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Based on the form-invariance of the thermal conduction equation different from wave equation, transformation thermodynamics has opened up a new area for the arbitrarily manipulating of heat fluxes at discretion by using thermal metamaterials. Moreover, it can help researchers to design different kinds of thermal devices with many unique properties that cannot be simply realized by natural materials, such as thermal cloaking, thermal concentrating, thermal rotating and thermal illusion. Among these devices, the conventional thermal cloak enabling heat fluxes to travel around the inner region, has attracted the most significant attention so far. At the present time, the studies of the thermal cloak mainly focus on two-dimensional space with arbitrary shape and three-dimensional space with regular shape, which appear to be far from enough to meet the engineering requirements. In this paper, we derive the general expression of the thermal conductivity for three-dimensional thermal cloak with arbitrary shape according to the transformation thermodynamics. In this paper, the thermal conductivity in the polar coordinate system is transformed into that in the Cartesian coordinate system by means of coordinate transformation. On the basis of the expression of the thermal conductivity, we adopt full-wave simulation by using the software COMSOL Multiphysics to analyze the cloaking performances of five designed thermal cloaks, i.e., spherical thermal cloak, ellipsoidal thermal cloak, three-dimensional conformal thermal cloak with arbitrary shapes, non-conformal thermal cloak with the sphere outside the ellipsoid, and three-dimensional non-conformal thermal cloak with arbitrary shapes. The results show that the heat fluxes travel around the protection area, and eventually return to their original paths. The temperature profile inside the thermal cloak keeps unchanged, and the temperature field outside the thermal cloak is not distorted, which proves that the cloak has a perfect thermal invisible effect. Both the conformal and non-conformal thermal cloak have perfect thermal protection and invisible function. In this paper, the transformation thermodynamics is extended from two-dimensional thermal cloak to three-dimensional thermal cloak with better universality. At the same time, this technology provides more flexibility in controlling heat flow and target temperature field, which will have potential applications in designing microchip, motor protection and target thermal stealth. It is believed that the method presented here can be applied to other branches of physics, such as acoustics, matter waves and elastic waves.
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Keywords:
- transformation thermodynamics /
- thermal cloak /
- three-dimensional arbitrary shape /
- temperature field
[1] Pendry J B, Schurig D, Smith D R 2006 Science 312 1780
[2] Leonhardt U 2006 Science 312 1777
[3] Pendry J B, Schurig D, Smith D R 2006 Opt. Express 14 9794
[4] Pendry J B, Schurig D, Smith D R 2007 Opt. Express 15 14772
[5] Liu Y, Zentgraf T, Bartal G, Zhang X 2010 Nano Lett. 10 1991
[6] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F, Smith D R 2006 Science 314 977
[7] Cummer S A, Popa B, Schurig D, Smith D R, Pendry J B 2006 Phys. Rev. E 74 036621
[8] Rahm M, Schurig D, Roberts D A, Cummer S A, Smith D R, Pendry J B 2008 Photon. Nanostruct. Fundam. Appl. 6 87
[9] Aubry A, Lei D Y, Fernándezdomínguez A I, Sonnefraud Y, Maier S A, Pendry J B 2010 Nano Lett. 10 2574
[10] Xu H X, Wang G M, Qi M Q, Li L, Cui T J 2013 Adv. Opt. Mater. 1 495
[11] Chen H, Chan C T 2007 Appl. Phys. Lett. 90 241105
[12] Enoch S, Tayeb G, Sabouroux P, Guérin N, Vincent P 2002 Phys. Rev. Lett. 89 213902
[13] Chen Y, Yang F, Xu J Y, Liu X J 2008 Appl. Phys. Lett. 92 151913
[14] Wei Q, Chen Y, Liu X J 2012 Appl. Phys. A 109 913
[15] Zhang S, Genov D A, Sun C, Zhang X 2008 Phys. Rev. Lett. 100 123002
[16] Farhat M, Guenneau S, Enoch S 2009 Phys. Rev. Lett. 103 024301
[17] Hu R, Wei X L, Hu J Y, Luo X B 2014 Sci. Rep. 4 3600
[18] Li T H, Zhu D L, Mao F C, Huang M, Yang J J, Li S B 2016 Front. Phys. 11 1
[19] Fan C Z, Gao Y, Huang J P 2008 Appl. Phys. Lett. 92 251907
[20] Guenneau S, Amra C, Veynante D 2012 Opt. Express 20 8207
[21] Schittny R, Kadic M, Guenneau S, Wegener M 2013 Phys. Rev. Lett. 110 195901
[22] Mao F C, Li T H, Huang M, Yang J J, Chen J C 2014 Acta Phys. Sin. 63 014401 (in Chinese) [毛春福, 李廷华, 黄铭, 杨晶晶, 陈俊昌 2014 63 014401]
[23] Qin C L, Yang J J, Huang M 2014 Acta Phys. Sin. 63 194402 (in Chinese) [秦春雷, 杨晶晶, 黄铭 2014 63 194402]
[24] Yang S M, Tao W Q 2006 Heat Transfer (4th Ed.) (Beijing: Higher Education Press) p43 (in Chinese) [杨世铭, 陶文铨 2006 传热学(第四版)(北京: 高等教育出版社) 第43页]
[25] Yang T Z, Huang L J, Chen F, Xu W K 2013 J. Phys. D: Appl. Phys. 46 305102
[26] Chen T, Weng C N, Tsai Y L 2015 J. Appl. Phys. 117 054904
[27] Wu Q, Zhang K, Meng F Y, Li L W 2010 Acta Phys. Sin. 59 6071 (in Chinese) [吴群, 张狂, 孟繁义, 李乐伟 2010 59 6071]
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[1] Pendry J B, Schurig D, Smith D R 2006 Science 312 1780
[2] Leonhardt U 2006 Science 312 1777
[3] Pendry J B, Schurig D, Smith D R 2006 Opt. Express 14 9794
[4] Pendry J B, Schurig D, Smith D R 2007 Opt. Express 15 14772
[5] Liu Y, Zentgraf T, Bartal G, Zhang X 2010 Nano Lett. 10 1991
[6] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F, Smith D R 2006 Science 314 977
[7] Cummer S A, Popa B, Schurig D, Smith D R, Pendry J B 2006 Phys. Rev. E 74 036621
[8] Rahm M, Schurig D, Roberts D A, Cummer S A, Smith D R, Pendry J B 2008 Photon. Nanostruct. Fundam. Appl. 6 87
[9] Aubry A, Lei D Y, Fernándezdomínguez A I, Sonnefraud Y, Maier S A, Pendry J B 2010 Nano Lett. 10 2574
[10] Xu H X, Wang G M, Qi M Q, Li L, Cui T J 2013 Adv. Opt. Mater. 1 495
[11] Chen H, Chan C T 2007 Appl. Phys. Lett. 90 241105
[12] Enoch S, Tayeb G, Sabouroux P, Guérin N, Vincent P 2002 Phys. Rev. Lett. 89 213902
[13] Chen Y, Yang F, Xu J Y, Liu X J 2008 Appl. Phys. Lett. 92 151913
[14] Wei Q, Chen Y, Liu X J 2012 Appl. Phys. A 109 913
[15] Zhang S, Genov D A, Sun C, Zhang X 2008 Phys. Rev. Lett. 100 123002
[16] Farhat M, Guenneau S, Enoch S 2009 Phys. Rev. Lett. 103 024301
[17] Hu R, Wei X L, Hu J Y, Luo X B 2014 Sci. Rep. 4 3600
[18] Li T H, Zhu D L, Mao F C, Huang M, Yang J J, Li S B 2016 Front. Phys. 11 1
[19] Fan C Z, Gao Y, Huang J P 2008 Appl. Phys. Lett. 92 251907
[20] Guenneau S, Amra C, Veynante D 2012 Opt. Express 20 8207
[21] Schittny R, Kadic M, Guenneau S, Wegener M 2013 Phys. Rev. Lett. 110 195901
[22] Mao F C, Li T H, Huang M, Yang J J, Chen J C 2014 Acta Phys. Sin. 63 014401 (in Chinese) [毛春福, 李廷华, 黄铭, 杨晶晶, 陈俊昌 2014 63 014401]
[23] Qin C L, Yang J J, Huang M 2014 Acta Phys. Sin. 63 194402 (in Chinese) [秦春雷, 杨晶晶, 黄铭 2014 63 194402]
[24] Yang S M, Tao W Q 2006 Heat Transfer (4th Ed.) (Beijing: Higher Education Press) p43 (in Chinese) [杨世铭, 陶文铨 2006 传热学(第四版)(北京: 高等教育出版社) 第43页]
[25] Yang T Z, Huang L J, Chen F, Xu W K 2013 J. Phys. D: Appl. Phys. 46 305102
[26] Chen T, Weng C N, Tsai Y L 2015 J. Appl. Phys. 117 054904
[27] Wu Q, Zhang K, Meng F Y, Li L W 2010 Acta Phys. Sin. 59 6071 (in Chinese) [吴群, 张狂, 孟繁义, 李乐伟 2010 59 6071]
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