On weak plane Couette and Poiseuille flows of rigid rod and platelet ensembles (with Z. Cui, M.G. Forest and Q. Wang), SIAM J. Appl. Math. , 66 (4), 2006 Abstract. Films and molds of nematic polymer materials are notorious for heterogeneity in the orientational distribution of the rigid rod or platelet macromolecules. Predictive tools for structure length scales generated by shear-dominated processing are vitally important: both during processing because of flow feedback phenomena such as shear thinning or thickening, and postprocessing since gradients in the rod or platelet ensemble translate to nonuniform composite properties and to residual stresses in the material. These issues motivate our analysis of two prototypes for planar shear processing: drag-driven Couette and pressure-driven Poiseuille flows. Hydrodynamic theories for high aspect ratio rod and platelet macromolecules in viscous solvents are well developed, which we apply in this paper to model the coupling between short-range excluded volume interactions, anisotropic distortional elasticity (unequal elasticity constants), wall anchoring conditions, and hydrodynamics. The goal of this paper is to generalize scaling properties of steady flow molecular structures in slow Couette flows with equal elasticity constants [M. G. Forest et al., J. Rheol., 48 (2004), pp. 175-192] in several ways: to contrast isotropic and anisotropic elasticity; to compare Couette versus Poiseuille flow; and to consider dynamics and stability of these steady states within the asymptotic model equations.

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Semi-dilute nano-rod dispersions interact nonlocally and nonlinearly through excluded-volume and distortional elasticity potentials. When driven by steady shear with confinement boundary conditions, remarkable behavior of the rod orientational distribution ensues: strong anisotropy; steady and unsteady responses; and gradient structure on (thus far) unpredictable lengthscales. Extreme variability and sensitivity of these features to experimental controls, coupled with nano-rod measurement limitations, continue to confound materials processing strategies. Thus, modeling and simulation play a critical role. In this paper, we present a hierarchy of 0-d, 1-d and 2-d physical space simulations of steady parallel-plate shear experiments, using a mesoscopic tensor model for the rod orientational distribution [45,8] and a spectral-Galerkin numerical algorithm [52]. We impose steady shear to focus on the orientational response of the nano-rod ensemble to two experimental controls: the Deborah number (De), or normalized imposed shear rate; and physical plate anchoring conditions on the rod ensemble. Our results yield dimensional robustness versus instability of sheared, semi-dilute, nano-rod dispersions: To begin, we present 0-d and 1-d phase diagrams that are consistent with results of the modeling community. Next, we present the first study of numerical stability (for all attractors in the phase diagrams) to 2-d perturbations in the flow-gradient and vorticity directions. The key findings are: time-periodic 1-d structure attractors at low-to-moderate De are robust to 2-d perturbations; period-doubling transitions at intermediate De to chaotic attractors in 0 and 1 space dimension are unstable to coherent 2-d morphology, but remain chaotic; as De increases, chaotic dynamics becomes regularized, first to periodic and then to steady structure attractors, along with a return to robust 1-d morphology; and finally, logrolling (vorticity-aligned) anchoring selects the most distinct attractors and De cascade with respect to other anchoring conditions.

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We present a kinetic model for flowing active suspensions and analyze the behavior of a suspension subjected to a weak steady shear. Asymptotic solutions are sought in Deborah number expansions. At the leading order, we explore the steady states and perform their stability analysis. We predict the rheology of active systems including an activity thickening or thinning behavior of the apparent viscosity and a negative apparent viscosity depending on the particle type, flow alignment, and the anchoring conditions, which can be tested on bacterial suspensions. We find remarkable dualities that show that flow-aligning rodlike contractile (extensile) particles are dynamically and rheologically equivalent to flow-aligning discoid extensile (contractile) particles for both tangential and homeotropic anchoring conditions. Another key prediction of this work is the role of the concentration of active suspensions in controlling the rheological behavior: the apparent viscosity may decrease with the increase of the concentration.