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March  2006, 6(2): 291-310. doi: 10.3934/dcdsb.2006.6.291

Mesoscale structures in flows of weakly sheared cholesteric liquid crystal polymers

Zhenlu Cui 1, , M. Carme Calderer 2, and  Qi Wang 3,

1. 

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, United States
2. 

School of Mathematics, University of Minnesota, 206 Church Street S.E, Minneapolis, MN 55455, United States
3. 

Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, United States

Received  March 2005 Revised  September 2005 Published  December 2005

We revisit the permeation flow issue in weakly sheared cholesteric liquid crystal polymers in plane Couette and Poiseuille flow geometries using a mesoscopic theory obtained from the kinetic theory for flows of cholesteric liquid crystal polymers [2]. We first present two classes of equilibrium solutions due to the order parameter variation and the director variation, respectively; then, study the permeation mode in weakly sheared flows of cholesteric liquid crystal polymers employing a coarse-grain approximation. We show that in order to solve the permeation flow problem correctly using the coarse-grain approximation, secondary flows must be considered, resolving a long standing inconsistency in the study of cholesteric liquid crystal flows [7]. Asymptotic solutions are sought in Deborah number expansions. The primary and secondary flow as well as the director dynamics are shown to dominate at leading order while the local nematic order fluctuations are higher order effects. The leading order solutions are obtained explicitly and analyzed with respect to the cholesteric pitch and other material parameters. The role of the anisotropic elasticity in equilibrium phase transition and permeation flows is investigated as well.
Keywords: cholesterics, phase-transitio., Couette flows, Poiseuille flows, Asymptotic expansions, liquid crystals, polymers.
Mathematics Subject Classification: 76A15, 76M4.
Citation: Zhenlu Cui, M. Carme Calderer, Qi Wang. Mesoscale structures in flows of weakly sheared cholesteric liquid crystal polymers. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 291-310. doi: 10.3934/dcdsb.2006.6.291
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