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Local versus nonlocal constitutive theories of nonequilibrium thermodynamics: the Guyer-Krumhansl equation as an example. (English) Zbl 1479.80003

On the example of the celebrated Grad’s 13-moment system of kinetic theory of rarefied gases and phonon hydrodynamics, it is proved that the constitutive equations of nonequilibrium thermodynamics must be nonlocal. A thermodynamic model of Guyer-Krumhansl heat-transport equation is derived within the frame of weakly nonlocal Rational Thermodynamics. The constitutive equation for the entropy flux is analyzed as well. Some nonlinear generalizations of Maxwell-Cattaneo equation are studied in view of the experiments on thermal wave propagation. The paper arose from a critique of the statement [T. Ruggeri, Q. Appl. Math. 70, No. 3, 597–611 (2012; Zbl 1421.74011)] that the constitutive equations of continuum thermodynamics cannot be nonlocal, and that the Navier-Stokes-Fourier system can be obtained by the Grad system by a limit procedure. For the exemplary Guyer-Krumhansl system, both the statements have been refuted by proving that (i) the limit process referred there to obtain the Navier-Stokes-Fourier system by the classical Grad system, holds only for a very small subset of all known fluids; (ii) both the Navier-Stokes-Fourier and the Guyer-Krumhansl systems can be obtained by some new closure processes of moments system which put the models beyond the Rational Extended Thermodynamics (RET). The models developed in this paper represent practical examples of the efficacy of nonlocal constitutive equations, which allow to represent both hyperbolic and parabolic situations, without needing of any approximation and regularization procedure.

MSC:

80A17 Thermodynamics of continua
80A10 Classical and relativistic thermodynamics
80A19 Diffusive and convective heat and mass transfer, heat flow
82C35 Irreversible thermodynamics, including Onsager-Machlup theory
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
78A25 Electromagnetic theory (general)
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35Q82 PDEs in connection with statistical mechanics
35Q20 Boltzmann equations
35Q60 PDEs in connection with optics and electromagnetic theory

Citations:

Zbl 1421.74011
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References:

[1] Grad, H., On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2, 331-407 (1949) · Zbl 0037.13104
[2] Sellitto, A.; Cimmelli, VA; Jou, D., Mesoscopic Theories of Heat Transport in Nanosystems (2016), Berlin: Springer, Berlin · Zbl 1339.80001
[3] Rogolino, P.; Sellitto, A.; Cimmelli, V., Influence of nonlinear effects on the efficiency of a thermoelectric generator, Z. Angew. Math. Phys., 66, 2829-2842 (2015) · Zbl 1326.80007
[4] Sellitto, A., Carlomagno, I., Di Domenico, M.: Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model. Z. Angew. Math. Phys. 72, Article: 7 (2021) · Zbl 1460.80001
[5] Cattaneo, C., Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3, 83-101 (1948) · Zbl 0035.26203
[6] Chen, PJ; Nunziato, JW, Thermodynamic restrictions on the relaxation functions of the theory of heat conduction with finite wave speeds, Z. Angew. Math. Phys., 25, 791-798 (1974)
[7] Guyer, RA; Krumhansl, JA, Solution of the linearized phonon Boltzmann equation, Phys. Rev., 148, 766-778 (1966)
[8] Fourier, JBJ, The Analytical Theory of Heat (2009), Cambridge: Cambridge University Press, Cambridge
[9] Cimmelli, VA, Different thermodynamic theories and different heat conduction laws, J. Non-Equilib. Thermodyn., 29, 299-333 (2004) · Zbl 1111.82305
[10] Cimmelli, VA; Jou, D.; Ruggeri, T.; Ván, P., Entropy principle and recent results in non-equilibrium theories, Entropy, 16, 1756-1807 (2014) · Zbl 1338.82034
[11] Müller, I.; Ruggeri, T., Rational Extended Thermodynamics (1998), New York: Springer, New York · Zbl 0895.00005
[12] Coleman, BD; Noll, W., The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal., 13, 167-178 (1963) · Zbl 0113.17802
[13] Jou, D.; Casas-Vázquez, J.; Lebon, G., Extended Irreversible Thermodynamics (2010), Berlin: Springer, Berlin · Zbl 1185.74002
[14] Fichera, G., Is the Fourier theory of heat propagation paradoxical?, Rend. Circ. Mat. Palermo, 41, 5-28 (1992) · Zbl 0756.35035
[15] Cimmelli, VA, On the causality requirement for diffusive-hyperbolic systems in non-equilibrium thermodynamics, J. Non-Equilib. Thermodyn., 34, 125-139 (2009) · Zbl 1195.80001
[16] Rogolino, P.; Kovács, R.; Ván, P.; Cimmelli, VA, Generalized heat-transport equations: parabolic and hyperbolic models, Continuum Mech. Thermodyn., 30, 1245-1258 (2018) · Zbl 1439.80007
[17] Rogolino, P., Cimmelli, V.A.: Differential consequences of balance laws in extended irreversible thermodynamics of rigid heat conductors. Proc. R. Soc. A 475, 20180482 (19 pages) (2018) · Zbl 1472.80002
[18] Öncü, TS; Moodie, TB, On the constitutive relations for second sound in elastic solids, Arch. Ration. Mech. Anal., 121, 87-99 (1992) · Zbl 0774.73007
[19] Cimmelli, VA; Sellitto, A.; Triani, V., A generalized Coleman-Noll procedure for the exploitation of the entropy principle, Proc. R. Soc. A, 466, 911-925 (2010) · Zbl 1195.82068
[20] Cimmelli, VA; Sellitto, A.; Triani, V., A new perspective on the form of the first and second laws in rational thermodynamics: Korteweg fluids as an example, J. Non-Equilib. Thermodyn., 35, 251-265 (2010) · Zbl 1213.80008
[21] Jackson, HE; Walker, CT, Thermal conductivity, second sound, and phonon-phonon interactions in NaF, Phys. Rev. Lett., 3, 1428-1439 (1971)
[22] Jackson, HE; Walker, CT; McNelly, TF, Second sound in NaF, Phys. Rev. Lett., 25, 26-28 (1970)
[23] Narayanamurti, V.; Dynes, RD, Observation of second sound in Bismuth, Phys. Rev. Lett., 28, 1461-1465 (1972)
[24] Struchtrup, H.; Torrilhon, M., Regularization of Grad 13 moment equations: derivation and linear analysis, Phys. Fluids, 15, 2668-2680 (2003) · Zbl 1186.76504
[25] Fryer, M.; Struchtrup, H., Moment model and boundary conditions for energy transport in the phonon gas, Continuum Mech. Thermodyn., 26, 593-618 (2014) · Zbl 1341.82037
[26] Reissland, JA, The Physics of Phonons (1973), London: Wiley, London
[27] Larecki, W., Banach, Z.: Consistency of the phenomenological theories of wave-type heat transport with the hydrodynamics of a phonon gas. J. Phys. A Math. Theor. 43, 385501 (24 pages) (2010) · Zbl 1446.82075
[28] Ruggeri, T., Can constitutive equations be represented by non-local equations?, Q. Appl. Math., LXX, 597-611 (2012) · Zbl 1421.74011
[29] Verhás, J., On the entropy current, J. Non-Equilib. Thermodyn., 8, 201-206 (1983)
[30] Sellitto, A., Cimmelli, V.A., Jou, D.: Entropy flux and anomalous axial heat transport at the nanoscale. Phys. Rev. B 87, 054302 (7 pages) (2013)
[31] Ván, P.; Fülöp, T., Universality in heat conduction theory: weakly nonlocal thermodynamics, Ann. Phys., 524, 470-478 (2012) · Zbl 1263.80010
[32] Cimmelli, VA; Frischmuth, K., Gradient generalization to the extended thermodynamic approach and diffusive-hyperbolic heat conduction, Phys. B, 400, 257-265 (2007)
[33] Lebon, G.; Jou, D.; Casas-Vázquez, J.; Muschik, W., Weakly nonlocal and nonlinear heat transport in rigid solids, J. Non-Equilib. Thermodyn., 23, 176-191 (1998) · Zbl 0912.73006
[34] Cimmelli, V.A., Sellitto, A., Jou, D.: Nonequilibrium temperatures, heat waves, and nonlinear heat transport equations. Phys. Rev. B 81, 054301 (9 pages) (2010) · Zbl 1213.80011
[35] Verhás, J., Thermodynamics and Rheology (1997), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 0878.73001
[36] Maugin, GA; Muschik, W., Thermodynamics with Internal Variables. Part I. General Concepts, J. Non-Equilib. Thermodyn., 19, 217-249 (1994) · Zbl 0808.73006
[37] Cimmelli, VA, Boundary conditions in the presence of internal variables, J. Non-Equilib. Thermodyn., 27, 327-334 (2002) · Zbl 1161.80301
[38] Cimmelli, VA, Weakly nonlocal thermodynamics of anisotropic rigid heat conductors revisited, J. Non-Equilib. Thermodyn., 36, 285-309 (2011) · Zbl 1268.80001
[39] Liu, I-S, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Ration. Mech. Anal., 46, 131-148 (1972) · Zbl 0252.76003
[40] Coleman, BD; Newman, DC, Implication of a nonlinearity in the theory of second sound in solids, Phys. Rev. B, 37, 1492-1498 (1988)
[41] Cimmelli, VA; Frischmuth, K., Determination of material functions through second sound measurements in a hyperbolic heat conduction theory, Math. Comput. Model., 24, 19-28 (1996) · Zbl 0883.65100
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