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.
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References
P.G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford University Press, Oxford, 1995)
J. Han, Y. Luo, W. Wang, P. Zhang, Z. Zhang, Arch. Ration. Mech. Anal. 215, 741 (2015)
W. Wang, P. Zhang, Z. Zhang, SIAM J. Math. Anal. 47, 127 (2015)
J.L. Ericksen, Arch. Ration. Mech. Anal. 9, 371 (1962)
F.M. Leslie, Arch. Ration. Mech. Anal. 28, 265 (1968)
F.M. Leslie, Adv. Liq. Cryst. 4, 1 (1979)
S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1992)
F.-H. Lin, C. Liu, Commun. Pure Appl. Math. 48, 501 (1995)
F.-H. Lin, C. Liu, Arch. Ration. Mech. Anal. 154, 135 (2000)
H. Sun, C. Liu, Discrete Contin. Dyn. Syst. 23, 455 (2008)
L. Peng, Y. Hu, L. Hong, Eur. Phys. J. E 42, 73 (2019)
F.-H. Lin, C. Liu, J. Partial Differ. Equ. 14, 289 (2001)
F.-H. Lin, C. Wang, Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 372, 20130361 (2014)
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)
M. Hieber, J. Prüss, Math. Ann. 369, 977 (2017)
E. Feireisl, E. Rocca, G. Schimperna, Nonlinearity 24, 243 (2011)
F.D. Anna, C. Liu, Arch. Ration. Mech. Anal. 231, 637 (2019)
F. Gay-Balmaz, C. Tronci, Proc. Math. Phys. Eng. Sci. 467, 1197 (2011)
A.N. Beris, B.J. Edwards, Thermodynamics of flowing systems: with internal microstructure (Oxford University Press on Demand, 1994)
T. Qian, P. Sheng, Phys. Rev. E 58, 7475 (1998)
E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Commun. Math. Sci. 12, 317 (2014)
E. Feireisl, G. Schimperna, E. Rocca, A. Zarnescu, Ann. Mat. Pura Appl. 194, 1269 (2015)
A.M. Sonnet, P.L. Maffettone, E.G. Virga, J. Non-Newton. Fluid Mech. 119, 51 (2004)
I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (Taylor and Francis, London and New York, 2004)
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)
H.N.W. Lekkerkerker, P. Coulon, R.V.D. Haegen, R. Deblieck, J. Chem. Phys. 80, 3427 (1984)
H. He, E.M. Sevick, D.R.M. Williams, J. Chem. Phys. 144, 2186 (2016)
D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics (Springer, 1996)
Y. Zhu, L. Hong, Z. Yang, W.-A. Yong, J. Non-Equilib. Thermodyn. 40, 67 (2015)
M. Sun, D. Jou, J. Zhang, J. Non-Newton. Fluid Mech. 229, 96 (2016)
W. Muschik, Arch. Ration. Mech. Anal. 66, 379 (1977)
J. Casas-Vázquez, D. Jou, Rep. Prog. Phys. 66, 1937 (2003)
J.L. Ericksen, Trans. Soc. Rheol. 5, 23 (1961)
F.C. Frank, Discuss. Faraday Soc. 25, 19 (1958)
J.M. Ball, Mol. Cryst. Liq. Cryst. 647, 1 (2017) https://doi.org/10.1080/15421406.2017.1289425
G. Durand, L. Leger, F. Rondelez, M. Veyssie, Phys. Rev. Lett. 23, 1361 (1969)
S. Lee, R.B. Meyer, J. Chem. Phys. 84, 3443 (1986)
M.P. Allen, D. Frenkel, Phys. Rev. A 37, 1813 (1988)
B.J. Edwards, J. Non-Newton. Fluid Mech. 36, 243 (1990)
M. Grmela, J. Phys. Commun. 2, 032001 (2018)
S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Publishing Company, Amsterdam, 1962)
M. Doi, Soft Matter Physics (Oxford University Press, Oxford, 2013)
M. Grmela, H.C. Öttinger, Phys. Rev. E 56, 6620 (1997)
C.W. Oseen, Trans. Faraday Soc. 29, 883 (1933)
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Supplemental Material: A comprehensive derivation on the tensorial models for non-isothermal flows of nematic liquid crystals
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Peng, L., Hu, Y. & Hong, L. Conservation-Dissipation Formalism for soft matter physics: II. Application to non-isothermal nematic liquid crystals. Eur. Phys. J. E 42, 74 (2019). https://doi.org/10.1140/epje/i2019-11839-2
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DOI: https://doi.org/10.1140/epje/i2019-11839-2