STATIONARY BIFURCATIONS AND TRICRITICALITY IN A CREEPING NEMATIC POLYMER FLOW

Numerical solutions to the Leslie-Ericksen equations for the transient and steady shear flows of a model rigid-rod nematic polymer are obtained using Galerkin finite elements, computational bifurcation methods, and gyroscopic torque balances. The properties of the model polymer are those of PBG (poly-gamma-benzyl-glutamate). The four-component solution vector consists of primary and secondary velocity components and of the tilt theta and twist phi angles of the director field. The two-component parameter vector P considered comprises the Ericksen number E, and the fixed director tilt anchoring angle theta(w) at the two bounding surfaces. According to the magnitude of P = P(E, theta(w)), ten types of solutions are found and fully characterized. The stability of these ten types of solutions is established using both computational bifurcation methods and dynamic simulations. These ten types of solutions are classified as in-shear-plane (IP) solution if the twist angle is zero every where (phi = 0), and out-of-shear-plane (OP) solutions if the twist angle does not vanish in the whole computational domain. Further categorization according to local stability differentiates locally stable IP solutions (IS) from unstable IP solutions (IU). The main structural changes between these solutions are orientational first-order (discontinuous) and second-order (continuous) transitions between IP and OP solutions; the transitions are mathematically described as stationary supercritical (second order, continuous) bifurcations and as stationary subcritical (first order, discontinuous) bifurcations. The type (IP or OP), stability (S or U) of the different solutions and the transition (first and second) order involved as a function of P are determined numerically and results are summarized in tabular form.