Continuum Mechanics and Thermodynamics

, Volume 26, Issue 4, pp 483–502 | Cite as

Analysis of the wave propagation processes in heat transfer problems of the hyperbolic type

  • Mikhail B. BabenkovEmail author
  • Elena A. Ivanova
Original Article


A number of problems for the interaction of laser radiation with a heat-conducting half-space and a layer are considered. The obtained solutions are compared with each other and with the solutions of the classic heat equation and the wave equation. A laser impulse is modelled by defining the heat flux at the boundary for the opaque medium, or by defining the distribution of heat sources in the volume for the semitransparent medium. The power of the laser pulse depends on time as the Dirac delta function or as the Heaviside function do. It allows for the simulation of instant and continuous laser exposure on the medium. Temperature distributions are obtained by using Green’s functions for a half-space and a layer with the Dirichlet and Neumann boundary conditions.


Hyperbolic heat conduction Heat flux relaxation constant Heat propagation speed Heat transport in nanosystems Heat pulse 


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  1. 1.
    Pop E., Sinha S., Goodson K.E.: Heat generation and transport in nanometer scale transistors. Proc. IEEE 94, 1587–1601 (2006)CrossRefGoogle Scholar
  2. 2.
    Tong X.C.: Development and Application of Advanced Thermal Management Materials. Springer, New York (2011)Google Scholar
  3. 3.
    Haque M.A., Saif M.T.A.: Thermo-mechanical properties of nano-scale freestanding aluminum films. Thin Solid Films 484(1–2), 364–368 (2005)ADSCrossRefGoogle Scholar
  4. 4.
    Jou D., Casas-Vazquez J., Lebon G.: Extended Irreversible Thermodynamics. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cattaneo C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Compte Rendus 247, 431–433 (1958)MathSciNetGoogle Scholar
  6. 6.
    Vernotte P.: Les paradoxes de la theorie continue de lequation de la chaleur. CR Acad. Sci. 246(22), 3154–3155 (1958)MathSciNetGoogle Scholar
  7. 7.
    Sieniutycz S.: The variational principles of classic type for non-coupled non-stationary irreversible transport processes with convective motion and relaxation. Int. J. Heat Mass Transf. 20(11), 1221–1231 (1977)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Müller I., Müller W.H.: Fundamentals of Thermodynamics and Applications: with Historical Annotations and Many Citations from Avogadro to Zermelo. Springer, Berlin (2009)Google Scholar
  9. 9.
    Germain P., Nguyen Q.-S., Suquet P.: Continuum thermodynamics. J. Appl. Mech. 50, 1010–1020 (1983)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Prigogine I.: Introduction to Thermodynamics of Irreversible Processes. Wiley, New York (1968)Google Scholar
  11. 11.
    Truesdell C.: Rational Thermodynamics: A Course of Lectures on Selected Topics. McGraw-Hill, New York (1969)Google Scholar
  12. 12.
    Chandrasekharaiah D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51(12), 705–729 (1998)ADSCrossRefGoogle Scholar
  13. 13.
    Wang C.-C.: Stress relaxation and the principle of fading memory. Arch. Ration. Mech. Anal. 18(2), 117–126 (1965)CrossRefzbMATHGoogle Scholar
  14. 14.
    Tzou D.Y.: Macro-to-Microscale Heat Transfer. The Lagging Behaviour. Taylor and Francis, New York (1997)Google Scholar
  15. 15.
    Levanov E.I., Sotskii E.N.: Some properties of the heat-transfer process in a motionless medium, taking account of heat-flux relaxation. J. Eng. Phys. 50(6), 733–740 (1986)CrossRefGoogle Scholar
  16. 16.
    Magunov A.N.: Laser thermometry of solids: state of the art and problems. Meas. Tech. 45(2), 173–181 (2002)CrossRefGoogle Scholar
  17. 17.
    Liu Y., Mandelis A.: Laser optical and photothermal thermometry of solids and thin films. Exp. Methods Phys. Sci. 42, 297–336 (2009)CrossRefGoogle Scholar
  18. 18.
    Novikov N.A.: Hyperbolic equation of thermal conductivity. Solution of the direct and inverse problems for a semiinfinite bar. J. Eng. Phys. 35(4), 1253–1257 (1978)CrossRefGoogle Scholar
  19. 19.
    Baumeister K.J., Hamill T.D.: Hyperbolic heat conduction equation—a solution for the semi-infinite problem. ASME J. Heat Transf. 91, 543–548 (1969)CrossRefGoogle Scholar
  20. 20.
    Polyanin A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman and Hall/CRC, Boca Raton (2001)CrossRefGoogle Scholar
  21. 21.
    Lewandowska M.: Hyperbolic heat conduction in the semi-infinite body with a time-dependent laser heat source. Heat Mass Transf. 37, 333–342 (2001)ADSCrossRefGoogle Scholar
  22. 22.
    Lam T.T.: Thermal propagation in solids due to surface laser pulsation and oscillation. Int. J. Therm. Sci. 49, 1639–1648 (2010)CrossRefGoogle Scholar
  23. 23.
    Isakovich M.A.: Obshhaja akustika (General Acoustics, in Russ.). Nauka, Moscow (1973)Google Scholar
  24. 24.
    Doetsch G.: Guide to the Applications of the Laplace and Z-Transforms. Van Nostrand-Reinhold, London (1971)zbMATHGoogle Scholar
  25. 25.
    Beaty H.W., Fink D.G.: Standard Handbook for Electrical Engineers. Mc Graw Hill, New Yourk (2012)Google Scholar
  26. 26.
    Nowacki W.: Dynamic Problems of Thermoelasticity. Springer, Warsaw (1975)Google Scholar
  27. 27.
    Berber S., Kwon Y.K., Tomanek D.: Unusually high thermal conductivity of carbon nanotubes. Phys. Rev. Lett. 84(20), 4613–4616 (2000)ADSCrossRefGoogle Scholar
  28. 28.
    Mo, Z., Anderson, J., Liu, J.: Integrating nano carbontubes with microchannel cooler. In: Proceeding of the Sixth IEEE CPMT Conference on High Density Microsystem Design and Packaging and Component Failure Analysis HDP’04, pp. 373–376 (2004)Google Scholar
  29. 29.
    Debye P.: Zur theorie der spezifischen wärmen. Ann. Phys. 344(14), 789–839 (1912)CrossRefGoogle Scholar
  30. 30.
    Matsunaga, R.H., Santos, I.: Measurement of the Thermal Relaxation Time in Agar-gelled Water. In: Proceedings of 34th Annual International Conference of the IEEE EMBS San Diego, California USA, pp. 5722–5725 (2012)Google Scholar
  31. 31.
    Kaminski W.: Hyperbolic heat conduction equation for materials with a non-homogeneous inner structure. ASME J. Heat Transf. 112, 555–560 (1990)CrossRefGoogle Scholar
  32. 32.
    Mitra K., Kumar S., Vedavarz A., Moallemi M.K.: Experimental evidence of hyperbolic heat conduction in processed meat. ASME J. Heat Transf. 117, 568–573 (1995)CrossRefGoogle Scholar
  33. 33.
    Grabmann A., Peters F.: Experimental investigation of heat conduction in wet sand. Heat Mass Transf. 35, 289–294 (1999)ADSCrossRefGoogle Scholar
  34. 34.
    Herwig H., Beckert K.: Experimental evidence about the controversy concerning Fourier or non-Fourier heat conduction in materials with a non-homogeneous inner structure. Heat Mass Transf. 36, 387–392 (2000)ADSCrossRefGoogle Scholar
  35. 35.
    Roetzela 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, 541–552 (2003)CrossRefGoogle Scholar
  36. 36.
    Chester M.: Second sound in solids. Phys. Rev. 131, 2013–2015 (1963)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Theoretical MechanicsSt. Petersburg State Polytechnical University (SPb-SPU)Saint PetersburgRussia
  2. 2.Institute for Problems in Mechanical EngineeringRussian Academy of SciencesSaint PetersburgRussia

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