Abstract
Dual-phase-lag equation for heat conduction is analyzed from the point of view of non-equilibrium thermodynamics. Its first-order Taylor series expansion is consistent with the second law as long as the two relaxation times are not negative.
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Ackerman, C.C., Bertman, B., Fairbank, H.A., Guyer, R.A.: Second sound in solid Helium. Phys. Rev. Lett. 16(18), 789–791 (1966)
Afrin, N., Zhang, Y., Chen, J.K.: Thermal lagging in living biological tissue based on nonequilibrium heat transfer between tissue, arterial and venous bloods. Int. J. Heat Mass Transf. 54(11), 2419–2426 (2011)
Afrin, N., Zhou, J., Zhang, Y., Tzou, D.Y., Chen, J.K.: Numerical simulation of thermal damage to living biological tissues induced by laser irradiation based on a generalized dual phase lag model. Numer. Heat Transf. Part A: Appl. 61(7), 483–501 (2012)
Akbarzadeh, A.H., Pasini, D.: Phase-lag heat conduction in multilayered cellular media with imperfect bonds. Int. J. Heat Mass Transf. 75, 656–667 (2014)
Antaki, P.J.: New interpretation of non-Fourier heat conduction in processed meat. J. Heat Transf. 127(2), 189–193 (2005)
Berezovski, A., Ván, P.: Microinertia and internal variables. Contin Mech Thermodyn 28, 1027–1037 arXiv preprint arXiv:1504.03485 (2016)
Both, S., Czél, B., Fülöp, T., Gróf, G., Gyenis, Á., Kovács, R., Ván, P., Verhás, J.: Deviation from the Fourier law in room-temperature heat pulse experiments. J. Non-Equilib. Thermodyn. 41(1), 41–48 (2016)
Cattaneo, C.: Sur une forme de lequation de la chaleur eliminant le paradoxe dune propagation instantanee. Compt. Rendu. Hebd. Des Seances De L’Acad. Des Sci. 247(4), 431–433 (1958)
Ciancio, V., Cimmelli, V.A., Ván, P.: On the evolution of higher order fluxes in non-equilibrium thermodynamics. Math. Comput. Modell. 45, 126–136 (2007). Cond-mat/0407530
Dreher, M., Quintanilla, R., Racke, R.: Ill-posed problems in thermomechanics. Appl. Math. Lett. 22(9), 1374–1379 (2009)
Dreyer, W., Struchtrup, H.: Heat pulse experiments revisited. Contin. Mech. Thermodyn. 5, 3–50 (1993)
Ezzat, M.A., El-Bary, A.A., Fayik, M.A.: Fractional Fourier law with three-phase lag of thermoelasticity. Mech. Adv. Mater. Struct. 20(8), 593–602 (2013)
Fabrizio, M., Franchi, F.: Delayed thermal models: stability and thermodynamics. J. Therm. Stress. 37(2), 160–173 (2014)
Fabrizio, M., Lazzari, B.: Stability and second law of thermodynamics in dual-phase-lag heat conduction. Int. J. Heat Mass Transf. 74, 484–489 (2014)
Fabrizio, M., Lazzari, B., Tibullo, V.: Stability and thermodynamic restrictions for a dual-phase-lag thermal model. J. Non-Equilib. Thermodyn. (2017). Published Online:2017/01/10
Guyer, R.A., Krumhansl, J.A.: Solution of the linearized phonon Boltzmann equation. Phys. Rev. 148(2), 766–778 (1966)
Guyer, R.A., Krumhansl, J.A.: Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals. Phys. Rev. 148, 778–788 (1966)
Herwig, H., Beckert, K.: Experimental evidence about the controversy concerning Fourier or non-Fourier heat conduction in materials with a nonhomogeneous inner structure. Heat Mass Transf 36(5), 387–392 (2000)
Jackson, H.E., Walker, C.T., McNelly, T.F.: Second sound in NaF. Phys. Rev. Lett. 25(1), 26–28 (1970)
Joseph, D.D., Preziosi, L.: Heat waves. Rev. Mod. Phys. 61(1), 41 (1989)
Joseph, D.D., Preziosi, L.: Addendum to the paper on heat waves. Rev. Mod. Phys. 62(2), 375–391 (1990)
Jou, D., Cimmelli, V.A.: Constitutive equations for heat conduction in nanosystems and non-equilibrium processes: an overview. Commun. Appl. Ind. Math. 7(2), 196–222 (2016)
Kovács, R., Ván, P.: Generalized heat conduction in heat pulse experiments. Int. J. Heat Mass Transf. 83, 613–620 (2015)
Lane, C.T., Fairbank, H.A., Fairbank, W.M.: Second sound in liquid Helium II. Phys. Rev. 71, 600–605 (1947)
Liu, K.C., Chen, H.T.: Investigation for the dual phase lag behavior of bio-heat transfer. Int. J. Therm. Sci. 49(7), 1138–1146 (2010)
Mariano, P.M.: Finite speed heat propagation as a consequence of microstructural events. In: 14th Joint European Thermodynamics Conference
Matolcsi, T.: Ordinary Thermodynamics. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest (2005)
Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. 157, 49–88 (1867)
Mitra, K., Kumar, S., Vedevarz, A., Moallemi, M.K.: Experimental evidence of hyperbolic heat conduction in processed meat. J. Heat Transf. 117(3), 568–573 (1995)
Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, Berlin (1998)
Nyíri, B.: On the extension of the governing principle of dissipative processes to nonlinear constitutive equations. Acta Phys. Hung. 66(1), 19–28 (1989)
Nyíri, B.: On the entropy current. J. Non-Equilib. Thermodyn. 16(2), 179–186 (1991)
Peshkov, V.: Second sound in Helium II. J. Phys. (Moscow) 8, 381 (1944)
Quintanilla, R., Racke, R.: Qualitative aspects in dual-phase-lag heat conduction. Proc. R. Soc. Lond A: Math. Phys. Eng. Sci. 463(2079), 659–674 (2007)
Roetzel, W., Putra, N., Das, S.K.: Experiment and analysis for non-Fourier conduction in materials with non-homogeneous inner structure. Int. J. Therm. Sci. 42(6), 541–552 (2003)
Rogers, S.J.: Transport of heat and approach to second sound in some isotropically pure alkali-halide crystals. Phys. Rev. B 3(4), 1440 (1971)
Rukolaine, S.A.: Unphysical effects of the dual-phase-lag model of heat conduction. Int. J. Heat Mass Transf. 78, 58–63 (2014)
Rukolaine, S.A.: Unphysical effects of the dual-phase-lag model of heat conduction: higher-order approximations. Int. J. Therm. Sci. 113, 83–88 (2017)
Sahoo, N., Ghosh, S., Narasimhan, A., Das, S.K.: Investigation of non-Fourier effects in bio-tissues during laser assisted photothermal therapy. Int. J. Therm. Sci. 76, 208–220 (2014)
Sellitto, A., Cimmelli, V.A., Jou, D.: Mesoscopic Theories of Heat Transport in Nanosystems, vol. 6. Springer, Berlin (2016)
Taitel, Y.: On the parabolic, hyperbolic and discrete formulation of the heat conduction equation. Int. J. Heat Mass Transf. 15, 369–371 (1972)
Tang, D., Araki, N., Yamagishi, N.: Transient temperature responses in biological materials under pulsed IR irradiation. Heat Mass Transf. 43(6), 579–585 (2007)
Tzou, D.Y.: A unified field approach for heat conduction from macro- to micro-scales. J. Heat Transf. 117(1), 8–16 (1995)
Tzou, D.Y.: Macro- to Micro-scale Heat Transfer: The Lagging Behavior. CRC Press, Boca Raton (1996)
Ván, P.: Other dynamic laws in thermodynamics. Phys. Essays 8(4), 457–465 (1995)
Ván, P.: Generic stability of dissipative non-relativistic and relativistic fluids. J. Stat. Mech. Theory Exp. p. 02054 (2009). ArXiv:0811.0257
Ván, P., Berezovski, A., Fülöp, T., Gróf, G., Kovács, R., Lovas, Á., Verhás, J.: Guyer–Krumhansl-type heat conduction at room temperature. EPL 118(5), 50,005 (2017). ArXiv:1704.00341v1
Ván, P., Bíró, T.S.: Relativistic hydrodynamics - causality and stability. Eur. Phys. J. Spec. Top. 155, 201–212 (2008). ArXiv:0704.2039v2
Verhás, J.: Thermodynamics and Rheology. Akadémiai Kiadó-Kluwer Academic Publisher, Alphen aan den Rijn (1997)
Vernotte, P.: Les paradoxes de la théorie continue de léquation de la chaleur. Comptes Rendus Hebd. Des Seances De L’Acad. Des Sci. 246(22), 3154–3155 (1958)
Ván, P.: Weakly nonlocal irreversible thermodynamics—the Guyer–Krumhansl and the Cahn–Hilliard equations. Phys. Lett. A 290(1–2), 88–92 (2001)
Ván, P., Fülöp, T.: Universality in heat conduction theory—weakly nonlocal thermodynamics. Ann. der Phys. (Berlin) 524(8), 470–478 (2012)
Wang, M., Yang, N., Guo, Z.Y.: Non-Fourier heat conductions in nanomaterials. J. Appl. Phys. 110(6), 064,310 (2011)
Xu, F., Seffen, K.A., Lu, T.J.: Non-Fourier analysis of skin biothermomechanics. Int. J. Heat Mass Transf 51(9), 2237–2259 (2008)
Zhang, Y.: Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues. Int. J. Heat Mass Transf 52(21), 4829–4834 (2009)
Zhou, J., Chen, J.K., Zhang, Y.: Dual-phase lag effects on thermal damage to biological tissues caused by laser irradiation. Comput. Biol. Med. 39(3), 286–293 (2009)
Zhukovsky, K.V.: Exact solution of Guyer–Krumhansl type heat equation by operational method. Int. J. Heat Mass Transf 96, 132–144 (2016)
Zhukovsky, K.V.: Operational approach and solutions of hyperbolic heat conduction equations. Axioms 5(4), 28 (2016)
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Communicated by Andreas Öchsner.
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Kovács, R., Ván, P. Thermodynamical consistency of the dual-phase-lag heat conduction equation. Continuum Mech. Thermodyn. 30, 1223–1230 (2018). https://doi.org/10.1007/s00161-017-0610-x
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DOI: https://doi.org/10.1007/s00161-017-0610-x