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Nonlinear heat wave propagation in a rigid thermal conductor

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Abstract

We present new nonlinear, one-dimensional equations of extended thermodynamics for temperature and heat flux that describe damped heat wave propagation and predict dependence of the second sound velocity on temperature and heat flux in a rigid thermal conductor. The aim of the present work is to investigate the implications of the considered nonlinearities on the process of heat propagation, especially that such nonlinear effects cannot be disregarded for nanosystems and in small-sized heat conductors, when temperature or heat flux variations are not always proportional to gradients. The characteristics of the nonlinear system of equations are investigated, and a formula for the breaking distance is obtained. For illustration, these equations are solved in the linear approximation by separation of variables, then in the nonlinear case by the homotopy perturbation method in the half-space, and numerically in a bounded interval under concrete initial and boundary conditions. The solutions are discussed in detail and presented graphically. They illustrate the effect of the different material parameters on the propagation of thermal waves and may be used to evaluate some material parameters. The proposed equations provide a closer insight into the thermal interactions in rigid thermal conductors.

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Abbreviations

C :

Specific heat capacity

K :

Coefficient of heat conduction

\(K_0\) :

Coefficient of heat conduction at reference temperature

\(K_1, K_2\) :

Coupling constants for the coefficient of heat conduction

\(L_0\) :

Characteristic length

Q :

Heat flux component

\(Q_0\) :

Characteristic heat flux

\(T_0\) :

Characteristic time

\(\mu _1, \mu _2\) :

Coupling constants

\(\theta \) :

Absolute temperature as measured from a reference temperature \(\varTheta _0\)

\(\varTheta _0\) :

Reference temperature

\(\rho \) :

Mass density

\(\tau _0\) :

Thermal relaxation time at reference temperature

\(\tau _1, \tau _2\) :

Coupling constants for the thermal relaxation time

References

  1. Jou, D., Casas-Vázquez, J., Lebon, G.: Extended irreversible thermodynamics. Rep. Prog. Phys. 51, 1105–1179 (1988)

    Article  MathSciNet  Google Scholar 

  2. Green, A.E., Naghdi, P.M.: A re-examination of basic postulates of thermomechanics. Proc. R. Soc. Ser. A 432, 171–194 (1991)

    MathSciNet  MATH  Google Scholar 

  3. Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253–264 (1992)

    Article  MathSciNet  Google Scholar 

  4. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)

    Article  MathSciNet  Google Scholar 

  5. Green, A.E., Naghdi, P.M.: A unified procedure for construction of theories of deformable media. I. Classical continuum physics. Proc. R. Soc. Lond. 448, 335–356 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Green, A.E., Naghdi, P.M.: A unified procedure for construction of theories of deformable media. II. Generalized continua. Proc. R. Soc. Lond. 448, 357–377 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Green, A.E., Naghdi, P.M.: A unified procedure for construction of theories of deformable media. III. Mixture of interacting continua. Proc. R. Soc. Lond. 448, 379–388 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Bargmann, S., Favata, A., Podio-Guidugli, P.: A revised exposition of the Green–Naghdi theory of heat propagation. J. Elast. 114, 143–154 (2014). https://doi.org/10.1007/s10659-013-9431-8

    Article  MathSciNet  MATH  Google Scholar 

  9. Cimmelli, V.A., Jou, D., Ruggeri, T., Ván, P.: Entropy principle and recent results in non-equilibrium theories. Entropy 16, 1756–1807 (2014). https://doi.org/10.3390/e16031756

    Article  MathSciNet  MATH  Google Scholar 

  10. Sellitto, A., Cimmelli, V.A.: Heat-pulse propagation in thermoelastic systems: application to graphene. Acta Mech. 230(1), 121–136 (2019). https://doi.org/10.1007/s00707-018-2274-4

    Article  MathSciNet  Google Scholar 

  11. Coleman, B.D., Mizel, V.: Thermodynamics and departures from Fourier’s Law of heat conduction. Arch. Ration. Mech. Anal. 13, 245–261 (1963)

    Article  MathSciNet  Google Scholar 

  12. Coleman, B.D., Fabrizio, M., Owen, D.R.: On the thermodynamics of second sound in dielectric crystals. Arch. Ration. Mech. Anal. 80(2), 135–158 (1982)

    Article  MathSciNet  Google Scholar 

  13. Coleman, B.D., Newman, D.C.: Implications of a nonlinearity in the theory of second sound in solids. Phys. Rev. B 37(4), 1492–1498 (1988)

    Article  Google Scholar 

  14. Ghaleb, A.F.: A model of continuous, thermoelastic media within the frame of extended thermodynamics. Int. J. Eng. Sci. 24(5), 765–771 (1986)

    Article  Google Scholar 

  15. Ghaleb, A.F.: Coupled thermoelectroelasticity in extended thermodynamics. In: Hetnarski, R.B. (ed.) Encyclopedia of Thermal Stresses, pp. 767–774. Springer, Berlin (2014)

    Chapter  Google Scholar 

  16. Ghaleb, A.F., Abou-Dina, M.S., Rawy, E.K., El-Dhaba’, A.R.: A model of nonlinear thermo-electroelasticity in extended thermodynamics. Int. J. Eng. Sci. 119, 29–39 (2017). https://doi.org/10.1016/j.ijengsci.2017.06.010

    Article  MathSciNet  MATH  Google Scholar 

  17. Fülöp, T., Kovács, R., Lovas, Á., Rieth, Á., Fodor, T., Szücs, M., Ván, P., Gróf, G.: Emergence of non-Fourier hierarchies. Entropy 20(11), 832 (2018). https://doi.org/10.3390/e20110832

    Article  Google Scholar 

  18. Chandrasekharaiah, D.S.: A generalized linear thermoelasticity theory for piezoelectric media. Acta Mech. 71, 39–49 (1988)

    Article  Google Scholar 

  19. Montanaro, A.: On the constitutive relations for second sound in thermo-electroelasticity. Arch. Mech. 63(3), 225–254 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Kuang, Z.-B.: Theory of Electroelasticity. Springer, Heidelberg (2014)

    Book  Google Scholar 

  21. Glass, D.E., Özisik, M.N., McRae, D.S.: Hyperbolic heat conduction with temperature-dependent thermal conductivity. J. Appl. Phys. 59, 1861–1865 (1986)

    Article  Google Scholar 

  22. Fusco, D., Manganaro, N.: Linearization of a hyperbolic model of nonlinear heat conduction through hodograph-like and Bäcklund transformations. J. Nonlinear Mech. 24, 99–103 (1989)

    Article  Google Scholar 

  23. Pascal, H.: A nonlinear model of heat conduction. J. Phys. A Math. Gen. 25, 939–948 (1992)

    Article  MathSciNet  Google Scholar 

  24. Marchant, T.R.: Thermal waves for nonlinear hyperbolic heat conduction. Math. Comput. Modell. 18(10), 111–121 (1993)

    Article  MathSciNet  Google Scholar 

  25. Stoner, R.J., Maris, H.J.: Temperature-dependence of the velocity of second sound and the determination of phonon lifetimes from thermal conductivity. In: Meissner, M., Pohl, R.O. (eds.) Phonon Scattering in Condensed Matter VII. Springer Series in Solid-State Sciences, vol. 112. Springer, Berlin (1993)

    Google Scholar 

  26. Bhagat, S.M., Davis, R.S.: Influence of a dc heat flux on the velocity of second sound near \(T_{\lambda }\). J. Low Temp. Phys. 7(1–2), 157–167 (1972)

    Article  Google Scholar 

  27. Coleman, B.D., Newman, D.C.: Implications of a nonlinearity in the theory of second sound in solids. Phys. Rev. B Condens. Matter 37(4), 1492–1498 (1988)

    Article  Google Scholar 

  28. Tarkenton, G.M., Cramer, M.S.: Nonlinear second sound in solids. Phys. Rev. B 49, 11794 (1994)

    Article  Google Scholar 

  29. Muracchini, A., Seccia, L.: Discontinuity waves, shock formation and critical temperature in crystals. J. Math. Anal. Appl. 240, 382–397 (1999)

    Article  MathSciNet  Google Scholar 

  30. Srivastava, V.K., Awasthi, M.K., Chaurasia, R.K., Tamsir, M.: The telegraph equation and its solution by reduced differential transform method. Modell. Simul. Eng. 2013, 746351. https://doi.org/10.1155/2013/746351 (2013)

  31. Acebrón, J.A., Ribeiro, M.A.: A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions. J. Comput. Phys. 305, 29–43 (2016). https://doi.org/10.1016/j.jcp.2015.10.027

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, B., Yu, W., Mascagni, M.: Revisiting Kac’s method: a Monte Carlo algorithm for solving the Telegrapher’s equations. Math. Comput. Simul. 156, 178–193 (2019)

    Article  MathSciNet  Google Scholar 

  33. Ayres Jr., F.: Schaum’s Outline of Theory and Problems of Differential Equations. McGraw Hill Book Company, New York (1952)

    Google Scholar 

  34. Nayfeh, A.H.: Perturbation Methods. Wiley-Inter Science, New York (1973)

    MATH  Google Scholar 

  35. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-Interscience, New York (1979)

    MATH  Google Scholar 

  36. Ji, H.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)

    Article  MathSciNet  Google Scholar 

  37. He, J.H.: A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Nonlinear Mech. 35(1), 37–43 (2000)

    Article  MathSciNet  Google Scholar 

  38. El-Dib, Y.O., Moatimid, G.M.: On the coupling of the homotopy perturbation and Frobenius method for exact solutions of singular nonlinear differential equations. Nonlinear Sci. Lett. A. 9(3), 220–230 (2018)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Cairo University, Cairo, Giza, 12613, Egypt

    W. Mahmoud, A. F. Ghaleb & M. S. Abou-Dina

  2. Department of Mathematics, Faculty of Education, Ain-Shams University, Cairo, Egypt

    G. M. Moatimid

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  1. W. Mahmoud
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  2. G. M. Moatimid
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  3. A. F. Ghaleb
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  4. M. S. Abou-Dina
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Correspondence to A. F. Ghaleb.

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Mahmoud, W., Moatimid, G.M., Ghaleb, A.F. et al. Nonlinear heat wave propagation in a rigid thermal conductor. Acta Mech 231, 1867–1886 (2020). https://doi.org/10.1007/s00707-020-02628-4

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  • DOI: https://doi.org/10.1007/s00707-020-02628-4

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