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Numerical Schemes for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Model in a Double-Layered Nanoscale Thin Film

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Abstract

This article proposes a time fractional dual-phase-lagging (DPL) heat conduction model in a double-layered nanoscale thin film with the temperature-jump boundary condition and a thermal lagging effect interfacial condition between layers. The model is proved to be well-posed. A finite difference scheme with second-order spatial convergence accuracy in maximum norm is then presented for solving the fractional DPL model. Unconditional stability and convergence of the scheme are proved by using the discrete energy method. A numerical example without exact solution is given to verify the accuracy of the scheme. Finally, we show the applicability of the time fractional DPL model by predicting the temperature rise in a double-layered nanoscale thin film, where a gold layer is on a chromium padding layer exposed to an ultrashort-pulsed laser heating.

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References

  1. Jamshidi, M., Ghazanfarian, J.: Dual-phase-lag analysis of CNT–MoS\(_2\)–ZrO\(_2\)–SiO\(_2\)–Si nano-transistor and arteriole in multi-layered skin. Appl. Math. Model. 60, 490–507 (2018)

    Article  MathSciNet  Google Scholar 

  2. Sun, H., Sun, Z.Z., Dai, W.: A second-order finite difference scheme for solving the dual-phase-lagging equation in a double-layered nanoscale thin film. Numer. Methods Partial Differ. Equ. 33, 142–173 (2017)

    Article  MathSciNet  Google Scholar 

  3. Tzou, D.Y.: Macro- To Microscale Heat Transfer: The Lagging Behavior, 2nd edn. Wiley, New York (2015)

    Google Scholar 

  4. Ghazanfarian, J., Shomali, Z., Abbassi, A.: Macro- to nanoscale heat and mass transfer: the lagging behavior. Int. J. Thermophys. 36, 1416–1467 (2015)

    Article  Google Scholar 

  5. Nasri, F., Aissa, MFBen, Belmabrouk, H.: Effect of second-order temperature jump in metal-oxide-semiconductor field effect transistor with dual-phase-lag model. Microelectron. J. 46, 67–74 (2015)

    Article  Google Scholar 

  6. Saghatchi, R., Ghazanfarian, J.: A novel SPH method for the solution of dual-phase-lag model with temperature-jump boundary condition in nanoscale. Appl. Math. Model. 39, 1063–1073 (2015)

    Article  MathSciNet  Google Scholar 

  7. Shomali, Z., Abbassi, A.: Investigation of highly non-linear dual-phase-lag model in nanoscale solid argon with temperature-dependent properties. Int. J. Therm. Sci. 83, 56–67 (2014)

    Article  Google Scholar 

  8. Dai, W., Han, F., Sun, Z.Z.: Accurate numerical method for solving dual-phase-lagging equation with temperature jump boundary condition in nanoheat conduction. Int. J. Heat Mass Transf. 64, 966–975 (2013)

    Article  Google Scholar 

  9. Ghazanfarian, J., Shomali, Z.: Investigation of dual-phase-lag heat conduction model in a nanoscale metal-oxide-semiconductor field-effect transistor. Int. J. Heat Mass Transf. 55, 6231–6237 (2012)

    Article  Google Scholar 

  10. Awad, E.: On the generalized thermal lagging behavior. J. Therm. Stress. 35, 193–325 (2012)

    Article  Google Scholar 

  11. Sherief, H.H., EI-Sayed, A.M.A., EI-Latief, A.M.A.: Fractional order theory of thermoelasticity. Int. J. Solid Struct. 47, 269–275 (2010)

    Article  Google Scholar 

  12. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent—II. Geophys. J. Int. 13, 529–539 (1967)

    Article  Google Scholar 

  13. Youssef, H.M.: Theory of fractional order generalized thermoelasticity. J. Heat Transf. 132, 061301 (2010)

    Article  Google Scholar 

  14. Povstenko, Y.Z.: Fractional heat conduction equation and associated thermal stress. J. Therm. Stress. 28, 83–102 (2005)

    Article  MathSciNet  Google Scholar 

  15. Yu, Y.J., Tian, X.G., Lu, T.J.: Fractional order generalized electro-magneto-thermo-elasticity. Eur. J. Mech. A Solids 42, 188–202 (2013)

    Article  MathSciNet  Google Scholar 

  16. Kim, P., Shi, L., Majumdar, A., McEuen, P.: Thermal transport measurements of individual multiwalled nanotubes. Phys. Rev. Lett. 87, 215502 (2001)

    Article  Google Scholar 

  17. Balandin, A.A.: Thermal properties of graphene and nanostructured carbon materials. Nat. Mater. 10, 569–581 (2011)

    Article  Google Scholar 

  18. Tzou, D.Y.: Nonlocal behavior in phonon transport. Int. J. Heat Mass Transf. 54, 475–481 (2011)

    Article  Google Scholar 

  19. Ji, C.C., Dai, W., Sun, Z.Z.: Numerical method for solving the time-fractional dual-phase-lagging heat conduction equation with the temperature-jump boundary condition. J. Sci. Comput. 75, 1307–1336 (2018)

    Article  MathSciNet  Google Scholar 

  20. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  21. Liao, M., Gan, Z.H.: New insight on negative bias temperature instability degradation with drain bias of 28 nm high-K metal gate p-MOSFET devices. Microelectron. Reliab. 54, 2378–2382 (2014)

    Article  Google Scholar 

  22. Ho, J.R., Kuo, C.P., Jiaung, W.S.: Study of heat transfer in multilayered structure within the framework of dual-phase-lag heat conduction model using lattice Boltzmann method. Int. J. Heat Mass Transf. 46, 55–69 (2003)

    Article  Google Scholar 

  23. Liu, K.C.: Analysis of dual-phase-lag thermal behaviour in layered films with temperature-dependent interface thermal resistance. J. Phys. D Appl. Phys. 38, 3722–3732 (2005)

    Article  Google Scholar 

  24. Shen, M., Keblinski, P.: Ballistic vs. diffusive heat transfer across nanoscopic films of layered crystals. J. Appl. Phys. 115, 144310 (2014)

    Article  Google Scholar 

  25. Pillers, M., Lieberman, M.: Rapid thermal processing of DNA origami on silicon creates embedded silicon carbide replicas. In: 13th Annual Conference on Foundations of Nanoscience, Snowbird, Utah, April 11–16 (2016)

  26. Tsai, T.W., Lee, Y.M.: Analysis of microscale heat transfer and ultrafast thermoelasticity in a multi-layered metal film with nonlinear thermal boundary resistance. Int. J. Heat Mass Transf. 62, 87–98 (2013)

    Article  Google Scholar 

  27. Sadasivam, S., Waghmare, U.V., Fisher, T.S.: Electron–phonon coupling and thermal conductance at a metal-semiconductor interface: first-principles analysis. J. Appl. Phys. 117, 134502 (2015)

    Article  Google Scholar 

  28. Ghazanfarian, J., Abbassi, A.: Effect of boundary phonon scattering on dual-phase-lag model to simulate micro- and nano-scale heat conduction. Int. J. Heat Mass Transf. 52, 3706–3711 (2009)

    Article  Google Scholar 

  29. Alikhanov, A.A.: A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46, 660–666 (2010)

    Article  MathSciNet  Google Scholar 

  30. Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MathSciNet  Google Scholar 

  31. Sun, Z.Z.: The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations. Science Press, Beijing (2009)

    Google Scholar 

  32. Feng, L.B., Liu, F., Turner, I., Zheng, L.C.: Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid. Fract. Calc. Appl. Anal. 21, 1073–1103 (2018)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, People’s Republic of China

    Cui-cui Ji

  2. Mathematics and Statistics, Louisiana Tech University, Ruston, LA, 71272, USA

    Weizhong Dai

  3. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Zhi-zhong Sun

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  1. Cui-cui Ji
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  2. Weizhong Dai
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  3. Zhi-zhong Sun
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Correspondence to Zhi-zhong Sun.

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Cui-cui Ji and Zhi-zhong Sun were supported by National Natural Science Foundation of China (No. 11671081).

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Ji, Cc., Dai, W. & Sun, Zz. Numerical Schemes for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Model in a Double-Layered Nanoscale Thin Film. J Sci Comput 81, 1767–1800 (2019). https://doi.org/10.1007/s10915-019-01062-6

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  • DOI: https://doi.org/10.1007/s10915-019-01062-6

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