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We report an analysis, using the tools of nonlinear dynamics and chaos theory, of the fluctuations in the stress determined from simulations of shear flow of Stokesian suspensions. The simulations are for shear between plane parallel walls of a suspension of rigid identical spheres in a Newtonian fluid, over a range of particle concentration. By analyzing the time series of the stress, we find that the dynamics underlying these fluctuations is deterministic, low-dimensional, and chaotic. We use the dynamic and metric invariants of the underlying dynamics as a means of characterizing suspension behavior. The dimension of the chaotic attractor increases with particle concentration, indicating the increasing influence of multiple-body interactions on the rheology of the suspension with rise in particle concentration. We use our analysis to make accurate predictions of the short-term evolution of a stress component from its preceding time series, and predict the evolution of one component of the stress using the time series of another. We comment on the physical origin of the chaotic stress fluctuations, and on the implications of our results on the relation between the microstructure and the stress.

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A Langevin approach to computing the orientation moments of a dilute suspension of spheroids in a simple shear flow at arbitrary Péclet number is presented. In this method we obtain the equations governing the time evolution of the orientation averages using a generalized Langevin equation approach and develop a computational technique for computing the evolution of the moments from these equations. These results are compared with those available in the literature obtained from other methods and show good agreement. The approach presented here can be easily generalized to a number of similar systems such as forced suspensions of dipolar particles in shear flows and can be applied to other flow problems governed by appropriate Fokker-Planck equations.

We demonstrate for the first time that the rheological parameters like the apparent viscosities and the first and second normal stress differences of suspensions of orientable particles can show chaotic behavior when the orientation vector evolves chaotically. We also demonstrate that the range of the values of the rheological parameters is about 10 000 times greater when the parameters evolve chaotically. This suggests that a wide range of properties may be obtained by small variations in controllable parameters. When coupled with suitable control of chaos algorithms, a wide range of suspension behavior is thus possible since a chaotic solution can be considered as an unlimited reservoir of periodic solutions of arbitrary period.

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