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

174 M. B. Mackaplow and E. S. G. ShaqfehRadial distancelxSi,0 0.5 1 .O 1.5 2I I I2. -. 0-.. =-..00 ,.0....*..*......- *...Single stokeslet....... 0 Simulation data- aligned suspension... ........ Simulation data- isotropic suspension....... Screened stokeslet, x ....... ..................... Screened stokeslet, x............ Screened stokeslet, izI I I I II 0.1 0.2 0.3 0.4 0.5 0.6Radial distance, 1FIGURE 7. Decay of the velocity field multiplied by radial distance, r x vz, created by a point forcein a unit cell as a function of radial distance, r, in the plane perpendicular to the point force. Thesimulation results for unit cells containing both aligned and isotropic fibre suspensions are shown.The unit cell contains 150 spheroids having an aspect ratio of 663, for an effective concentration ofn13 = 7. Also shown are the decay of an isolated stokeslet, and stokeslets screened on length scalesof Xsim and XDA.5.2. Comparison of simulation results to experimental dataWe will now compare our simulations to published experimental data. However, allthese studies involved suspensions of cylindrical fibres. We could not directly simulatesuspensions of cylinders. Instead, we simulated suspensions of spheroids havingthe same aspect ratios and n13 value as the suspensions used in the experiments.These simulation results were then scaled using the ratio of the dilute theoreticalprediction for cylinders (equation (16)), to that for spheroids (equation (15)). Thisphysically corresponds to an isolated cylinder having the same particle stresslet as aspheroid with a larger aspect ratio, A,ff, where Aeff > A. However, the percentageenhancement of the stresslet due to interparticle interactions will still depend onlyon n13. We have based this conversion method on the findings of our analogousstudy of heat and mass transport through fibre suspensions (Mackaplow et al. 1994),where converged numerical results were obtained for both spheroidal and cylindricalfibres.We have not attempted comparisons to falling-ball rheometry studies, such as thoseby Milliken et al. (1989). Such investigations determine the ‘effective viscosity’ offibre suspensions by measuring the fall speed of a sphere sedimenting through thesuspension. The flow field created by a sedimenting sphere will induce a local fibreordering that decays away from the sphere. Since it is the fibres closest to the spherethat will have the greatest effect on its fall velocity, comparisons of these results to oursimulations of isotropic suspensions are not meaningful. Additionally, the numericalsimulations of falling-ball rheometry by Harlen, Sundarajakumar & Koch (1995)show that even in the dilute regime, fibre contacts, which we neglect, substantiallyincreases the ‘effective viscosity’. We note this was not found to be the case in shearedsuspensions.