Abstract
In this paper the heat conduction in viscous fluids is described by using the theory of classical irreversible thermodynamics with internal variables. In this theory, the deviation from the local equilibrium is characterized by vectorial internal variables and a generalized entropy current density expressed in terms of so-called current multipliers. Cross effects between heat conduction and viscosity are also considered and some phenomenological generalizations of Fourier’s and Newton’s laws are obtained.
Funding statement: This paper was supported by National Group of Mathematical Physics GNFM-INdAM (Italy).
Acknowledgment
The authors thank the anonymous referee for helpful comments and suggestions.
References
[1] W. Muschik, Why so many “schools” of thermodynamics, Atti Accad. Peloritana dei Pericolanti, Cl. Sci. Mat. Fis. Nat. LXXXVI suppl. 1 (2008), 1–24.10.1007/s10010-007-0053-9Search in Google Scholar
[2] L.Onsager, Reciprocal relations in irreversible processes, Phys. Rev. 37 (1931), 405–426; 38 (1931), 2265–2279.10.1103/PhysRev.37.405Search in Google Scholar
[3] S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962, pp. 239–303.Search in Google Scholar
[4] C. Truesdell, Rational Thermodynamics, chap. 5, McGraw Hill, New York, 1969.Search in Google Scholar
[5] I. Müller and T. Ruggeri, Rational Extended Thermodynamics, chaps. 3, 5, 2nd ed. Springer, Berlin, 1998.10.1007/978-1-4612-2210-1Search in Google Scholar
[6] T. Ruggeri and M. Sugiytama, Rational Extended Thermodynamics beyond the Monatomic Gas, Springer, Cham, New York, Dordrecht, London, 2015, ISBN 978-3-319-13340-9.10.1007/978-3-319-13341-6Search in Google Scholar
[7] D. Jou, J. Casas-Vazquez and G. Lebon, Extended Irreversible Thermodynamics, chaps. 2, 10, 16, 4th ed. Springer-Verlag, Berlin, 2010.10.1007/978-90-481-3074-0Search in Google Scholar
[8] G. Lebon, D. Jou and J. Casas-Vazquez, Understanding Non-Equilibrium Thermodynamics, Springer-Verlag, Berlin, Heidelberg, 2008.10.1007/978-3-540-74252-4Search in Google Scholar
[9] G. Lebon, M. Grmela and D. Lhuiller, A comparative study of the coupling of flow with non-Fickean thermodiffusion. Part I: Extended irreversible thermodynamics, J. Non-Equilib. Thermodyn. 28 (2003), 1–22.10.1515/JNETDY.2003.001Search in Google Scholar
[10] H. C. Öttinger and M. Grmela, Dynamics and thermodynamics of complex fluids.II. Illustrations of a GENERIC formalism, Phys. Rev. E56 (1997), 6633–6655.10.1103/PhysRevE.56.6633Search in Google Scholar
[11] M. Grmela, G. Lebon and D. Lhuiller, A comparative study of the coupling of flow with non-Fickean thermodiffusion. Part II: Generic, J. Non-Equilib. Thermodyn. 28 (2003), 23–50.10.1515/JNETDY.2003.002Search in Google Scholar
[12] D. Lhuiller, M. Grmela and G. Lebon, A comparative study of the coupling of flow with non-Fickean thermodiffusion. Part III: Internal variables, J. Non-Equilib. Thermodyn. 28 (2003), 51–68.10.1515/JNETDY.2003.003Search in Google Scholar
[13] G. A. Kluitenberg, On heat dissipation due to irreversible mechanical phenomena in continuous media. Physica A 35 (1967), 117–192.10.1016/0031-8914(67)90064-XSearch in Google Scholar
[14] G. A. Kluitenberg, On vectorial internal variables and dielectric and magnetic relaxation phenomena, Physica A 109 (1981), 91–122.10.1016/0378-4371(81)90039-XSearch in Google Scholar
[15] G. A. Kluitenberg, Plasticity and Non-Equilibrium Thermodynamics, CISM Lecture Notes, Springer-Verlag, Wien, New York, 1984.10.1007/978-3-7091-2636-3_4Search in Google Scholar
[16] V. Ciancio, On the generalized Debye equation for media with dielectric relaxation phenomena described by vectorial internal variables, J. Non-Equilib. Thermodyn. 14 (1989), 239–250.10.1515/jnet.1989.14.3.239Search in Google Scholar
[17] V. Ciancio and G. A. Kluitenberg, On electromagnetic waves in isotropic media with dielectric relaxation, Acta Phys. Hung. 66(1–4) (1989), 251–276.10.1007/BF03155797Search in Google Scholar
[18] V. Ciancio, L. Restuccia and G. A. Kluitenberg, A thermodynamic derivation of equations for dielectric relaxation phenomena in anisotropic polarizable media, J. Non-Equilib. Thermodyn. 15 (1990), 151–171.10.1515/jnet.1990.15.2.157Search in Google Scholar
[19] G. A. Maugin, Internal variables and dissipative structures, J. Non-Equilib. Thermodyn. 15 (1990), 173–192.10.1515/jnet.1990.15.2.173Search in Google Scholar
[20] A. Palumbo, C. Papenfuss and P. Rogolino, A mesoscopic approach to diffusion phenomena in mixtures, J. Non-Equilib. Thermodyn. 30 (2005), 401–419.10.1515/JNETDY.2005.028Search in Google Scholar
[21] M. Francaviglia, A. Palumbo and P. Rogolino, Thermodynamics of mixture as a problem with internal variables. The general theory, J. Non-Equilib. Thermodyn. 30 (2006), 419–429.10.1515/JNETDY.2006.018Search in Google Scholar
[22] M. Francaviglia, A. Palumbo and P. Rogolino, Internal variables thermodynamics of two component mixtures under linear constitutive hypothesis with an application to superfluid helium, J. Non-Equilib. Thermodyn. 30 (2006), 419–429.10.1515/JNETDY.2008.007Search in Google Scholar
[23] G. A. Kluitenberg and V. Ciancio, On linear dynamical equations of state for isotropic media I- General formalism, Physica A 93 (1978), 273–286.10.1016/0378-4371(78)90221-2Search in Google Scholar
[24] V. Ciancio and G. A. Kluitenberg, On linear dynamical equations of state for isotropic media II. Some cases of special interest, Physica A 99 (1979), 592–600.10.1016/0378-4371(79)90074-8Search in Google Scholar
[25] V. Ciancio and J. Verhás, A thermodynamic theory for radiating heat transfer, J. Non-Equilib. Thermodyn. 15 (1990), 33–43.10.1515/jnet.1990.15.1.33Search in Google Scholar
[26] V. Ciancio and J. Verhás, A thermodynamic theory for heat radiation through the atmosphere, J. Non-Equilib. Thermodyn. 16 (1991), 57–65.10.1515/jnet.1991.16.1.57Search in Google Scholar
[27] V. Ciancio, F. Farsaci and G. A. Bartolotta, Phenomenological and state coefficients in viscoanelastic medium of order one (with memory), Comput. Sci. Appl. ICCSA 3980 (2006), 821–827.10.1007/11751540_89Search in Google Scholar
[28] V. Ciancio, A. Bartolotta and F.Farsaci, Experimental confirmation on a thermodynamical theory for viscoanelastic media with memory, Physica B, Condens. Matter, 394(1–2) (2007), 8–13.10.1016/j.physb.2007.01.031Search in Google Scholar
[29] A.Ciancio, V.Ciancio and F.Farsaci, Wave propagation in media obeying a thermoviscoanelastic model, Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 69(4) (2007), 69–79. ISSN: 1223-7027.Search in Google Scholar
[30] V.Ciancio, A.Ciancio and F.Farsaci, On general properties of phenomenological and state coefficients for isotropic viscoanelastic media, Physica B, Condens. Matter 403 (2008), 3221–3227. ISSN: 0921-4526.10.1016/j.physb.2008.04.021Search in Google Scholar
[31] A. Ciancio, An approximate evaluation of the phenomenological and state coefficients for visco-anelastic media with memory, Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 73(4) (2011), 3–14.Search in Google Scholar
[32] V. Ciancio and L. Restuccia, On heat equation in the framework of classic irreversible thermodynamics with internal variables, Int. J. Geom. Methods Mod. Phys. 13 (2016), (11 pages), DOI: 10.1142/S021988781640003X.Search in Google Scholar
[33] V. Ciancio and A. Palumbo, A thermodynamic theory with hidden vectorial variables on possible interactions among heat conduction, diffusion phenomena, viscous flow and chemical reaction in fluid mixture, in: Thermocon 2016, International Conference and Summerschool, Thermal Theories of Continua: Survey and Developments, Messina April 19–22, 2016, Italy (2016), (18 pages).Search in Google Scholar
[34] P. Ván and T. Fülöp, Universality in heat conduction theory: weakly nonlocal thermodynamics, Ann. Phys. 524(8), (2012), 470–478.10.1002/andp.201200042Search in Google Scholar
[35] I. Müller, On the entropy inequality, Arch. Ration. Mech. Anal. 26(2) (1967), 118–141.10.1007/BF00285677Search in Google Scholar
[36] G. Lebon, Heat conduction at micro and nanoscales: A review through the prism of extended irreversible thermodynamic, J. Non-Equilib. Thermodyn. 39(1) (2014), 35–59.10.1515/jnetdy-2013-0029Search in Google Scholar
[37] D. Jou and L. Restuccia, Mesoscopic transport equations and contemporary thermodynamics: An introduction, Contemp. Phys. 52(5) (2011), 465–474.10.1080/00107514.2011.595596Search in Google Scholar
[38] G. Lebon, D. Jou and J. Casas-Vazquez, Heat conduction at low temperature a non-linear generalization of the Guyer–Krumhansl equation, Period. Polytechn. Ser. Chem. Eng., 41(2) (1997), 185–196.Search in Google Scholar
[39] N. Kalospiros, B. J. Edwards and A. N. Beris, Internal variables for relaxation phenomena in heat and mass transfer, Int. J. Heat Mass Transf. 36 (1993), 1191–1200.10.1016/S0017-9310(05)80089-4Search in Google Scholar
[40] J. Verhás, On the entropy current, J. Non-Equilib. Thermodyn. 8 (1983), 201–206.10.1515/jnet.1983.8.3.201Search in Google Scholar
[41] B.Nyíri, On the entropy current, J. Non-Equilib. Thermodyn. 16 (1991), 179–186.10.1515/jnet.1991.16.2.179Search in Google Scholar
[42] J. Fourier, Analytical Theory of Heat, Cambridge University Press, Cambridge, 1878.Search in Google Scholar
[43] C. Cattaneo, Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena 3 (1948), 83–101.10.1007/978-3-642-11051-1_5Search in Google Scholar
[44] P. Vernott, La veritable équation de la chaleur, Philos. Trans. R. Soc. Lond. 157 (1867), 49–88.Search in Google Scholar
[45] R. A. Guyer and J. A. Krumhansl, Solution on the linearized Boltzmann phonon equation, Phys. Rev. 148 (1966), 766–778.10.1103/PhysRev.148.766Search in Google Scholar
[46] R. A. Guyer and J. A. Krumhansl, Thermal conductivity, second sound and phonon hydrodynamic phenomena in non-metallic crystals, Phys. Rev. 148 (1966), 778–788.10.1103/PhysRev.148.778Search in Google Scholar
[47] D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys. 61, 41–74 (1989); 62 (1990), 375–392.10.1103/RevModPhys.61.41Search in Google Scholar
[48] A. E. Green and P. M. Naghdi, A re-examination of basic postulates of thermomechanics, R. Soc. Math. Phys. Sci. 61 (1989), 41–74; 62 (1990), 375–392.Search in Google Scholar
[49] M.Morse, Relation between the critical points of a real function of n independent variables, Trans. Am. Math. Soc. 27 (1925), 345–396.10.1090/S0002-9947-1925-1501318-XSearch in Google Scholar
[50] I. Gyarmati, On the wave approach of thermodynamics and some problems of non-linear theories, J. Non-Equilib. Thermodyn. 2 (1977), 233–260.10.1515/jnet.1977.2.4.233Search in Google Scholar
[51] G. F. Smith, On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int. J. Eng. Sci. 9 (1971), 899–916.10.1016/0020-7225(71)90023-1Search in Google Scholar
[52] H. Jeffreys, Cartesian Tensor, Cambridge University Press, Cambridge, 1957.Search in Google Scholar
[53] J. C. Maxwell, On the dynamical theory of gases, C. R. Acad. Sci. Paris 2477 (1958), 2103–2107.Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Editorial
- Research Articles
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- Ergodicity, Maximum Entropy Production, and Steepest Entropy Ascent in the Proofs of Onsager’s Reciprocal Relations
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- Novikov Engine with Fluctuating Heat Bath Temperature
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- Non-Equilibrium Dislocation Dynamics in Semiconductor Crystals and Superlattices
- A Thermodynamical Theory with Internal Variables Describing Thermal Effects in Viscous Fluids
- A Thermodynamically Consistent Approach to Phase-Separating Viscous Fluids
Articles in the same Issue
- Frontmatter
- Editorial
- Research Articles
- GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems
- Ergodicity, Maximum Entropy Production, and Steepest Entropy Ascent in the Proofs of Onsager’s Reciprocal Relations
- Entropy Generation Minimization in Dimethyl Ether Synthesis: A Case Study
- Systematic Constraint Selection Strategy for Rate-Controlled Constrained-Equilibrium Modeling of Complex Nonequilibrium Chemical Kinetics
- General Properties for an Agrawal Thermal Engine
- Novikov Engine with Fluctuating Heat Bath Temperature
- From Finite Time to Finite Physical Dimensions Thermodynamics: The Carnot Engine and Onsager’s Relations Revisited
- Non-Equilibrium Dislocation Dynamics in Semiconductor Crystals and Superlattices
- A Thermodynamical Theory with Internal Variables Describing Thermal Effects in Viscous Fluids
- A Thermodynamically Consistent Approach to Phase-Separating Viscous Fluids