American Institute of Mathematical Sciences

Advanced
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 and Continuous Dynamical Systems - B, 2006, 6 (2) : 291-310. doi: 10.3934/dcdsb.2006.6.291
[1]

Hong Zhou, M. Gregory Forest. Anchoring distortions coupled with plane Couette & Poiseuille flows of nematic polymers in viscous solvents: Morphology in molecular orientation, stress & flow. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 407-425. doi: 10.3934/dcdsb.2006.6.407
[2]

Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 45-60. doi: 10.3934/dcdsb.2011.15.45
[3]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure and Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201
[4]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
[5]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
[6]

Junyu Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 739-755. doi: 10.3934/dcds.2013.33.739
[7]

Ke Xu, M. Gregory Forest, Xiaofeng Yang. Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 457-473. doi: 10.3934/dcdsb.2011.15.457
[8]

Mohamed Tij, Andrés Santos. Non-Newtonian Couette-Poiseuille flow of a dilute gas. Kinetic and Related Models, 2011, 4 (1) : 361-384. doi: 10.3934/krm.2011.4.361
[9]

Muhammad Bilal Riaz, Naseer Ahmad Asif, Abdon Atangana, Muhammad Imran Asjad. Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 645-664. doi: 10.3934/dcdss.2019041
[10]

Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks and Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197
[11]

Eduard Feireisl, Antonín Novotný. Two phase flows of compressible viscous fluids. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022091
[12]

Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic and Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691
[13]

Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[14]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455
[15]

Chun Liu, Jie Shen. On liquid crystal flows with free-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 307-318. doi: 10.3934/dcds.2001.7.307
[16]

Xiaoli Li, Boling Guo. Well-posedness for the three-dimensional compressible liquid crystal flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1913-1937. doi: 10.3934/dcdss.2016078
[17]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623
[18]

Fatih Bayazit, Britta Dorn, Marjeta Kramar Fijavž. Asymptotic periodicity of flows in time-depending networks. Networks and Heterogeneous Media, 2013, 8 (4) : 843-855. doi: 10.3934/nhm.2013.8.843
[19]

Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325
[20]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy