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

182 M. B. Mackaplow and E. S. G. ShaqfehComparison of our numerical simulations to experiments and theoretical predictionsfor steady state shear and extensional viscosities also support the conclusionthat the semi-dilute suspension screening length is independent of the fibre orientationdistribution in the suspension. This conclusion is powerful since it allows oneto convert knowledge of the transient fibre orientation distribution in a suspensionto the transient rheological properties of the suspension. This is particularly valuablein a rapidly changing flow field where steady-state conditions are not reached sincedirect measurement of transient rheological properties are difficult.Slender-body theory will faithfully capture the effect of fibre-fibre interactionson the hydrodynamic stress in an isotropic suspension for concentrations up tox A/5, beyond which it underestimates their effect. Interestingly, although theaverage steady-state closest approach distance between fibres in shear flow is muchlarger than for those in an isotropic suspension, the few fibres that make the dominantcontribution to the suspension stress experience closest approach distances that areapproximately the same as those in an isotropic suspension. Consequently, in suchflows the same upper concentration limit for the ability of slender-body theory tocapture the full effect of fibre-fibre interactions is observed. This limit is presumablymuch higher in aligned suspensions, since for a given suspension concentration theaverage closest approach distance is much larger.Our algorithm can easily be modified to determine the sedimentation characteristicsof fibre suspensions. This is done by changing the right-hand side of (8) to specifythat the net force exerted by a fibre on the fluid is equal to the gravitational bodyforce exerted on the fibre. Monte Carlo and dynamic studies of fibre sedimentationare in progress.The authors would like to acknowledge support for this work from both a PresidentialYoung Investigator Award, Grant No. CTS-90557284, to ESGS, as well asa Merck Fellowship to MBM. Computer resources (CRAY C90) were supplied by agrant from the San Diego Supercomputer Center. This material is based on worksupported by Cornell University through the National Science Foundation Grant No.DDM-9212582.Appendix. Calculations of the correction to the effective viscosity of a fibresuspension for two-body interactionsA.l. General developmentA method for calculating expressions for the effective viscosity in a random dispersionof rigid fibres in a Newtonian fluid which accounts for multi-body interactions hasbeen carefully presented by Shaqfeh & Fredrickson (1990). We shall refer to thispublicaton for the details and only outline the mathematical steps necessary to derivethe results quoted in 54 - namely the first correction to the predicted effective viscosityof a fibre solution in the dilute limit which accounts for two-body interactions.The general theory developed by Shaqfeh & Fredrickson (1990) for the wavenumberdependentpropagator, Gjk(q) in suspensions of randomly positioned fibres can besummarized in the following set of integral equations :