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Controlling Heat Conduction with Nonlinear-Transformation Cloaking

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Abstract

Recently, coordinate transformation method can be used to design invisibility cloaks for many types of waves, including heat conduction. The main difficulties encountered implementing the cloaking devices may be due to singular and anisotropic thermodynamic properties after coordinate transformation. This paper designs a nonlinear cloak with tunable thermodynamic parameters from a thermal diffusion path perspective to control thermal conduction. A theory of nonlinear-transformation thermodynamics is given and a nonlinear thermal diffusion path equation for thermal conduction is derived. A broadband thermal cloak is constructed with layered homogeneous and isotropic materials based on a nonlinear transformation. The cloaking performance of the nonlinear-transformation cloak can be adjusted by varying a design parameter, which affects thermal diffusion in the inner region of the cloak. Both the cloaks can exhibit detouring thermal diffusion around the inclusion, while the cloak designed in this paper has more flexibility in guiding heat conduction by changing the value of the design parameter. Numerical simulations demonstrate that the nonlinear-transformation cloak is effective in shielding the heat diffusion from outer region of the cloak. The methodology developed in this paper provides a useful approach to flexibly adjusting the thermal diffusion path according to different engineering requirements.

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References

  1. J.B. Pendry, D. Schurig, D.R. Smith, Science 312, 1780 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  2. U. Leonhardt, Science 312, 1777 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  3. A. Greenleaf, M. Lassas, G. Uhlmann, Math. Res. Lett. 10, 685 (2003)

    Article  MathSciNet  Google Scholar 

  4. S.A. Cummer, D. Schurig, New J. Phys. 9, 45 (2007)

    Article  ADS  Google Scholar 

  5. D. Schurig, J.J. Mock, B.J. Justice, S.A. Cummer, J.B. Pendry, A.F. Star, D.R. Smith, Science 314, 977 (2006)

    Article  ADS  Google Scholar 

  6. S. Guenneau, R.C. McPhedran, S. Enoch, A.B. Movchan, M. Farhat, N.-A.P. Nicorovici, J. Opt. 13, 024014 (2011)

    Article  ADS  Google Scholar 

  7. P.L. Gao, L.Z. Wu, Proc. R. Soc. A 472, 20150348 (2016)

    Article  ADS  Google Scholar 

  8. J. O’Neill, O. Selsil, R.C. Mcphedran, A.B. Movchan, N.V. Movchan, J. Mech. Appl. Math. 68, 263 (2015)

    Article  Google Scholar 

  9. A.N. Norris, Proc. R. Soc. A 464, 2411 (2008)

    Article  ADS  Google Scholar 

  10. J. Qin, W. Luo, P. Yang, B. Wang, T. Deng, T.C. Han, Int. J. Heat Mass Tran. 141, 487 (2019)

    Article  Google Scholar 

  11. S. Hoseinzadeh, P.S. Heyns, A.J. Chamkha, A. Shirkhani, J. Therm. Anal. Calorim. 138, 727 (2019)

    Article  Google Scholar 

  12. H. Jordaan, P.S. Heyns, S. Hoseinzadeh, J. Thermal Sci. Eng. Appl. 13, 041017 (2021)

    Article  Google Scholar 

  13. S. Hoseinzadeh, P.S. Heyns, Eng. Fail. Anal. 113, 104548 (2020)

    Article  Google Scholar 

  14. S. Huberman, R.A. Duncan, K. Chen, B. Song, V. Chiloyan, Z. Ding, A.A. Maznev, G. Chen, K.A. Nelson, Science 364, 375–379 (2019)

    Article  ADS  Google Scholar 

  15. M. Gandolfi, G.L. Celardo, F. Borgonovi, G. Ferrini, A. Avella, F. Banfi, C. Giannetti, Phys. Scr. 92, 034004 (2017)

    Article  ADS  Google Scholar 

  16. M. Gandolfi, C. Giannetti, F. Banfi, Phys. Rev. Lett. 125, 265901 (2020)

    Article  ADS  Google Scholar 

  17. M. Gandolfi, G. Benetti, C. Glorieux, C. Giannetti, F. Banfi, Int. J. Heat Mass Tran. 143, 118553 (2019)

    Article  Google Scholar 

  18. C.Z. Fan, Y. Gao, J.P. Huang, Appl. Phys. Lett. 92, 251907 (2018)

    Article  ADS  Google Scholar 

  19. L.J. Xu, J.P. Huang, Int. J. Heat Mass Tran. 159, 120133 (2020)

    Article  Google Scholar 

  20. J. Guo, Z.G. Qu, Int. J. Heat Mass Tran. 127, 1212 (2018)

    Article  Google Scholar 

  21. J. Wang, G.L. Dai, J.P. Huang, SiScience 23, 101637 (2020)

    ADS  Google Scholar 

  22. S. Guenneau, C. Amra, D. Veynante, Opt. Exp. 20, 8207 (2012)

    Article  ADS  Google Scholar 

  23. X.B. Yuan, G.C. Lin, Y.S. Wang, AIP Adv. 6, 045013 (2016)

    Article  ADS  Google Scholar 

  24. M. Moccia, G. Castaldi, S. Savo, Y. Sato, V. Galdi, Phys. Rev. X 4, 021025 (2014)

    Google Scholar 

  25. L.J. Xu, J. Wang, G.L. Dai, S. Yang, F.B. Yang, G. Wang, J.P. Huang, Int. J. Heat Mass Tran. 165, 120659 (2021)

    Article  Google Scholar 

  26. A.N. Norris, C. Vemula, J. Sound Vib. 181, 115 (1995)

    Article  ADS  Google Scholar 

  27. N. Stenger, M. Wilhelm, M. Wegener, Phys. Rev. Lett. 108, 014301 (2012)

    Article  ADS  Google Scholar 

  28. A. Zareei, M.R. Alam, J. Fluid Mech. 778, 273 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  29. A. Zareei, M.R. Alam, Phys. Rev. E 95, 063002 (2017)

    Article  ADS  Google Scholar 

  30. Q.X. Ji, G.D. Fang, J. Liang, AIP Adv. 8, 045316 (2018)

    Article  ADS  Google Scholar 

  31. M. Liu, W.D. Zhu, J. Appl. Mech. 85, 081002 (2018)

    Article  ADS  Google Scholar 

  32. M. Liu, W.D. Zhu, Wave Motion 92, 102421 (2020)

    Article  MathSciNet  Google Scholar 

  33. M. Liu, W.D. Zhu, J. Sound Vib. 445, 270 (2019)

    Article  ADS  Google Scholar 

  34. M.W. McCall, P. Kinsler, A. Favaro, Proc. of SPIE 8093, 80930 (2011)

    Article  ADS  Google Scholar 

  35. R. Adler, M. Bazin, M. Schiffer, Introduction to General Relativity (McGraw-Hill, New York, 1975).

    MATH  Google Scholar 

  36. R.A. Crudo, J.G. O’Brien, Phys. Rev. A 80, 033824 (2009)

    Article  ADS  Google Scholar 

  37. D. Torrent, J. Sánchez-Dehesa, Wave Motion 48, 497 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the support from Natural Science Foundation of Jiangsu Province through Grant No. BK20200884, Natural Science Foundation of Colleges and Universities in Jiangsu Province through Grant No. 20KJB130004, and Jiangsu's Mass Entrepreneurship and Innovation Program.

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Authors and Affiliations

  1. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Mao Liu, Lijun Zhuo, Wenan Jiang, Xindong Ma, Meng Han & Mitao Song

Authors
  1. Mao Liu
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  2. Lijun Zhuo
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  3. Wenan Jiang
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  4. Xindong Ma
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  5. Meng Han
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  6. Mitao Song
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Correspondence to Mao Liu.

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Liu, M., Zhuo, L., Jiang, W. et al. Controlling Heat Conduction with Nonlinear-Transformation Cloaking. Int J Thermophys 42, 89 (2021). https://doi.org/10.1007/s10765-021-02849-2

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  • DOI: https://doi.org/10.1007/s10765-021-02849-2

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