Some mathematical issues in the modeling of flow phenomena of polymeric liquid crystals

Abstract

We consider shear flow of polymeric liquid crystals for large and small values of the Ericksen number ℰ. The model consists of governing equations for the velocity field v, the pressure p, the director n, and the order parameter s. The constitutive functions for the Leslie coefficients, αi, derived from the molecular theory of Doi play a crucial role in the modeling. One of the goals of the analysis is to examine the role of s in describing singularities as well as in obtaining new regimes which cannot be predicted by the previous Leslie–Ericksen model. In particular, solutions are obtained that correspond to domain structures parallel to the flow. The domains are separated by singular lines across which the director experiences jumps of ±45°. Such flows correspond to regimes with large values of ℰ. Another type of configuration analyzed in the article corresponds to periodic stripes parallel to the flow and such that the optic fields vary along the direction perpendicular to the plane of shear. Such configurations occur in the limit of small ℰ.