Near-equilibrium dynamics of Dio models for liquid crystal polymer flows: catastrophic and regularized behavior

Doi models for flows of concentrated solutions of homogeneous liquid crysta l polymers (LCPs) are analyzed in the quadratic closure approximation. Our purpose is to clarify a remarkable near-equilibrium behavior of these equat ions which has apparently gone unnoticed; these results are important for a ny numerical or experimental interpretations of LCP flows based on Doi mode ls near mechanical and nematic equilibria. To reveal this behavior, we anal ytically solve the linearized Doi nematodynamic equations; this calculation explicitly captures the coupling between the pure nematic and pure hydrody namic linearized dynamics. The original Doi model without solvent viscosity is analyzed first: the low concentration (N<3) isotropic phase and the hig h concentration (N>8/3) prolate nematic phase yield well-posed linearized d ynamic; at higher concentration both the isotropic phase (N>3) and the obla te nematic phase (N>3) yield catastrophic linearized dynamics, with exponen tial growthrates proportional to the amplitude of the wavevector of the lin earized disturbance. This result implies that there is unbounded growth in vanishingly small length scales for data near these equilibria. We then exp lore three physical regularizations of the original Doi model: solvent visc osity, finite-range intermolecular interactions, and spatial inhomogeneity in LCP concentration. Each effect yields well-posed linearized dynamics of all now-nematic equilibria, with bounded growth rates. Solvent viscosity an d spatial inhomogeneity alone are not sufficient to produce a finite wavele ngth instability cutoff of the high-concentration isotropic and oblate nema tic equilibria, whereas a finite-range intermolecular potential alone yield s a finite cutoff. The flow-orientation interactions for unstable nematic p hases produce a spatial wavevector dependence of the instability from which we reveal a flow-induced spatially anisotropic (or directional) instabilit y of the oblate phase. (C) 1999 Elsevier Science B.V. All rights reserved.