Advertisement

Conservation-Dissipation Formalism for soft matter physics: I. Augmentation to Doi's variational approach

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

Abstract.

In the first paper of this series, we prove that by choosing the proper variational function and variables, the variational approach proposed by Doi in soft matter physics is equivalent to the Conservation-Dissipation Formalism. To illustrate the correspondence between these two theories, several novel examples in soft matter physics, including the particle diffusion in dilute solutions, polymer phase separation dynamics and nematic liquid crystal flows, are carefully examined. Based on our work, a deep connection among the generalized Gibbs relation, the second law of thermodynamics and the variational principle in non-equilibrium thermodynamics is revealed.

Graphical abstract

Keywords

Soft Matter: Polymers and Polyelectrolytes 

References

  1. 1.
    Y. Zhu, L. Hong, Z. Yang, W.-A. Yong, J. Non-Equilib. Thermodyn. 40, 67 (2015)CrossRefGoogle Scholar
  2. 2.
    I. Müller, T. Ruggeri, Extended Thermodynamics, Vol. 37 (Springer Science and Business Media, 2013)Google Scholar
  3. 3.
    D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics (Springer, 1996)Google Scholar
  4. 4.
    D. Jou, J. Casas-Vázquez, M. Criado-Sancho, Thermodynamics of Fluids under Flow (Springer-Verlag, Berlin, 2011)Google Scholar
  5. 5.
    W.-A. Yong, J. Math. Phys. 49, 033503 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Peng, Y. Zhu, L. Hong, Phys. Rev. E 97, 062123 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    S.K. Godunov, Dokl. Akad. Nauk SSSR 2, 521 (1961)Google Scholar
  8. 8.
    K.O. Friedrichs, P.D. Lax, Proc. Natl. Acad. Sci. U.S.A. 68, 1686 (1971)CrossRefGoogle Scholar
  9. 9.
    X. Huo, Modeling and analysis of viscoelastic fluids, PhD Thesis, Tsinghua University, China, 2017Google Scholar
  10. 10.
    X. Huo, W.-A. Yong, J. Differ. Equ. 261, 1264 (2016)CrossRefGoogle Scholar
  11. 11.
    Z. Yang, W.-A. Yong, Y. Zhu, Generalized hydrodynamics and the classical hydrodynamic limit, arXiv:1809.0161 (2018)Google Scholar
  12. 12.
    J. Liu, W.-A. Yong, Geophys. J. Int. 204, 535 (2016)CrossRefGoogle Scholar
  13. 13.
    H. Yan, W.-A. Yong, J. Hyperbolic Differ. Equ. 9, 325 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    L. Hong, Z. Yang, Y. Zhu, W.-A. Yong, J. Non-Equilib. Thermodyn. 40, 247 (2015)CrossRefGoogle Scholar
  15. 15.
    V. Lakshminarayanan, A. Ghatak, K. Thyagarajan, Lagrangian Optics (Springer Science and Business Media, 2013)Google Scholar
  16. 16.
    I.M. Gelfand, R.A. Silverman, Calculus of Variations (Courier Corporation, 2000)Google Scholar
  17. 17.
    H. Goldstein, Classical Mechanics (Pearson Education India, 2011)Google Scholar
  18. 18.
    L. Rayleigh, Proc. Math. Soc. (London) 4, 363 (1873)Google Scholar
  19. 19.
    M. Doi, Soft Matter Physics (Oxford University Press, 2013)Google Scholar
  20. 20.
    M. Doi, J. Phys.: Condens. Matter 23, 284118 (2011)Google Scholar
  21. 21.
    H. Qian, J. Math. Phys. 54, 053302 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. Grmela, J. Phys. Commun. 2, 032001 (2018)CrossRefGoogle Scholar
  23. 23.
    M. Grmela, G. Lebon, J. Phys. A: Gen. Phys. 23, 3341 (1990)CrossRefGoogle Scholar
  24. 24.
    M. Grmela, D. Jou, J. Casas-Vázquez, J. Chem. Phys. 108, 7937 (1998)CrossRefGoogle Scholar
  25. 25.
    M. Sun, D. Jou, J. Zhang, J. Non-Newton. Fluid Mech. 229, 96 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    L. Onsager, Phys. Rev. 37, 405 (1931)CrossRefGoogle Scholar
  27. 27.
    X. Xu, U. Thiele, T. Qian, J. Phys.: Conden. Matter 27, 085005 (2015)Google Scholar
  28. 28.
    J. Zhou, M. Doi, Phys. Rev. Fluids 3, 084004 (2018)CrossRefGoogle Scholar
  29. 29.
    W. Jiang, Q. Zhao, T. Qian, D.J. Srolovitz, W. Bao, Acta Mater. 163, 154 (2019)CrossRefGoogle Scholar
  30. 30.
    X. Xu, T. Qian, Proc. IUTAM 20, 144 (2017)CrossRefGoogle Scholar
  31. 31.
    S. Hu, Y. Wang, X. Man, M. Doi, Langmuir 33, 5965 (2017)CrossRefGoogle Scholar
  32. 32.
    M. Herty, W.-A. Yong, Automatica 69, 12 (2016)CrossRefGoogle Scholar
  33. 33.
    W.-A. Yong, Automatica 101, 252 (2019)CrossRefGoogle Scholar
  34. 34.
    I. Peshkov, M. Pavelka, E. Romenski, M. Grmela, Continuum Mech. Thermodyn. 30, 1343 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    M. Grmela, L. Hong, D. Jou, G. Lebon, M. Pavelka, Phys. Rev. E 95, 033121 (2017)CrossRefGoogle Scholar
  36. 36.
    K. Sekimoto, Stochastic Energetics, Vol. 799 (Springer, 2010)Google Scholar
  37. 37.
    M. Grmela, H.C. Öttinger, Phys. Rev. E 56, 6620 (1997)MathSciNetCrossRefGoogle Scholar
  38. 38.
    H.C. Öttinger, M. Grmela, Phys. Rev. E 56, 6633 (1997)MathSciNetCrossRefGoogle Scholar
  39. 39.
    W.-A. Yong, Arch. Ration. Mech. Anal. 172, 247 (2004)MathSciNetCrossRefGoogle Scholar
  40. 40.
    W.-A. Yong, SIAM J. Appl. Math. 64, 1737 (2004)MathSciNetCrossRefGoogle Scholar
  41. 41.
    D. Zhou, P. Zhang, W.E, Phys. Rev. E 73, 061801 (2006)CrossRefGoogle Scholar
  42. 42.
    J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)CrossRefGoogle Scholar
  43. 43.
    M. Doi, S.-F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988)Google Scholar
  44. 44.
    H. Tanaka, T. Koyama, T. Araki, J. Phys.: Condens. Matter 15, S387 (2002)Google Scholar
  45. 45.
    J.L. Ericksen, Trans. Soc. Rheol. 5, 23 (1961)CrossRefGoogle Scholar
  46. 46.
    F.M. Leslie, Adv. Liq. Cryst. 4, 1 (1979)CrossRefGoogle Scholar
  47. 47.
    F.-H. Lin, C. Liu, Commun. Pure Appl. Math. 48, 501 (1995)CrossRefGoogle Scholar
  48. 48.
    F.-H. Lin, C. Liu, Arch. Ration. Mech. Anal. 154, 135 (2000)MathSciNetCrossRefGoogle Scholar
  49. 49.
    H. Sun, C. Liu, Discrete Contin. Dyn. Syst. 23, 455 (2009)MathSciNetCrossRefGoogle Scholar
  50. 50.
    R. Temam, A. Miranville, Mathematical Modeling in Continuum Mechanics (Cambridge University Press, 2005)Google Scholar
  51. 51.
    J.G. Hu, Z.C. Ouyang, Phys. Rev. E 47, 461 (1993)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

Personalised recommendations