Skip to main content
SpringerLink
Log in

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

  • Regular Article
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Y. Zhu, L. Hong, Z. Yang, W.-A. Yong, J. Non-Equilib. Thermodyn. 40, 67 (2015)

    Article  ADS  Google Scholar 

  2. I. Müller, T. Ruggeri, Extended Thermodynamics, Vol. 37 (Springer Science and Business Media, 2013)

  3. D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics (Springer, 1996)

  4. D. Jou, J. Casas-Vázquez, M. Criado-Sancho, Thermodynamics of Fluids under Flow (Springer-Verlag, Berlin, 2011)

  5. W.-A. Yong, J. Math. Phys. 49, 033503 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  6. L. Peng, Y. Zhu, L. Hong, Phys. Rev. E 97, 062123 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  7. S.K. Godunov, Dokl. Akad. Nauk SSSR 2, 521 (1961)

    Google Scholar 

  8. K.O. Friedrichs, P.D. Lax, Proc. Natl. Acad. Sci. U.S.A. 68, 1686 (1971)

    Article  ADS  Google Scholar 

  9. X. Huo, Modeling and analysis of viscoelastic fluids, PhD Thesis, Tsinghua University, China, 2017

  10. X. Huo, W.-A. Yong, J. Differ. Equ. 261, 1264 (2016)

    Article  ADS  Google Scholar 

  11. Z. Yang, W.-A. Yong, Y. Zhu, Generalized hydrodynamics and the classical hydrodynamic limit, arXiv:1809.0161 (2018)

  12. J. Liu, W.-A. Yong, Geophys. J. Int. 204, 535 (2016)

    Article  ADS  Google Scholar 

  13. H. Yan, W.-A. Yong, J. Hyperbolic Differ. Equ. 9, 325 (2012)

    Article  MathSciNet  Google Scholar 

  14. L. Hong, Z. Yang, Y. Zhu, W.-A. Yong, J. Non-Equilib. Thermodyn. 40, 247 (2015)

    Article  Google Scholar 

  15. V. Lakshminarayanan, A. Ghatak, K. Thyagarajan, Lagrangian Optics (Springer Science and Business Media, 2013)

  16. I.M. Gelfand, R.A. Silverman, Calculus of Variations (Courier Corporation, 2000)

  17. H. Goldstein, Classical Mechanics (Pearson Education India, 2011)

  18. L. Rayleigh, Proc. Math. Soc. (London) 4, 363 (1873)

    Google Scholar 

  19. M. Doi, Soft Matter Physics (Oxford University Press, 2013)

  20. M. Doi, J. Phys.: Condens. Matter 23, 284118 (2011)

    Google Scholar 

  21. H. Qian, J. Math. Phys. 54, 053302 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  22. M. Grmela, J. Phys. Commun. 2, 032001 (2018)

    Article  Google Scholar 

  23. M. Grmela, G. Lebon, J. Phys. A: Gen. Phys. 23, 3341 (1990)

    Article  ADS  Google Scholar 

  24. M. Grmela, D. Jou, J. Casas-Vázquez, J. Chem. Phys. 108, 7937 (1998)

    Article  ADS  Google Scholar 

  25. M. Sun, D. Jou, J. Zhang, J. Non-Newton. Fluid Mech. 229, 96 (2016)

    Article  MathSciNet  Google Scholar 

  26. L. Onsager, Phys. Rev. 37, 405 (1931)

    Article  ADS  Google Scholar 

  27. X. Xu, U. Thiele, T. Qian, J. Phys.: Conden. Matter 27, 085005 (2015)

    ADS  Google Scholar 

  28. J. Zhou, M. Doi, Phys. Rev. Fluids 3, 084004 (2018)

    Article  ADS  Google Scholar 

  29. W. Jiang, Q. Zhao, T. Qian, D.J. Srolovitz, W. Bao, Acta Mater. 163, 154 (2019)

    Article  Google Scholar 

  30. X. Xu, T. Qian, Proc. IUTAM 20, 144 (2017)

    Article  Google Scholar 

  31. S. Hu, Y. Wang, X. Man, M. Doi, Langmuir 33, 5965 (2017)

    Article  Google Scholar 

  32. M. Herty, W.-A. Yong, Automatica 69, 12 (2016)

    Article  Google Scholar 

  33. W.-A. Yong, Automatica 101, 252 (2019)

    Article  Google Scholar 

  34. I. Peshkov, M. Pavelka, E. Romenski, M. Grmela, Continuum Mech. Thermodyn. 30, 1343 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  35. M. Grmela, L. Hong, D. Jou, G. Lebon, M. Pavelka, Phys. Rev. E 95, 033121 (2017)

    Article  ADS  Google Scholar 

  36. K. Sekimoto, Stochastic Energetics, Vol. 799 (Springer, 2010)

  37. M. Grmela, H.C. Öttinger, Phys. Rev. E 56, 6620 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  38. H.C. Öttinger, M. Grmela, Phys. Rev. E 56, 6633 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  39. W.-A. Yong, Arch. Ration. Mech. Anal. 172, 247 (2004)

    Article  MathSciNet  Google Scholar 

  40. W.-A. Yong, SIAM J. Appl. Math. 64, 1737 (2004)

    Article  MathSciNet  Google Scholar 

  41. D. Zhou, P. Zhang, W.E, Phys. Rev. E 73, 061801 (2006)

    Article  ADS  Google Scholar 

  42. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)

    Article  ADS  Google Scholar 

  43. M. Doi, S.-F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988)

  44. H. Tanaka, T. Koyama, T. Araki, J. Phys.: Condens. Matter 15, S387 (2002)

    Google Scholar 

  45. J.L. Ericksen, Trans. Soc. Rheol. 5, 23 (1961)

    Article  MathSciNet  Google Scholar 

  46. F.M. Leslie, Adv. Liq. Cryst. 4, 1 (1979)

    Article  Google Scholar 

  47. F.-H. Lin, C. Liu, Commun. Pure Appl. Math. 48, 501 (1995)

    Article  Google Scholar 

  48. F.-H. Lin, C. Liu, Arch. Ration. Mech. Anal. 154, 135 (2000)

    Article  MathSciNet  Google Scholar 

  49. H. Sun, C. Liu, Discrete Contin. Dyn. Syst. 23, 455 (2009)

    Article  MathSciNet  Google Scholar 

  50. R. Temam, A. Miranville, Mathematical Modeling in Continuum Mechanics (Cambridge University Press, 2005)

  51. J.G. Hu, Z.C. Ouyang, Phys. Rev. E 47, 461 (1993)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, 100084, Beijing, P. R. China

    Liangrong Peng, Yucheng Hu & Liu Hong

  2. Department of Mathematics, Minjiang University, 350121, Fuzhou, P. R. China

    Liangrong Peng

Authors
  1. Liangrong Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yucheng Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liu Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liu Hong.

Additional information

Publisher's Note

The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peng, L., Hu, Y. & Hong, L. Conservation-Dissipation Formalism for soft matter physics: I. Augmentation to Doi's variational approach. Eur. Phys. J. E 42, 73 (2019). https://doi.org/10.1140/epje/i2019-11847-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2019-11847-2

Keywords

Search

Navigation