Exact harmonic solutions to Guyer–Krumhansl-type equation and application to heat transport in thin films

  • K. Zhukovsky
  • D. Oskolkov
Original Article


A system of hyperbolic-type inhomogeneous differential equations (DE) is considered for non-Fourier heat transfer in thin films. Exact harmonic solutions to Guyer–Krumhansl-type heat equation and to the system of inhomogeneous DE are obtained in Cauchy- and Dirichlet-type conditions. The contribution of the ballistic-type heat transport, of the Cattaneo heat waves and of the Fourier heat diffusion is discussed and compared with each other in various conditions. The application of the study to the ballistic heat transport in thin films is performed. Rapid evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow evolution of its diffusive counterpart. The effect of the ballistic quasi-temperature component on the evolution of the complete quasi-temperature is explored. In this context, the influence of the Knudsen number and of Cauchy- and Dirichlet-type conditions on the evolution of the temperature distribution is explored. The comparative analysis of the obtained solutions is performed.


Non-Fourier heat conduction Guyer–Krumhansl equation Thin film Knudsen number 


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  1. 1.
    Fourier, J.P.J.: The Analytical Theory of Heat. Cambridge University Press, London (1878)zbMATHGoogle Scholar
  2. 2.
    Kovács, R., Ván, P.: Models of ballistic propagation of heat at low temperatures. Int. J. Thermophys. 37(9), 95 (2016)ADSCrossRefGoogle Scholar
  3. 3.
    Both, S., Czél, B., Fülöp, T., Ván, P., Verhás, J.: Deviation from the Fourier law in room-temperature heat pulse experiments. J. Nonequilib. Thermodyn. 41(1), 41–48 (2016)ADSCrossRefGoogle Scholar
  4. 4.
    Van, P., Berezovski, A., Fulop, T., Grof, Gy, Kovacs, R., Lovas, A., Verhas, J.: Guyer–Krumhansl-type heat conduction at room temperature. EPL 118(5), 50005 (2017)ADSCrossRefGoogle Scholar
  5. 5.
    Onsager, L.: Reciprocal relations in irreversible processes. Phys. Rev. 37, 119 (1931)CrossRefzbMATHGoogle Scholar
  6. 6.
    Peshkov, V.: Second sound in Helium II. J. Phys. (Mosc.) 8, 381 (1944)Google Scholar
  7. 7.
    Ackerman, C.C., Guyer, R.A.: Temperature pulses in dielectric solids. Ann. Phys. 50(1), 128–185 (1968)ADSCrossRefGoogle Scholar
  8. 8.
    Ackerman, C.C., Overton, W.C.: Second sound in solid helium-3. Phys. Rev. Lett. 22(15), 764 (1969)ADSCrossRefGoogle Scholar
  9. 9.
    McNelly, T.F., Rogers, S.J., Channin, D.J., Rollefson, R., Goubau, W.M., Schmidt, G.E., Krumhansl, J.A., Pohl, R.O.: Heat pulses in NaF: onset of second sound. Phys. Rev. Lett. 24(3), 100 (1970)ADSCrossRefGoogle Scholar
  10. 10.
    Narayanamurti, V., Dynes, R.D.: Observation of second sound in Bismuth. Phys. Rev. Lett. 26, 1461–1465 (1972)ADSCrossRefGoogle Scholar
  11. 11.
    Cattaneo, C.: Sur une forme de l’equation de la chaleur eliminant le paradoxe d’une propagation instantanee. C. R. Acad. Sci. Paris 247, 431–433 (1958)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Terman, Frederick Emmons: Radio Engineers’ Handbook, 1st edn. McGraw-Hill, New York (1943)Google Scholar
  13. 13.
    Moosaie, A.: Non-Fourier heat conduction in a finite medium with insulated boundaries and arbitrary initial conditions. Int. Commun. Heat Mass Transf. 35, 103–111 (2008)CrossRefGoogle Scholar
  14. 14.
    Ahmadikia, H., Rismanian, M.: Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions. J. Mech. Sci. Technol. 25(11), 2919–2926 (2011)CrossRefGoogle Scholar
  15. 15.
    Yen, C.C., Wu, C.Y.: Modelling hyperbolic heat conduction in a finite medium with periodic thermal disturbance and surface radiation. Appl. Math. Model. 27, 397–408 (2003)CrossRefzbMATHGoogle Scholar
  16. 16.
    Lewandowska, M.: Hyperbolic heat conduction in the semi-infinite body with a time-dependent laser heat source. Heat Mass Transf. 37(4–5), 333–342 (2001)ADSCrossRefGoogle Scholar
  17. 17.
    Lewandowska, M., Malinowski, L.: An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides. Int. Commun. Heat Mass Transf. 33, 61–69 (2006)CrossRefGoogle Scholar
  18. 18.
    Saedodin, S., Torabi, M.: Analytical solution of non-Fourier heat conduction in cylindrical coordinates. Int. Rev. Mech. Eng. 3, 726–732 (2009)Google Scholar
  19. 19.
    Challamel, N., Grazide, C., Picandet, V., Perrot, A., Zhang, Y.: A nonlocal Fourier’s law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices. C. R. Mec. 344, 388–401 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    Saedodin, S., Torabi, M.: Algebraically explicit analytical solution of three-dimensional hyperbolic heat conduction equation. Adv. Theor. Appl. Mech. 3(8), 369–383 (2010)zbMATHGoogle Scholar
  21. 21.
    Guyer, R.A., Krumhansl, J.A.: Solution of the linearized phonon Boltzmann equation. Phys. Rev. 148, 766–778 (1966)ADSCrossRefGoogle Scholar
  22. 22.
    Guyer, R.A., Krumhansl, J.A.: Thermal conductivity, second sound and phonon hydrodynamic phenomena in non-metallic crystals. Phys. Rev. 148, 778–788 (1966)ADSCrossRefGoogle Scholar
  23. 23.
    Lebon, G., Machrafi, H., Gremela, M., Dubois, Ch.: An extended thermodynamic model of transient heat conduction at sub-continuum scales. Proc. R. Soc. A 467, 3241–3256 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  25. 25.
    Zhukovsky, K.V.: Operational method of solution of linear non-integer ordinary and partial differential equations. SpringerPlus 5, 119 (2016). CrossRefGoogle Scholar
  26. 26.
    Zhukovsky, K.: Operational approach and solutions of hyperbolic heat conduction equations. Axioms 5, 28 (2016)CrossRefGoogle Scholar
  27. 27.
    Zhukovsky, K.V., Srivastava, H.M.: Analytical solutions for heat diffusion beyond Fourier law. Appl. Math. Comput. 293, 423–437 (2017)MathSciNetGoogle Scholar
  28. 28.
    Zhukovsky, K.V.: Violation of the maximum principle and negative solutions with pulse propagation in Guyer–Krumhansl model. Int. J. Heat Mass Transf. 98, 523–529 (2016)CrossRefGoogle Scholar
  29. 29.
    Zhukovsky, K.V.: Exact solution of Guyer–Krumhansl type heat equation by operational method. Int. J. Heat Mass Transf. 96, 132–144 (2016)CrossRefGoogle Scholar
  30. 30.
    Zhukovsky, K.: Exact negative solutions for Guyer–Krumhansl type equation and the violation of the maximum principle. Entropy 19, 440 (2017)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhukovsky, K.V.: A harmonic solution for the hyperbolic heat conduction equation and its relationship to the Guyer–Krumhansl equation. Mosc. Univ. Phys. Bull. 73(1), 45–52 (2018). Google Scholar
  32. 32.
    Zhukovsky, K.: Exact harmonic solution to ballistic type heat propagation in thin films and wires. Int. J. Heat Mass Transf. 120, 944–955 (2018)CrossRefGoogle Scholar
  33. 33.
    Boucetta, A., Ghodbane, H., Ayad, M.Y., Bahri, M.: A review on the performance and modelling of proton exchange membrane fuel cells. AIP Conf. Proc. 1758, 030019 (2016)CrossRefGoogle Scholar
  34. 34.
    Arato, E., Pinna, M., Mazzoccoli, M., Bosio, B.: Gas-phase mass-transfer resistances at polymeric electrolyte membrane fuel cells electrodes: theoretical analysis on the effectiveness of interdigitated and serpentine flow arrangements. Energies 9(4), 229 (2016)CrossRefGoogle Scholar
  35. 35.
    Veltzke, T., Kiewidt, L., Thöming, J.: Multicomponent gas diffusion in nonuniform tubes. AIChE J. 61(4), 1404–1412 (2015)CrossRefGoogle Scholar
  36. 36.
    Maidhily, M., Rajalakshmi, N., Dhathathreyan, K.S.: Electrochemical impedance spectroscopy as a diagnostic tool for the evaluation of flow field geometry in polymer electrolyte membrane fuel cells. Renew. Energy 51, 79–84 (2013)CrossRefGoogle Scholar
  37. 37.
    St-Pierre, J.: Hydrogen mass transport in fuel cell gas diffusion electrodes. Fuel Cells 11(2), 263–273 (2011)CrossRefGoogle Scholar
  38. 38.
    Misran, E., Daud, W.R.W., Majlan, E.H.: Review on serpentine flow field design for PEM fuel cell system. Key Eng. Mater. 447(448), 559–563 (2010)Google Scholar
  39. 39.
    Kim, S., Hong, I.: Effect of flow field design on the performance of a proton exchange membrane fuel cell (PEMFC). J. Ind. Eng. Chem. 13(5), 864–869 (2007)Google Scholar
  40. 40.
    Zhukovsky, K., Pozio, A.: Maximum current limitations of the PEM fuel cell with serpentine gas supply channels. J. Power Sources 130, 95–105 (2004)ADSCrossRefGoogle Scholar
  41. 41.
    Zhukovsky, K.V.: Three dimensional model of gas transport in a porous diffuser of a polymer electrolyte fuel cell. AIChE J. 49(12), 3029–3036 (2003)CrossRefGoogle Scholar
  42. 42.
    Zhukovsky, K.: Modeling of the current limitations of PEFC. AIChE J. 52(7), 2356–2366 (2006)CrossRefGoogle Scholar
  43. 43.
    Weber, A.Z., Newman, J.: Modeling transport in polymer-electrolyte fuel cells. Chem. Rev. 104(10), 4679–4726 (2004)CrossRefGoogle Scholar
  44. 44.
    Kawase, M., Sato, K., Mitsui, R., Asonuma, H., Kageyama, M., Yamaguchi, K., Inoue, G.: Electrochemical reaction engineering of polymer electrolyte fuel cell. AIChE J. 63(1), 249–256 (2017)CrossRefGoogle Scholar
  45. 45.
    Zhukovsky, K.V.: A method of inverse differential operators using ortogonal polynomials and special functions for solving some types of differential equations and physical problems. Mosc. Univ. Phys. Bull. 70(2), 93–100 (2015)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Zhukovsky, K.: Solution of some types of differential equations: operational calculus and inverse differential operators. Sci. World J. 2014, 1–8 (2014). (Article ID 454865)CrossRefGoogle Scholar
  47. 47.
    Zhukovsky, K.V.: Solving evolutionary-type differential equations and physical problems using the operator method. Theor. Math. Phys. 190(1), 52–68 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Zhukovsky, K.: Operational solution for some types of second order differential equations and for relevant physical problems. J. Math. Anal. Appl. 446(11), 628–647 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Dattoli, G., Srivastava, H.M., Zhukovsky, K.V.: Operational methods and differential equations with applications to initial-value problems. Appl. Math. Comput. 184, 979–1001 (2007)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Zhukovsky, K.V.: Operational solution of differential equations with derivatives of non-integer order, Black–Scholes type and heat conduction. Mosc. Univ. Phys. Bull. 71(3), 237–244 (2016)ADSCrossRefGoogle Scholar
  51. 51.
    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Wiley, New York (1984)zbMATHGoogle Scholar
  52. 52.
    Dattoli, G., Srivastava, H.M., Zhukovsky, K.V.: Orthogonality properties of the Hermite and related polynomials. J. Comput. Appl. Math. 182(1), 165–172 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Dattoli, G., Srivastava, H.M., Zhukovsky, K.V.: A new family of integral transforms and their applications. Integral Transform. Spec. Funct. 17(1), 31–37 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Taitel, Y.: On the parabolic, hyperbolic and discrete formulation of the heat conduction equation. Int. J. Heat Mass Transf. 15, 369–371 (1972)CrossRefGoogle Scholar
  55. 55.
    Barletta, A., Zanchini, E.: Hyperbolic heat conduction and local equilibrium: a second law analysis. Int. J. Heat Mass Transf. 40(5), 1007–1016 (1997)CrossRefzbMATHGoogle Scholar
  56. 56.
    Zanchini, E.: Hyperbolic heat conduction theories and nondecreasing entropy. Phys. Rev. B 60(2), 991–997 (1999)ADSCrossRefGoogle Scholar
  57. 57.
    Körner, C., Bergmann, H.W.: The physical defects of the hyperbolic heat conduction equation. Appl. Phys. A 67, 397–401 (1998)ADSCrossRefGoogle Scholar
  58. 58.
    Bright, T.J., Zhang, Z.M.: Common misperceptions of the hyperbolic heat equation. J. Thermophys. Heat Transf. 23, 601–607 (2009)CrossRefGoogle Scholar
  59. 59.
    Jha, K.K., Narasimhan, A.: Three-dimensional bio-heat transfer simulation of sequential and simultaneous retinal laser irradiation. Int. J. Therm. Sci. 50, 1191–1198 (2011)CrossRefGoogle Scholar
  60. 60.
    Zhang, L., Shang, X.: Analytical solution to non-Fourier heat conduction as a laser beam irradiating on local surface of a semi-infinite medium. Int. J. Heat Mass Transf. 85, 772–780 (2015)CrossRefGoogle Scholar
  61. 61.
    Sasmal, A., Mishra, S.C.: Analysis of non-Fourier conduction and radiation in a differentially heated 2-D square cavity. Int. J. Heat Mass Transf. 79, 116–125 (2014)CrossRefGoogle Scholar
  62. 62.
    Narasimhan, A., Sadasivam, S.: Non-Fourier bio heat transfer modelling of thermal damage during retinal laser irradiation. Int. J. Heat Mass Transf. 60, 591–597 (2013)CrossRefGoogle Scholar
  63. 63.
    Zhukovskij, K.V.: Gas flow in long microchannels. Vestn. Mosk. Univ. Ser. 3 Fiz. Astron. 3, 49–54 (2001)ADSGoogle Scholar
  64. 64.
    Parker, W.J., Jenkins, R.J., Butler, C.P., Abbott, G.L.: Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity. J. Appl. Phys. 32(9), 1679 (1961)ADSCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of PhysicsM.V. Lomonosov Moscow State UniversityMoscowRussia

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