A numerical study of the rheological properties of suspensions of ...

168 M. B. Mackaplow and E. S. G. ShaqfehVolume fraction of fibres, q0 0.005 0.010 0.015 0.020 0.025Q0 2 4 6 8 10 12 14n13FIGURE 2. Normalized extra particle stress, Q, of a suspension of aligned, spheroidal fibres withaspect ratio, A = 50, as a function of the suspension volume fraction. Shown are the predictionsof numerical simulations, the dilute theory (with and without two-body corrections), and thesemi-dilute theory of Shaqfeh & Fredrickson (1990) (both the original and modified versions).ln2(2A) = 30. The ability of dilute theories that incorporate the effects of two-bodyinteractions to predict suspension behaviour into the semi-dilute concentration regimewas also observed in our previous study of heat transfer through suspensions of highlyconducting fibres (Mackaplow et al. 1994).Figure 2 shows a plot of Q us. n13 for suspensions of fibres with an aspect ratioof 50. All of the fibres are aligned with the principal axis of extension of the flowfield. As in figure l(b), we see that by using the adjusted coefficients in the semi-dilutetheory of Shaqfeh & Fredrickson (1990), an excellent agreement between simulationsand theory can be achieved for n13 >> 1.Figure 3 shows a plot of Q us. n13 for suspensions with a fixed inclusion volumefraction, 4, of 6.67 x . Again, there is a positive deviation from dilute theoryfor n13 2 O( 1). Note that Q approaches a constant in the semi-dilute regime. This isconsistent with the prediction of the semi-dilute theory that the non-dimensionalizedscreening length in a semi-dilute suspension, X/b, is only a function of the volumefraction of the suspension. Whereas the published semi-dilute theory of Shaqfeh &Fredrickson (1990) quantitatively underestimates this value by approximately 20%,by utilizing the adjusted coefficients, excellent quantitative agreement is achieved. Asfor the previous sets of simulations, the dilute theory with two-body corrections does agood job of predicting the suspension rheology over the entire range of concentrationsstudied.In our previous study of heat and mass transfer through fibre suspensions, similarscalings of the suspension screening length in the semi-dilute regime were observed, aspredicted by the theory of Fredrickson & Shaqfeh (1989). However, in this previousstudy we found that using the theoretically predicted value of C” and neglecting the