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Non-linear numerical analysis of the Leslie-Ericksen (L-E) equations for transient and steady rectilinear simple shear flows is presented for a typical non-aligning rigid-rod low molecular weight nematic (8CBP (4-n-octyl-4'-cyanobiphenyl)). Instead of the fixed torque balance used by previous investigators, a computationally efficient and accurate form of the torque balance, gyroscopic torque balance, is formulated and implemented. A series of experimentally measured material constant sets is used to investigate the effect of the reactive parameter lambda on the three-dimensional orientation field. Stable steady state solutions for both parallel and homeotropic orientation anchorings are obtained. Dynamic simulations show the stability properties of the steady state solutions and the routes to the stable steady states. The exchange of stability in the orientation and velocity fields is described as a supercritical bifurcation. Two families of stable steady state solutions are found: in-shear-plane (IP) solutions and out-of-shear-plane (OP) solutions; the former are characterized by the confinement of the director within the shear plane while the latter are characterized by three-dimensional orientation. It is found that the stability of the steady states is controlled by the magnitude of the Ericksen number (E). Orientational transitions from IP to OP are shown to occur when E exceeds a certain critical value Eco, that depends on the magnitude of the reactive parameter and the director wall anchoring. The non-linear stability theory of Landau was used to describe these transition phenomena. The calculated apparent viscosity and first normal stress difference for the OP solutions are shown to be smooth functions of the Ericksen number E.
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