A continuum model for flows of cholesteric liquid crystal polymers and permeation flows

We present a hydrodynamic theory for flows of cholesteric liquid crystal polymers (CLCPs) following the continuum mechanics formulation of McMillan’s second order tensor theory for liquid crystals [E.H. MacMillan, Ph.D. thesis, A Theory of Anisotropic Fluid Department of Mechanics, University of Minnesota, 1987], in which anisotropic distortional elastic coefficients are prescribed as functions of the local nematic order. We explore two classes of spatially inhomogeneous equilibrium solutions associated with the variation of the uniaxial order parameter and the helical director pattern. In the latter solution, we show that the chirality effectively couples the distortional elasticity and the cholesteric pitch to impact on the local nematic order and vice versa. When the model is used to study permeation flows of weakly sheared cholesteric liquid crystal polymers, we find that secondary flows are strongly coupled to the primary flow and the director dynamics. At the first order of the coarse-grained asymptotic scheme, the flow and orientational director dynamics dominate while the dynamics of the orientation order parameters show up at the next order. The leading order solutions exhibit boundary layers of thickness in the order of O(q−1)O(q−1), where q   is the dimensionless wave number in the base cholesteric equilibrium. In plane Couette flows, the velocity components are nearly constants outside the boundary layers and the director angle varies linearly across the shear cell while the apparent viscosity scales like O(q)O(q); whereas in weak Poiseuille flows, the angle is quadratic outside the boundary layers and velocity components barely change while the apparent viscosity scales like O(q2)O(q2).