansion up to third order in \textbf{A} is given, and illustrative calculations are made for oscillatory simple shear based on parameters determined as in [1]. Desirably, the model offers a frame-indifferent description of the effects of streamline curvature on particle flux; and it admits transiently negative particle diffusivities following shear reversal, indicating dominance of Stokesian reversibility over shear-induced memory loss. The main drawback, is the plethora of scalar parameters, and possible simplifications inspired by previous models are discussed briefly. \newline [1] J.~D. Goddard, \textit{J. Fluid. Mech.}, 568:1--17, 2006. \newline [2] G.~P. Krishnan, et al., \textit{J. Fluid Mech.}, 321:371--93, 1996. \newline [3] R.~J. Phillips, et al., \textit{Phys. Fluids A}, 4(1): 30--40, 1992.