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Semi-Discrete Energy-Stable Schemes for a Tensor-Based Hydrodynamic Model of Nematic Liquid Crystal Flows

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Abstract

In this paper, we develop a first-order and a second-order coupled energy stable numerical scheme respectively for a Q-tensor based hydrodynamic model of nematic liquid crystal flows. We then extend the first order coupled scheme to a decoupled scheme and show that it is energy stable as well. The fully coupled schemes are implemented in 2-dimensional space and time, with which we study defect dynamics in flows of nematic liquid crystals in a channel. The numerical schemes are shown to be efficient in solving the Q-tensor based liquid crystal model. The methodology developed here also provides a paradigm for developing energy-stable schemes for more general hydrodynamic models of complex fluids which obey an energy dissipation law.

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References

  1. Beris, A.N., Edwards, B.: Thermodynamics of Flowing Systems. Oxford Science Publications, New York (1994)

    Google Scholar 

  2. Blow, M.L., Thampi, S.P., Yeomans, J.M.: Biphasic lyotropic active nematics. Phys. Rev. Lett. 113, 248303 (2014)

    Article  Google Scholar 

  3. Boyer, F., Lapuerta, C.: Study of a three component Cahn–Hilliard flow model. ESAIM Math. Model. Numer. Anal. 40(4), 653–687 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals. Oxford University Press, Oxford (1993)

    Google Scholar 

  5. Denniston, C., Orlandini, E., Yeomans, J.M.: Lattice boltzmann simulations of liquid crystal hydrodynamics. Phys. Rev. E 63(5), 056702 (2001)

    Article  Google Scholar 

  6. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford Science Publication, New York (1986)

    Google Scholar 

  7. Fan, J., Ozawa, T.: Regularity criteria for a coupled Navier–Stokes and q-tensor system. Int. J. Anal. 2013, 718173 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Forest, M.G., Wang, Q.: Hydrodynamic theories for mixtures of polymers and rodlike liquid crystalline polymers. Phys. Rev. E 72, 041805 (2005)

    Article  Google Scholar 

  9. Forest, M.G., Wang, Q., Zhou, H.: Homogeneous biaxial patterns and director instabilities of liquid crystal polymers in axial and planar elongation. Phys. Fluids 12, 490–498 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guillenn-Gonzalez, F., Rodriguez-Bellido, M.A.: Weak time regularity and uniqueness for a q-tensor model. SIAM J. Math. Anal. 46(5), 3540–3567 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guillenn-Gonzalez, F., Rodriguez-Bellido, M.A.: Weak solutions for an initial boundary q tensor problem related to liquid crystals. Nonlinear Anal. 112, 84–104 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Han, D., Wang, X.: A second order in time uniquely solvable unconditionally stable numerical schemes for Cahn–Hilliard–Navier–Stokes equation. J. Comput. Phys. 290(1), 139–156 (2015)

    Article  MathSciNet  Google Scholar 

  13. Leslie, F.M.: The theory of flow phenomena in liquid crystals. Adv. Liq. Cryst. 4, 1–81 (1979)

    Article  Google Scholar 

  14. Longa, L., Trebin, H.R.: Spontaneous polarization in chiral biaxial liquid crystals. Phys. Rev. A 42(6), 3453 (1990)

    Article  MathSciNet  Google Scholar 

  15. MacDonald, C.S., Mackenzie, J.A., Ramage, A.: Efficient moving mesh method for q-tensor models of nematic liquid crystals. SIAM J. Sci. Comput. 37(2), 215–238 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Marenduzzo, D., Orlandini, E., Yeomans, J.M.: Hydrodynamics and rheology of active liquid crystals a numerical investigation. Phys. Rev. Lett. 98, 118102 (2007)

    Article  Google Scholar 

  17. Paicu, M., Zarnescu, A.: Energy dissipaiton and regularity for a coupled Navier–Stokes and q-tensor system. Arch. Ration. Mech. Anal. 203, 45–67 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schopohl, N., Sluckin, T.J.: Defect core structure in nematic liquid crystals. Phys. Rev. Lett. 59(22), 2582 (1987)

    Article  Google Scholar 

  19. Shen, J., Wang, C., Wang, S., Wang, X.: Second-order convex splitting schemes for gradient flows with Ehrlich–Schwoebel type energy: application to thin film epitaxy. SIAM J. Numer. Anal. 50(1), 105–125 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shen, J., Yang, X.: Numerical approximation of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. Ser. B 28(4), 1669–1691 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shen, J., Yang, X.: Decoupled energy stable schems for phase field models of two phase complex fluids. SIAM J. Sci. Comput. 36(1), 122–145 (2014)

    Article  MathSciNet  Google Scholar 

  22. Sonnet, A.M., Maffettone, P.L., Virga, E.G.: Continuum theory for nematic liquid crystals with tensorial order. J. Non-Newton. Fluid Mech. 119, 51–59 (2004)

    Article  MATH  Google Scholar 

  23. Sulaiman, N., Marenduzzo, D., Yeomans, J.M.: Lattice boltzmann algorithm to simulate isotropic–nematic emulsions. Phys. Rev. E 74, 041708 (2006)

    Article  Google Scholar 

  24. Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. American Mathematical Society, Providence (2001)

    Book  MATH  Google Scholar 

  25. Tsuji, T., Rey, A.D.: Effect of long range order on sheared liquid crystalline materials, part I: compatability between tumbiling and behavior and fixed anchoring. J. Non-Newton. Fluid Mech. 73, 127–152 (1997)

    Article  Google Scholar 

  26. Wang, C., Wise, S.M.: An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 49, 945–969 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, Q.: Biaxial steady states and their stability in shear flows of liquid crystal polymers. J. Rheol. 41, 943–970 (1997)

    Article  Google Scholar 

  28. Wang, Q.: A hydrodynamic theory of nematic liquid crystalline polymers of different configurations. J. Chem. Phys. 116, 9120–9136 (2002)

    Article  Google Scholar 

  29. Wise, S.M., Wang, C., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47(3), 2269–2288 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, X.: Error analysis of stabilized semi-implicit method of Allen–Cahn equation. Discrete Contin. Dyn. Syst. Ser. B 11, 1057–1070 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang, X., Feng, J.J., Liu, C., Shen, J.: Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. J. Comput. Phys. 218, 417–428 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, X., Forest, M.G., Mullins, W., Wang, Q.: 2-D lid-driven cavity flow of nematic polymers: an unsteady sea of defects. Soft Matter 6, 1138–1156 (2010)

    Article  Google Scholar 

  33. Yang, X., Forest, M.G., Wang, Q.: Near equilibrium dynamics and one-dimensional spatial–temporal structures of polar active liquid crystals. Chin. Phys. B 23(11), 118701 (2014)

    Article  Google Scholar 

  34. Yang, X., Wang, Q.: Capillary instability of axisymmetric active liquid crystal jets. Soft Matter 10(35), 6758–6776 (2014)

    Article  Google Scholar 

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Acknowledgments

Jia Zhao and Qi Wang are partially supported by NSF-DMS-1200487 and NSF-DMS-1517347, NIH-2R01GM078994-05A1, AFOSR-FA9550-12-1-0178, and an SC EPSCOR/IDEA award. In addition, Jia Zhao is also supported by a Dissertation Fellowship from the Provost Office of USC.

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Authors and Affiliations

  1. Department of Mathematics, Interdisciplinary Mathematics Institute and NanoCenter at USC, University of South Carolina, Columbia, SC, 29208, USA

    Jia Zhao & Qi Wang

  2. Beijing Computational Science Research Center, Beijing, 100193, China

    Qi Wang

  3. School of Mathematics, Nankai University, Tianjin, 300071, China

    Qi Wang

Authors
  1. Jia Zhao
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  2. Qi Wang
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Correspondence to Qi Wang.

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Zhao, J., Wang, Q. Semi-Discrete Energy-Stable Schemes for a Tensor-Based Hydrodynamic Model of Nematic Liquid Crystal Flows. J Sci Comput 68, 1241–1266 (2016). https://doi.org/10.1007/s10915-016-0177-x

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  • DOI: https://doi.org/10.1007/s10915-016-0177-x

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