A constitutive model for a semi-concentrated suspension of rigid fibers in a non-Newtonian fluid is derived in the present study. This work is extended from a previous work by Dinh and Armstrong which counted rigid fibers suspended in a Newtonian fluid. To investigate the effect of the sheardependent suspending fluid on the shear viscosity of suspension, the Ellis fluid is assumed to model such a non-linear matrix. It is shown that the present derivation, via a cell model, gives an analytic form to calculate the drag coefficient of fibers and to illustrate the influence of material parameters of matrix fluid. The resulting equation gives the bulk stress in terms of an integral over a fourth-order orientation vector, e, similar to the expression of Doi and Edwards for dense macromolecular fluid. For the purpose of evaluation, the present model attempts to predict the rheological behavior in the inception of flow, steady simple shear flow and stress-growth experiments.

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A quasi-static asymptotic analysis is employed to investigate the elastic effects of fluids on the shear viscosity of highly concentrated suspensions at low and high shear rates. First a brief discussion is presented on the difference between a quasi-static analysis and the periodic-dynamic approach. The critical point is based on the different order-of-contact time between particles. By considering the motions between a particle with N near contact point particles in a two-dimensional "cell" structure and incorporating the concept of shear-dependent maximum packing fraction reveals the structural evolution of the suspension under shear and a newly asymptotic framework is devised. In order to separate the influence of different elastic mechanisms, the second-order Rivlin-Ericksen fluid assumption for describing normal-stress coefficients at low shear rates and Harnoy's constitutive equation for accounting for the stress relaxation mechanism at high shear rates are employed. The derived formulation shows that the relative shear viscosity is characterized by a recoverable shear strain, SR, at low shear rates if the second normal-stress difference can be neglected, and Deborah number, De, at high shear rates. The predicted values of the viscosities increase with SR, but decrease with De. The role of S R in the matrix is more pronounced than that of De. These tendencies are significant when the maximum packing fraction is considered to be shear-dependent. The results are consistent with that of Frankel and Acrivos in the case of a Newtonian suspension, except for when the different divergent threshhold is given as [ 1 -(~/qSm)l/2 ] --1.

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