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Conservation-Dissipation Formalism for soft matter physics: II. Application to non-isothermal nematic liquid crystals

  • Liangrong Peng
  • Yucheng Hu
  • Liu HongEmail author
Regular Article
  • 7 Downloads

Abstract.

For most existing non-equilibrium theories, the modeling of non-isothermal processes is a hard task. Intrinsic difficulties involve the non-equilibrium temperature, the coexistence of conserved energy and dissipative entropy, etc. In this paper, by taking the non-isothermal flow of nematic liquid crystals as a typical example, we illustrate that thermodynamically consistent models in either vectorial or tensorial forms can be constructed within the framework of the Conservation-Dissipation Formalism (CDF). And the classical isothermal Ericksen-Leslie model and Qian-Sheng model are shown to be special cases of our new vectorial and tensorial models in the isothermal, incompressible and stationary limit. Most importantly, from the above examples, it is known that CDF can easily solve the issues relating with non-isothermal situations in a systematic way. The first and second laws of thermodynamics are satisfied simultaneously. The non-equilibrium temperature is defined self-consistently as a partial derivative of the entropy function. Relaxation-type constitutive relations are constructed, which give rise to classical linear constitutive relations, like Newton's law and Fourier's law, in stationary limits. Therefore, CDF is expected to have a broad scope of applications in soft matter physics, especially under complicated situations, such as non-isothermal, compressible and nanoscale systems.

Graphical abstract

Keywords

Flowing Matter: Liquids and Complex Fluids 

Supplementary material

10189_2019_11839_MOESM1_ESM.pdf (221 kb)
Supplemental Material: A comprehensive derivation on the tensorial models for non-isothermal flows of nematic liquid crystals

References

  1. 1.
    P.G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford University Press, Oxford, 1995)Google Scholar
  2. 2.
    J. Han, Y. Luo, W. Wang, P. Zhang, Z. Zhang, Arch. Ration. Mech. Anal. 215, 741 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    W. Wang, P. Zhang, Z. Zhang, SIAM J. Math. Anal. 47, 127 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    J.L. Ericksen, Arch. Ration. Mech. Anal. 9, 371 (1962)CrossRefGoogle Scholar
  5. 5.
    F.M. Leslie, Arch. Ration. Mech. Anal. 28, 265 (1968)CrossRefGoogle Scholar
  6. 6.
    F.M. Leslie, Adv. Liq. Cryst. 4, 1 (1979)CrossRefGoogle Scholar
  7. 7.
    S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1992)Google Scholar
  8. 8.
    F.-H. Lin, C. Liu, Commun. Pure Appl. Math. 48, 501 (1995)CrossRefGoogle Scholar
  9. 9.
    F.-H. Lin, C. Liu, Arch. Ration. Mech. Anal. 154, 135 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. Sun, C. Liu, Discrete Contin. Dyn. Syst. 23, 455 (2008)CrossRefGoogle Scholar
  11. 11.
    L. Peng, Y. Hu, L. Hong, Eur. Phys. J. E 42, 73 (2019)Google Scholar
  12. 12.
    F.-H. Lin, C. Liu, J. Partial Differ. Equ. 14, 289 (2001)Google Scholar
  13. 13.
    F.-H. Lin, C. Wang, Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 372, 20130361 (2014)CrossRefGoogle Scholar
  14. 14.
    M. Hieber, J. Prüss, Thermodynamical Consistent Modeling and Analysis of Nematic Liquid Crystal Flows, Vol. 183, Mathematical Fluid Dynamics, Present and Future, Springer Proceedings in Mathematics and Statistics (Springer, Tokyo, 2016)Google Scholar
  15. 15.
    M. Hieber, J. Prüss, Math. Ann. 369, 977 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    E. Feireisl, E. Rocca, G. Schimperna, Nonlinearity 24, 243 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    F.D. Anna, C. Liu, Arch. Ration. Mech. Anal. 231, 637 (2019)MathSciNetCrossRefGoogle Scholar
  18. 18.
    F. Gay-Balmaz, C. Tronci, Proc. Math. Phys. Eng. Sci. 467, 1197 (2011)CrossRefGoogle Scholar
  19. 19.
    A.N. Beris, B.J. Edwards, Thermodynamics of flowing systems: with internal microstructure (Oxford University Press on Demand, 1994)Google Scholar
  20. 20.
    T. Qian, P. Sheng, Phys. Rev. E 58, 7475 (1998)CrossRefGoogle Scholar
  21. 21.
    E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Commun. Math. Sci. 12, 317 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    E. Feireisl, G. Schimperna, E. Rocca, A. Zarnescu, Ann. Mat. Pura Appl. 194, 1269 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    A.M. Sonnet, P.L. Maffettone, E.G. Virga, J. Non-Newton. Fluid Mech. 119, 51 (2004)CrossRefGoogle Scholar
  24. 24.
    I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (Taylor and Francis, London and New York, 2004)Google Scholar
  25. 25.
    P.J. Flory, Molecular theory of liquid crystals, in Liquid Crystal Polymers I. Advances in Polymer Science, edited by N.A. Platé, Vol. 59 (Springer, 1984)Google Scholar
  26. 26.
    H.N.W. Lekkerkerker, P. Coulon, R.V.D. Haegen, R. Deblieck, J. Chem. Phys. 80, 3427 (1984)CrossRefGoogle Scholar
  27. 27.
    H. He, E.M. Sevick, D.R.M. Williams, J. Chem. Phys. 144, 2186 (2016)Google Scholar
  28. 28.
    D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics (Springer, 1996)Google Scholar
  29. 29.
    Y. Zhu, L. Hong, Z. Yang, W.-A. Yong, J. Non-Equilib. Thermodyn. 40, 67 (2015)CrossRefGoogle Scholar
  30. 30.
    M. Sun, D. Jou, J. Zhang, J. Non-Newton. Fluid Mech. 229, 96 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    W. Muschik, Arch. Ration. Mech. Anal. 66, 379 (1977)CrossRefGoogle Scholar
  32. 32.
    J. Casas-Vázquez, D. Jou, Rep. Prog. Phys. 66, 1937 (2003)CrossRefGoogle Scholar
  33. 33.
    J.L. Ericksen, Trans. Soc. Rheol. 5, 23 (1961)CrossRefGoogle Scholar
  34. 34.
    F.C. Frank, Discuss. Faraday Soc. 25, 19 (1958)CrossRefGoogle Scholar
  35. 35.
    J.M. Ball, Mol. Cryst. Liq. Cryst. 647, 1 (2017)  https://doi.org/10.1080/15421406.2017.1289425 CrossRefGoogle Scholar
  36. 36.
    G. Durand, L. Leger, F. Rondelez, M. Veyssie, Phys. Rev. Lett. 23, 1361 (1969)CrossRefGoogle Scholar
  37. 37.
    S. Lee, R.B. Meyer, J. Chem. Phys. 84, 3443 (1986)CrossRefGoogle Scholar
  38. 38.
    M.P. Allen, D. Frenkel, Phys. Rev. A 37, 1813 (1988)CrossRefGoogle Scholar
  39. 39.
    B.J. Edwards, J. Non-Newton. Fluid Mech. 36, 243 (1990)CrossRefGoogle Scholar
  40. 40.
    M. Grmela, J. Phys. Commun. 2, 032001 (2018)CrossRefGoogle Scholar
  41. 41.
    S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Publishing Company, Amsterdam, 1962)Google Scholar
  42. 42.
    M. Doi, Soft Matter Physics (Oxford University Press, Oxford, 2013)Google Scholar
  43. 43.
    M. Grmela, H.C. Öttinger, Phys. Rev. E 56, 6620 (1997)MathSciNetCrossRefGoogle Scholar
  44. 44.
    C.W. Oseen, Trans. Faraday Soc. 29, 883 (1933)CrossRefGoogle Scholar

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Zhou Pei-Yuan Center for Applied MathematicsTsinghua UniversityBeijingP. R. China
  2. 2.Department of MathematicsMinjiang UniversityFuzhouP. R. China

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