Chevron patterns in liquid crystal flows

Abstract

We study plane shear and Poiseuille flows of uniaxially nematic liquid crystals for large values of the Ericksen number, ε. This is a non-dimensional quantity that renders rigorous notions of fast flow regimes. (The former corresponds to large values of ε.) The model employed is that due to Ericksen for liquid crystals with variable degree of orientation (Ericksen, 1991). The constitutive equations derived by Kuzuu and Doi (1984) and Marrucci (1981), play an essential role in the analysis. Variables of the problem include the velocity field, v, the pressure, p, the scalar order parameter, s, and the director, n. Recent experimental work involving, both, low molecular weight liquid crystals as well as polymers, has pointed out that the positivity of the ratio α 2 α 3 of the Leslie coefficients does not necessarily guarantee uniform molecular alignment in shear flow regimes, as previously believed. In particular, for fast regimes, the flow breaks up into stripped configurations, chevrons parallel to the velocity field. Taking such observations into account, here, we consider regimes that fulfill the alignment condition, and show that for large values of ε, solutions exhibiting disclinations in the flow region can be found. Moreover we point out that across singular lines, the director experiences jump discontinuities of approximately ±45° to the direction of flow. We construct solutions that present domain structures in the form of stripes parallel to the flow direction, on the plane of shear. The interface between two such stripes corresponds to a nearly isotropic layer.