Prediction of rheological properties of structured fluids in homogeneous shear based on a realizable model for the orientation dyad

Non-spherical particles dispersed in a fluid have a tendency to align in shear flows because of particle-fluid drag. This phenomenon is opposed by rotary diffusion. At high concentrations and in the absence of hydrodynamic couples, self-alignment can also occur because excluded volume forces prevent the return-to-isotropy of anisotropic states by rotary Brownian motion. The balance between microhydrodynamic and diffusive (i.e., Brownian and excluded volume) torques at the microscale has a direct impact on the rheological properties of rigid rod fluids (particulate suspensions and liquid crystalline polymers) at the continuum scale.; Over the past sixty years, important characteristics of the microstructure associated with the foregoing alignment phenomenon have been quantified in terms of the low order moments of the orientation density function governed by the rotary Smoluchowski equation. In this research, a closed model for the second order moment < pp > (orientation dyad) has been identified based on the condition that in the absence of an external field all realizable anisotropic states must relax to stable equilibrium states. A key step in the development of the new closure is the use of an algebraic pre-closure for the orientation tetrad < p p p p > in terms of the orientation dyad < p p > that preserves the six-fold symmetry and contraction properties of the original orientation tetrad.; In the presence of a simple shear flow, the microstructure and the rheological characteristics predicted for rigid-rod fluids agree with previous theoretical and experimental results for a wide range of Peclet numbers. In addition to the Peclet number (i.e., Pe ≡ ||∇u||/(6 DoR )), the orientation director also depends on three other dimensionless groups: a tumbling parameter, lambda; an excluded volume coefficient, U; and, a dimensionless time t ≡ 6 DoR tˆ. The rotary diffusion coefficient for dilute solutions, DoR , is used to scale time. Unlike other closure models, the approach developed hereinafter predicts that all two-dimensional and three-dimensional realizable anisotropic states relax to either a steady state (isotropic or anisotropic) or a periodic state, depending on Pe, lambda; and U. The model predicts the existence of shear thinning and shear thickening phenomena, Newtonian plateau regions at low and high Peclet numbers, positive (and negative) first normal stress differences, and negative (and positive) second normal stress differences. For Pe = 0, multiple equilibrium states exist for 4.72 ≤ U ≤ 5.00. For Pe > 0 and initial directors located in the flow-deformation plane, the predominant feature for U < 25 is the existence of a unique nematic-like microstructure with a steady alignment of the director that becomes completely aligned with the velocity as Pe → infinity. For lambda < 1 and U > 25, tumbling and wagging of the director occur at low to moderate values of the Peclet number. If the initial director has a component in the direction of the vorticity, then director kayaking and director log-rolling may occur. The coexistence of stable anisotropic states (or texture) predicted by the model may provide an explanation of why micro defects occur during the processing of some structured fluids.