International Journal of Thermophysics

, Volume 36, Issue 7, pp 1416–1467 | Cite as

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

Article

Abstract

The classical model of the Fourier’s law is known as the most common constitutive relation for thermal transport in various engineering materials. Although the Fourier’s law has been widely used in a variety of engineering application areas, there are many exceptional applications in which the Fourier’s law is questionable. This paper gathers together such applications. Accordingly, the paper is divided into two parts. The first part reviews the papers pertaining to the fundamental theory of the phase-lagging models and the analytical and numerical solution approaches. The second part wrap ups the various applications of the phase-lagging models including the biological materials, ultra-high-speed laser heating, the problems involving moving media, micro/nanoscale heat transfer, multi-layered materials, the theory of thermoelasticity, heat transfer in the material defects, the diffusion problems we call as the non-Fick models, and some other applications. It is predicted that the interest in the field of phase-lagging heat transport has grown incredibly in recent years because they show good agreement with the experiments across a wide range of length and time scales.

Keywords

Biological materials CV and DPL models Fluid flow Macro/nanoscale Non-Fourier and non-Fick models Thermoelasticity 

Nomenclature

B

Phase-lag ratio

C

Volumetric heat capacity (\({\hbox {J}}\)\({\cdot }\)\({\hbox {m}^{-3}}\)\({{\cdot }}\)\({\hbox {K}^{-1}}\))

Kn

Knudsen number

k

Thermal conductivity (\(\hbox {W}\)\({\cdot }\)\(\hbox {m}^{-1}\)\({\cdot }\)\(\hbox {K}^{-1}\))

\(L_{\mathrm{c}}\)

Characteristic length (m)

Q

Volumetric heat generation (\(\hbox {W}\)\({\cdot }\)\(\hbox {m}^{-3}\))

\(\overrightarrow{q}\)

Heat flux vector (W\({\cdot }\)\(\hbox {m}^{-2}\))

\(\overrightarrow{r}\)

Displacement vector (m)

T

Temperature (K)

\(T_0\)

Initial temperature (K)

\(T_{\mathrm{c}}\)

Characteristic temperature (K)

t

Time (s)

\(\overrightarrow{v}\)

Phonon group velocity (\(\hbox {m}\)\({\cdot }\)\(\hbox {s}^{-1}\))

\(\alpha \)

Thermal diffusivity (\(\hbox {m}^2\)\({\cdot }\)\(\hbox {s}^{-1}\))

\(\lambda \)

Mean-free-path of carriers (m)

\(\tau _T\)

Temperature phase-lag (s)

\(\tau _q\)

Heat flux phase-lag (s)

References

  1. 1.
    C. Cattaneo, Compte Rendus 247, 431 (1958)MathSciNetGoogle Scholar
  2. 2.
    P. Vernotte, Compte Rendus 246, 3154 (1958)MathSciNetGoogle Scholar
  3. 3.
    D.Y. Tzou, J. Heat Transf.-Trans. ASME 117, 8 (1995)CrossRefGoogle Scholar
  4. 4.
    D.Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, 2nd edn. (Wiley, New York, 2014)Google Scholar
  5. 5.
    C.L. Tien, A. Majumdar, F.M. Gerner, Microscale Energy Transport (Taylor & Francis, Washington, 1998)Google Scholar
  6. 6.
    L.Q. Wang, X.S. Zhou, Dual-Phase-Lagging Heat-Conduction Equations (Shandong University Press, Jinan, 2000)Google Scholar
  7. 7.
    L.Q. Wang, X.S. Zhou, Dual-Phase-Lagging Heat-Conduction Equations: Problems and Solutions (Shandong University Press, Jinan, China, 2001)Google Scholar
  8. 8.
    K.R. Sharma, Damped Wave Transport and Relaxation (Elsevier, Amsterdam, 2005)Google Scholar
  9. 9.
    D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, 4th edn. (Springer, Berlin, 2010)Google Scholar
  10. 10.
    Z.M. Zhang, Nano/Microscale Heat Transfer (McGraw-Hill, New York, 2007)Google Scholar
  11. 11.
    L. Wang, X. Zhou, X. Wei, Heat Conduction (Springer, Berlin, 2008)MATHGoogle Scholar
  12. 12.
    L. Wang, M. Xu, X. Wei, in Emerging Topics in Heat and Mass Transfer in Porous Media: Dual-Phase-Lagging and Porous-Medium Heat Conduction Processes, vol. 22, ed. by P. Vadász (Springer, Berlin, 2008), p. 11Google Scholar
  13. 13.
    M.N. Özişik, D.Y. Tzou, J. Heat Transf.-Trans. ASME 116, 526 (1994)CrossRefGoogle Scholar
  14. 14.
    D.S. Chandrasekharaiah, Appl. Mech. Rev.-Trans. ASME 51, 705 (1998)ADSCrossRefGoogle Scholar
  15. 15.
    F. Xu, T.J. Lu, K.A. Seffen, E.Y.K. Ng, Appl. Mech. Rev.-Trans. ASME 62, 050801 (2009)ADSCrossRefGoogle Scholar
  16. 16.
    A. Bhowmik, R. Singh, R. Repaka, S.C. Mishra, J. Therm. Biol. 38, 107 (2013)CrossRefGoogle Scholar
  17. 17.
    A.A. Balandin, Nat. Mater. 10, 569 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    H.H. Pennes, J. Appl. Physiol. 1, 93 (1948)ADSGoogle Scholar
  19. 19.
    J. Liu, Z. Ren, C. Wang, Chin. Sci. Bull. 40, 1493 (1995)Google Scholar
  20. 20.
    J. Liu, X. Chen, L.X. Xu, IEEE Trans. Biomed. Eng. 46, 420 (1999)CrossRefGoogle Scholar
  21. 21.
    J. Marciak-Kozlowska, M. Kozlowski, Nova J. Eye Res. 1, 49 (2011)Google Scholar
  22. 22.
    T.C. Shih, T.L. Horng, H.W. Huang, K.C. Ju, T.-C. Huang, P.Y. Chen, Y.J. Ho, W.L. Lin, Int. J. Heat Mass Transf. 55, 3763 (2012)CrossRefGoogle Scholar
  23. 23.
    F. Xu, K.A. Seffen, T.J. Lu, Int. J. Heat Mass Transf. 51, 2237 (2008)CrossRefGoogle Scholar
  24. 24.
    K.-C. Liu, H.-T. Chen, Int. J. Heat Mass Transf. 52, 1185 (2009)CrossRefGoogle Scholar
  25. 25.
    J. Zhou, J.K. Chen, Y. Zhang, Comput. Biol. Med. 39, 286 (2009)CrossRefGoogle Scholar
  26. 26.
    J. Fan, L. Wang, Int. J. Heat Mass Transf. 54, 722 (2011)CrossRefGoogle Scholar
  27. 27.
    K.C. Liu, Y.N. Wang, Y.S. Chen, Int. J. Therm. Sci. 58, 29 (2012)CrossRefGoogle Scholar
  28. 28.
    B.R. Barron, W. Dai, Numer. Heat Transf. B 49, 437 (2006)ADSCrossRefGoogle Scholar
  29. 29.
    P.J. Chen, M.E. Gurtin, Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 19, 614 (1968)ADSCrossRefGoogle Scholar
  30. 30.
    P.J. Chen, M.E. Gurtin, W.O. Williams, Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 19, 969 (1968)ADSCrossRefGoogle Scholar
  31. 31.
    P.J. Chen, M.E. Gurtin, W.O. Williams, Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 20, 107 (1969)ADSCrossRefGoogle Scholar
  32. 32.
    R. Quintanilla, P.M. Jordan, Mech. Res. Commun. 36, 796 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    J.K. Chen, J.E. Beraun, Numer. Heat Transf. A 40, 1 (2001)ADSGoogle Scholar
  34. 34.
    J.K. Chen, D.Y. Tzou, J.E. Beraun, Int. J. Heat Mass Transf. 49, 307 (2006)CrossRefGoogle Scholar
  35. 35.
    T.Q. Qiu, C.L. Tien, Int. J. Heat Mass Transf. 35, 719 (1992)ADSCrossRefGoogle Scholar
  36. 36.
    H. Wang, W. Dai, L.G. Hewavitharana, Int. J. Therm. Sci. 47, 7 (2008)CrossRefGoogle Scholar
  37. 37.
    W. Dai, G. Li, Numer. Methods Partial Diff. Equ. 22, 1396 (2006)MathSciNetCrossRefGoogle Scholar
  38. 38.
    I.K. Kaba, W. Dai, J. Comput. Appl. Math. 181, 125 (2005)MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    H. Wang, W. Dai, L.G. Hewavitharana, J. Thermophys. Heat Transf. 22, 530 (2008)CrossRefGoogle Scholar
  40. 40.
    T.Q. Qiu, C.L. Tien, Int. J. Heat Mass Transf. 37, 2789 (1994)CrossRefGoogle Scholar
  41. 41.
    T.Q. Qiu, C.L. Tien, ASME J. Heat Transf. 115, 835 (1993)CrossRefGoogle Scholar
  42. 42.
    M.A. Al-Nimr, O.M. Haddad, V.S. Arpaci, Heat Mass Transf. 35, 459 (1999)ADSCrossRefGoogle Scholar
  43. 43.
    M.A. Al-Nimr, V.S. Arpaci, Int. J. Heat Mass Transf. 43, 2021 (2000)CrossRefGoogle Scholar
  44. 44.
    M. Al-Odat, M.A. Al-Nimr, M. Hamdan, Int. J. Numer. Methods Heat Fluid Flow 12, 173 (2002)CrossRefGoogle Scholar
  45. 45.
    M. Naji, M.A. Al-Nimr, M. Hader, Int. J. Thermophys. 24, 545 (2003)CrossRefGoogle Scholar
  46. 46.
    M.A. Al-Nimr, M.K. Alkam, Int. J. Thermophys. 24, 577 (2003)CrossRefGoogle Scholar
  47. 47.
    J.K. Chen, J.E. Beraun, C.L. Tham, J. Opt. A: Pure Appl. Opt. 4, 650 (2002)ADSCrossRefGoogle Scholar
  48. 48.
    J.K. Chen, W.P. Latham, J.E. Beraun, Numer. Heat Transf. B 42, 1 (2002)ADSCrossRefGoogle Scholar
  49. 49.
    J.K. Chen, J.E. Beraun, C.L. Tham, Numer. Heat Transf. A 44, 705 (2003)ADSCrossRefGoogle Scholar
  50. 50.
    W. Dai, T. Niu, Nonlinear Anal. Hybrid Syst. 2, 121 (2008)MathSciNetCrossRefGoogle Scholar
  51. 51.
    T. Niu, W. Dai, Int. J. Therm. Sci. 48, 34 (2009)CrossRefGoogle Scholar
  52. 52.
    W. Dai, H. Song, S. Su, R. Nassar, Int. J. Numer. Methods Heat Fluid Flow 16, 693 (2006)CrossRefGoogle Scholar
  53. 53.
    E. Awad, J. Therm. Stress. 35, 293 (2012)CrossRefGoogle Scholar
  54. 54.
    G. Chen, J. Heat Transf.-Trans. ASME 124, 320 (2001)CrossRefGoogle Scholar
  55. 55.
    G. Chen, Phys. Rev. Lett. 86, 2297 (2001)ADSCrossRefGoogle Scholar
  56. 56.
    R. Yang, G. Chen, M. Laroche, Y. Taur, J. Heat Transf.-Trans. ASME 127, 298 (2005)CrossRefGoogle Scholar
  57. 57.
    T. Yamada, S. Hamian, B. Sunden, K. Park, M. Faghri, Int. J. Heat Mass Transf. 61, 287 (2013)CrossRefGoogle Scholar
  58. 58.
    Y. Ma, in Equation of Phonon Hydrodynamics for Non-Fourier Heat Conduction. In: Proceedings of the 44th AIAA Thermophysics Conference (San Diego, CA, 2013)Google Scholar
  59. 59.
    Z.Y. Guo, J. Eng. Thermophys. 27, 631 (2006)Google Scholar
  60. 60.
    Y. Dong, B.-Y. Cao, Z.-Y. Guo, Phys. Rev. E 87, 032150 (2013)ADSCrossRefGoogle Scholar
  61. 61.
    M. Wang, Z.-Y. Guo, Phys. Lett. A 374, 4312 (2010)ADSCrossRefGoogle Scholar
  62. 62.
    M. Wang, N. Yang, Z.-Y. Guo, J Appl. Phys. 110, 064310 (2011)ADSCrossRefGoogle Scholar
  63. 63.
    D.Y. Tzou, Z.Y. Guo, Int. J. Therm. Sci. 49, 1133 (2010)CrossRefGoogle Scholar
  64. 64.
    T.T. Lam, Int. J. Heat Mass Transf. 56, 653 (2013)CrossRefGoogle Scholar
  65. 65.
    S. Su, W. Dai, P.M. Jordan, R.E. Mickens, Int. J. Heat Mass Transf. 48, 22233 (2005)CrossRefGoogle Scholar
  66. 66.
    S. Su, W. Dai, Int. J. Heat Mass Transf. 49, 2793 (2005)CrossRefGoogle Scholar
  67. 67.
    W. Dai, R. Nassar, Int. J. Heat Mass Transf. 45, 1585 (2002)CrossRefGoogle Scholar
  68. 68.
    C.S. Tsai, Y.C. Lin, C.I. Hung, Heat Mass Transf. 41, 709 (2005)ADSCrossRefGoogle Scholar
  69. 69.
    A. Moosaie, Arch. Appl. Mech. 79, 679 (2009)ADSCrossRefGoogle Scholar
  70. 70.
    R. Shirmohammadi, A. Moosaie, Int. Commun. Heat Mass Transf. 36, 827 (2009)CrossRefGoogle Scholar
  71. 71.
    F. Jiang, Heat Mass Transf. 42, 1083 (2006)ADSCrossRefGoogle Scholar
  72. 72.
    K.K. Tamma, S.B. Railkar, Numer. Heat Transf. B 15, 211 (1989)ADSCrossRefGoogle Scholar
  73. 73.
    K.K. Tamma, J.F. D’Costa, Numer. Heat Transf. B 19, 49 (1991)ADSCrossRefGoogle Scholar
  74. 74.
    H. Rahideh, P. Malekzadeh, M.R. Golbahar Haghighi, ISRN Mech. Eng. 2011, 321605 (2011)CrossRefGoogle Scholar
  75. 75.
    S.C. Mishra, T.P. Pavan Kumar, J. Heat Transf.-Trans. ASME 131, 111302 (2009)CrossRefGoogle Scholar
  76. 76.
    D.E. Glass, M.N. Özişik, D.S. McRae, B. Vick, J. Appl. Phys. 59, 1861 (1986)ADSCrossRefGoogle Scholar
  77. 77.
    A.D. Polyanin, A.I. Zhurov, Int. J. Non-Linear Mech. 54, 115 (2013)ADSCrossRefGoogle Scholar
  78. 78.
    E. Fong, T.T. Lam, in Application of Solution Structure Theorems for Asymmetric Heating in Thin Films. In: Proceedings of 44th AIAA Thermophysics Conference (San Diego, CA, 2013)Google Scholar
  79. 79.
    B. Shen, P. Zhang, Int. J. Heat Mass Transf. 51, 1713 (2008)CrossRefGoogle Scholar
  80. 80.
    V.V. Kulish, V.B. Novozhilov, Microscale Thermophys. Eng. 8, 7 (2004)CrossRefGoogle Scholar
  81. 81.
    M.A. Hader, M.A. Al-Nimr, Heat Transf. Eng. 23, 35 (2002)ADSCrossRefGoogle Scholar
  82. 82.
    L. Wang, X. Wei, Int. J. Heat Mass Transf. 51, 1751 (2008)CrossRefGoogle Scholar
  83. 83.
    M. Xu, J. Guo, L. Wang, L. Cheng, Int. J. Therm. Sci. 50, 825 (2011)CrossRefGoogle Scholar
  84. 84.
    M. Xu, L. Wang, Int. J. Heat Mass Transf. 45, 1055 (2002)CrossRefGoogle Scholar
  85. 85.
    D.Y. Tzou, W. Dai, Int. J. Heat Mass Transf. 52, 1206 (2009)CrossRefGoogle Scholar
  86. 86.
    D.Y. Tzou, Ultrafast Heat Transport: The Lagging Behavior. In: Proceedings of the SPIE Conference on Terahertz and Gigahertz Photonics (Denver, Colorado, 1999), pp. 234–247Google Scholar
  87. 87.
    D.Y. Tzou, Y. Zhang, Int. J. Eng. Sci. 33, 1449 (1995)CrossRefGoogle Scholar
  88. 88.
    M.A. Hader, M.A. Al-Nimr, B.A. Abu Nabah, Int. J. Thermophys. 23, 1669 (2002)CrossRefGoogle Scholar
  89. 89.
    M.A. Al-Nimr, A.F. Khadrawi, M. Hammad, Heat Mass Transf. 39, 69 (2002)ADSCrossRefGoogle Scholar
  90. 90.
    C.-Y. Yang, Appl. Math. Model. 33, 2907 (2009)MathSciNetCrossRefGoogle Scholar
  91. 91.
    G.E. Cossali, Int. J. Therm. Sci. 43, 347 (2004)CrossRefGoogle Scholar
  92. 92.
    J. Ghazanfarian, A. Abbassi, Int. J. Heat Mass Transf. 52, 3706 (2009)CrossRefGoogle Scholar
  93. 93.
    J. Ghazanfarian, A. Abbassi, Int. J. Thermophys. 33, 552 (2012)ADSCrossRefGoogle Scholar
  94. 94.
    J. Ghazanfarian, Z. Shomali, Int. J. Heat Mass Transf. 55, 6231 (2012)CrossRefGoogle Scholar
  95. 95.
    J. Fangming, L. Dengying, J. Therm. Sci. 10, 1 (2001)CrossRefGoogle Scholar
  96. 96.
    K. Ramadan, Int. Commun. Heat Mass Transf. 35, 1177 (2008)CrossRefGoogle Scholar
  97. 97.
    S.I. Serdyukov, Phys. Lett. A 281, 16 (2001)ADSCrossRefGoogle Scholar
  98. 98.
    S.I. Serdyukov, N.M. Voskresenskii, V.K. Belnov, I.I. Karpov, J. Non-Equilib. Thermodyn. 28, 207 (2003)ADSGoogle Scholar
  99. 99.
    S.K. Serdyukov, Phys. A 391, 5871 (2012)CrossRefGoogle Scholar
  100. 100.
    M.A. Al-Nimr, M. Naji, V.S. Arbaci, J. Heat Transf.-Trans. ASME 122, 217 (2000)CrossRefGoogle Scholar
  101. 101.
    D. Joua, M. Criado-Sancho, Phys. Lett. A 248, 172 (1998)ADSCrossRefGoogle Scholar
  102. 102.
    F. Vázquez, J.A. del Río, J. Appl. Phys. 112, 123707 (2012)ADSCrossRefGoogle Scholar
  103. 103.
    A. Sellitto, V.A. Cimmelli, D. Jou, Phys. D 241, 1344 (2012)CrossRefGoogle Scholar
  104. 104.
    M. Xu, L. Wang, Int. J. Heat Mass Transf. 48, 5616 (2005)CrossRefGoogle Scholar
  105. 105.
    J. Shiomi, S. Maruyama, Phys. Rev. B 73, 205420 (2006)ADSCrossRefGoogle Scholar
  106. 106.
    W.-S. Jiaung, J.-R. Ho, Phys. Rev. E. 77, 066710 (2008)ADSCrossRefGoogle Scholar
  107. 107.
    R.S. Samian, A. Abbassi, J. Ghazanfarian, Int. J. Mod. Phys. C 24, 1350064 (2013)ADSCrossRefGoogle Scholar
  108. 108.
    R.S. Samian, A. Abbassi, J. Ghazanfarian, Int. J. Mod. Phys. C 25, 1350103 (2013)MathSciNetADSCrossRefGoogle Scholar
  109. 109.
    M. Xu, Q. Cheng, Int. J. Thermophys. 34, 306 (2013)ADSCrossRefGoogle Scholar
  110. 110.
    J. Ghazanfarian, A. Abbassi, Phys. Rev. E. 82, 026307 (2010)ADSCrossRefGoogle Scholar
  111. 111.
    H. Basirat Tabrizi, S. Andarwa, Int. Commun. Heat Mass Transf. 36, 186 (2009)CrossRefGoogle Scholar
  112. 112.
    K.-C. Liu, C.T. Lin, Numer. Heat Transf. A 58, 802 (2010)ADSCrossRefGoogle Scholar
  113. 113.
    D.Y. Tzou, J. Thermophys. Heat Transf. 9, 686 (1995)CrossRefGoogle Scholar
  114. 114.
    H. Herwig, K. Beckert, J. Heat Transf.-Trans. ASME 122, 363 (2000)CrossRefGoogle Scholar
  115. 115.
    W. Roetzel, N. Putra, S.K. Das, Int. J. Therm. Sci. 42, 541 (2003)CrossRefGoogle Scholar
  116. 116.
    R. Quintanilla, Appl. Math. Comput. 216, 2759 (2010)MathSciNetCrossRefGoogle Scholar
  117. 117.
    R. Quintanilla, J. Non-Equilib. Thermodyn. 27, 217 (2002)ADSGoogle Scholar
  118. 118.
    C.O. Horgan, R. Quintanilla, Math. Methods Appl. Sci. 28, 43 (2005)MathSciNetADSCrossRefGoogle Scholar
  119. 119.
    R. Quintanilla, R. Racke, Int. J. Heat Mass Transf. 49, 1209 (2006)CrossRefGoogle Scholar
  120. 120.
    P.M. Jordan, W. Dai, R.E. Mickens, Mech. Res. Commun. 35, 414 (2008)MathSciNetCrossRefGoogle Scholar
  121. 121.
    R. Quintanilla, Z. Angew, Math. Phys. 61, 381 (2010)Google Scholar
  122. 122.
    R. Quintanilla, R. Racke, Int. J. Heat Mass Transf. 51, 24 (2008)CrossRefGoogle Scholar
  123. 123.
    R. Quintanilla, Appl. Math. Comput. 213, 153 (2009)MathSciNetCrossRefGoogle Scholar
  124. 124.
    R. Quintanilla, J. Therm. Stress. 32, 1270 (2009)CrossRefGoogle Scholar
  125. 125.
    R. Quintanilla, Mech. Res. Commun. 38, 355 (2011)CrossRefGoogle Scholar
  126. 126.
    L. Wang, M. Xu, X. Zhou, Int. J. Heat Mass Transf. 44, 1659 (2001)CrossRefGoogle Scholar
  127. 127.
    L. Wang, M. Xu, Int. J. Heat Mass Transf. 45, 1165 (2002)CrossRefGoogle Scholar
  128. 128.
    R. Quintanilla, R. Racke, SIAM J. Appl. Math. 66, 977 (2006)MathSciNetCrossRefGoogle Scholar
  129. 129.
    R. Quintanilla, R. Racke, Proc. R. Soc. A 463, 659 (2007)MathSciNetADSCrossRefGoogle Scholar
  130. 130.
    R. Quintanilla, J. Therm. Stress. 31, 260 (2008)CrossRefGoogle Scholar
  131. 131.
    M. Dreher, R. Quintanilla, R. Racke, Appl. Math. Lett. 22, 1374 (2009)MathSciNetCrossRefGoogle Scholar
  132. 132.
    R. Cai, C. Gou, H. Li, Arch. Appl. Mech. 45, 893 (2006)Google Scholar
  133. 133.
    J. Escolano, F. Rodríguez, M.A. Castro, F. Vives, J.A. Martín, Math. Comput. Model. 54, 1841 (2011)CrossRefGoogle Scholar
  134. 134.
    S.M. Lin, J. Mech. Med. Biol. 13, 1350063 (2013)ADSCrossRefGoogle Scholar
  135. 135.
    M.H. Saad, C.Y. Cha, Int. J. Non-Linear Mech. 17, 129 (1982)CrossRefGoogle Scholar
  136. 136.
    A.F. Khadrawi, M.A. Al-Nimr, M. Tahat, Int. Commun. Heat Mass Transf. 31, 1015 (2004)CrossRefGoogle Scholar
  137. 137.
    P.J. Antaki, Int. J. Heat Mass Transf. 41, 2253 (1998)CrossRefGoogle Scholar
  138. 138.
    N.M. Abdel-Jabbar, M.A. Al-Nimr, Heat Transf. Eng. 24, 47 (2003)ADSCrossRefGoogle Scholar
  139. 139.
    M.A. Hader, M.A. Al-Nimr, V.A. Hammoudeh, Int. J. Thermophys. 27, 665 (2006)ADSCrossRefGoogle Scholar
  140. 140.
    R.K. Rahmani, T.G. Keith, R. Cottam, JSME Int. J. 46, 519 (2003)ADSCrossRefGoogle Scholar
  141. 141.
    Y. Zhang, C. Zheng, Y. Liu, L. Shao, C. Gou, Acta Mech. Sin. 25, 205 (2009)ADSCrossRefGoogle Scholar
  142. 142.
    V. Mohammadi-Fakhar, S.H. Momeni-Masuleh, Phys. Lett. A 374, 595 (2010)ADSCrossRefGoogle Scholar
  143. 143.
    J. Ordónez-Miranda, J.J. Alvarado-Gil, Mech. Res. Commun. 37, 276 (2010)CrossRefGoogle Scholar
  144. 144.
    R.T. Al-Khairy, Int. J. Therm. Sci. 50, 1369 (2011)CrossRefGoogle Scholar
  145. 145.
    D.W. Tang, N. Araki, Int. J. Heat Mass Transf. 42, 855 (1999)CrossRefGoogle Scholar
  146. 146.
    A. Haji-Sheikh, F. de Monte, J.V. Beck, Int. J. Heat Mass Transf. 62, 78 (2013)CrossRefGoogle Scholar
  147. 147.
    R. Alkhairy, Appl. Math. 3, 1170 (2012)CrossRefGoogle Scholar
  148. 148.
    F. Han, W. Dai, Appl. Math. Model. 37, 7940 (2013)MathSciNetADSCrossRefGoogle Scholar
  149. 149.
    W. Dai, L. Shen, R. Nassar, T. Zhu, Int. J. Heat Mass Transf. 41, 1817 (2004)CrossRefGoogle Scholar
  150. 150.
    W. Dai, L. Shen, T. Zhu, Numer. Heat Transf. B 46, 121 (2004)ADSCrossRefGoogle Scholar
  151. 151.
    W. Dai, L. Shen, R. Nassar, Numer. Methods Partial Differ. Equ. 20, 60 (2004)MathSciNetCrossRefGoogle Scholar
  152. 152.
    W. Dai, R. Nassar, J. Comput. Appl. Math. 132, 431 (2001)MathSciNetADSCrossRefGoogle Scholar
  153. 153.
    W. Dai, F. Han, Z. Sun, Int. J. Heat Mass Transf. 64, 966 (2013)CrossRefGoogle Scholar
  154. 154.
    W. Dai, R. Nassar, J. Comput. Appl. Math. 145, 247 (2002)MathSciNetADSCrossRefGoogle Scholar
  155. 155.
    A. Malek, Z.K. Bojdi, P.N.N. Golbarg, J. Heat Transf.-Trans. ASME 134, 094504 (2012)CrossRefGoogle Scholar
  156. 156.
    J.M. McDonough, I. Kunadian, R.R. Kumar, T. Yang, J. Comput. Phys. 219, 163 (2006)MathSciNetADSCrossRefGoogle Scholar
  157. 157.
    J. Cabrera, M.A. Castro, F. Rodríguez, J.A. Martín, Math. Comput. Model. 57, 1625 (2013)CrossRefGoogle Scholar
  158. 158.
    H. Wang, W. Dai, R. Melnik, Int. J. Therm. Sci. 45, 1179 (2006)CrossRefGoogle Scholar
  159. 159.
    Y.J. Tao, X.L. Huai, Z.G. Li, Chin. Phys. Lett. 23, 2487 (2006)ADSCrossRefGoogle Scholar
  160. 160.
    J.-R. Ho, C.-P. Kuo, W.-S. Jiaung, Int. J. Heat Mass Transf. 46, 55 (2003)CrossRefGoogle Scholar
  161. 161.
    Y. Chou, R.-J. Yang, Int. J. Heat Mass Transf. 51, 3525 (2008)CrossRefGoogle Scholar
  162. 162.
    Y. Chou, R.-J. Yang, Int. J. Heat Mass Transf. 52, 239 (2009)CrossRefGoogle Scholar
  163. 163.
    Q.-M. Fan, W.Q. Lu, Int. J. Heat Mass Transf. 45, 2815 (2002)CrossRefGoogle Scholar
  164. 164.
    W.-T. Ang, Eng. Comput. 19, 467 (2002)CrossRefGoogle Scholar
  165. 165.
    R. Saghatchi, J. Ghazanfarian, Appl. Math. Model. 39, 1063 (2015)MathSciNetCrossRefGoogle Scholar
  166. 166.
    J. Ghazanfarian, R. Saghatchi, D.V. Patil, Appl. Math. Comput. 259, 21 (2015)MathSciNetCrossRefGoogle Scholar
  167. 167.
    K.-C. Liu, P.-C. Chang, Appl. Math. Model. 31, 369 (2007)CrossRefGoogle Scholar
  168. 168.
    H.-L. Lee, W.-L. Chen, W.-J. Chang, E.-J. Wei, Y.-C. Yang, Appl. Therm. Eng. 52, 275 (2013)CrossRefGoogle Scholar
  169. 169.
    T.-M. Chen, Int. J. Heat Mass Transf. 50, 4424 (2007)CrossRefGoogle Scholar
  170. 170.
    K.-C. Liu, Comput. Phys. Commun. 177, 307 (2007)ADSCrossRefGoogle Scholar
  171. 171.
    F.S. Loureiro, L.C. Wrobel, W.J. Mansur, J. Braz. Soc. Mech. Sci. Eng. 34, 459 (2012)CrossRefGoogle Scholar
  172. 172.
    D.Y. Tzou, J. Heat Transf.-Trans. ASME 134, 051006 (2012)CrossRefGoogle Scholar
  173. 173.
    L. Wang, J. Fan, J. Heat Transf. 133, 011010 (2011)CrossRefGoogle Scholar
  174. 174.
    J. Zhou, Y. Zhang, J.K. Chen, Int. J. Therm. Sci. 48, 1477 (2009)CrossRefGoogle Scholar
  175. 175.
    J. Zhou, J.K. Chen, Y. Zhang, in Nonclassical Heat Transfer Models for Laser-Induced Thermal Damage in Biological Tissues. In: Proceedings of the ASME 2011 International Mechanical Engineering Congress & Exposition (Denver, Colorado, 2011), pp. 249–260Google Scholar
  176. 176.
    K.-C. Liu, P.-J. Cheng, D.T.W. Lin, Adv. Sci. Lett. 9, 925 (2011)ADSCrossRefGoogle Scholar
  177. 177.
    N. Afrin, J. Zhou, Y. Zhang, D.Y. Tzou, J.K. Chen, Numer. Heat Transf. A 61, 483 (2012)CrossRefGoogle Scholar
  178. 178.
    P. Antaki, J. Heat Mass Transf.-Trans. ASME 127, 189 (2005)CrossRefGoogle Scholar
  179. 179.
    F. Xu, T. Lu, K.A. Seffen, in Dual-Phase-Lag Model of Skin Bioheat Transfer. In: Proceedings of 2008 International Conference on BioMedical Engineering and Informatics (Sanya, Hainan, 2008), pp. 505–511Google Scholar
  180. 180.
    W. Dai, H. Wang, P.M. Jordan, R.E. Mickens, A. Bejan, Int. J. Heat Mass Transf. 51, 5497 (2008)CrossRefGoogle Scholar
  181. 181.
    F. Xu, M. Lin, T.J. Lu, Comput. Biol. Med. 40, 478 (2010)CrossRefGoogle Scholar
  182. 182.
    K.-C. Liu, P.-J. Cheng, Y.-N. Wang, Therm. Sci. 15, S61 (2011)CrossRefGoogle Scholar
  183. 183.
    H. Ahmadiki, A. Moradi, R. Fazlali, A.B. Parsa, J. Mech. Sci. Technol. 26, 1937 (2012)CrossRefGoogle Scholar
  184. 184.
    C. Liu, B.Q. Li, C.C. Mi, IEEE Trans. Nanobiosci. 8, 271 (2009)ADSCrossRefGoogle Scholar
  185. 185.
    K.-C. Liu, Int. J. Heat Mass Transf. 54, 2829 (2011)CrossRefGoogle Scholar
  186. 186.
    C. Liu, C.C. Mi, B.Q. Li, IEEE Trans. Nanotechnol. 8, 697 (2009)ADSCrossRefGoogle Scholar
  187. 187.
    K.-C. Liu, H.-T. Chen, Int. J. Heat Therm. Sci. 49, 1138 (2010)ADSCrossRefGoogle Scholar
  188. 188.
    Y. Zhang, Int. J. Heat Mass Transf. 52, 4829 (2009)CrossRefGoogle Scholar
  189. 189.
    N. Afrin, Y. Zhang, J.K. Chen, Int. J. Heat Mass Transf. 54, 2419 (2011)CrossRefGoogle Scholar
  190. 190.
    A. Narasimhan, S. Sadasivam, Int. J. Heat Mass Transf. 60, 591 (2013)CrossRefGoogle Scholar
  191. 191.
    P. Han, D. Tang, L. Zhou, Int. J. Eng. Sci. 44, 1510 (2006)CrossRefGoogle Scholar
  192. 192.
    R.T. Al-Khairy, Phys. Wave Phenom. 17, 277 (2009)ADSCrossRefGoogle Scholar
  193. 193.
    K.-C. Tseng, J.-R. Tsai, Numer. Heat Transf. A 53, 726 (2008)ADSCrossRefGoogle Scholar
  194. 194.
    D.Y. Tzou, K.S. Chiu, Int. J. Heat Mass Transf. 44, 1725 (2006)CrossRefGoogle Scholar
  195. 195.
    K. Ramadan, W.R. Tyfour, M.A. Al-Nimr, J. Heat Transf.-Trans. ASME 131, 111301 (2009)CrossRefGoogle Scholar
  196. 196.
    J.K. Chen, D.Y. Tzou, J.E. Beraun, Int. J. Heat Mass Transf. 48, 501 (2005)CrossRefGoogle Scholar
  197. 197.
    F. Jiang, D. Liu, J. Zhou, Microscale Thermophys. Eng. 6, 331 (2002)CrossRefGoogle Scholar
  198. 198.
    C.I. Christov, P.M. Jordan, Phy. Rev. Lett. 94, 154301 (2005)ADSCrossRefGoogle Scholar
  199. 199.
    L. Cheng, M. Xu, L. Wang, J. Heat Transf.-Trans. ASME 130, 121302 (2008)CrossRefGoogle Scholar
  200. 200.
    C.I. Christov, Mech. Res. Commun. 36, 481 (2009)MathSciNetCrossRefGoogle Scholar
  201. 201.
    H.S. Costa Mattos, R.A.C. Dias, Int. Commun. Heat Mass Transf. 42, 38 (2013)CrossRefGoogle Scholar
  202. 202.
    A.D. Polyanin, A.I. Zhurov, Int. J. Non-Linear Mech. 57, 116 (2013)ADSCrossRefGoogle Scholar
  203. 203.
    B. Straughan, F. Franchi, Benard convection and the Cattaneo law of heat conduction. Proc. R Soc. Edinb. 96, 175 (1984)MathSciNetMATHCrossRefGoogle Scholar
  204. 204.
    J.J. Vadasz, S. Govender, P. Vadasz, Int. J. Heat Mass Transf. 48, 2673 (2005)CrossRefGoogle Scholar
  205. 205.
    R.E. Khayat, D. Stranges, R. Khorasany, M. Niknami, B. Albaalbaki, in Thermal Convection of Nanofluids: A Non-Fourier Perspective and Linear Stability Analysis. In: Proceedings of the World Congress on Engineering (London, 2012)Google Scholar
  206. 206.
    A.F. Khadrawi, A. Othman, M.A. Al-Nimr, Int. J. Thermophys. 26, 905 (2005)ADSCrossRefGoogle Scholar
  207. 207.
    A.F. Khadrawi, M.A. Al-Nimr, Int. J. Thermophys. 28, 1387 (2007)ADSCrossRefGoogle Scholar
  208. 208.
    D.Y. Tzou, Int. J. Heat Mass Transf. 38, 3231 (1995)CrossRefGoogle Scholar
  209. 209.
    H. Pourmohamadian, H.B. Tabrizi, in Transient Heat Conduction for Micro Sphere. In: Proceedings of the 4th WSEAS International Conference on Heat and Mass Transfer (Queensland, 2007), pp. 119–120Google Scholar
  210. 210.
    H. Basirat, J. Ghazanfarian, P. Forooghi, in Implementation of Dual-Phase-Lag Model at Different Knudsen Numbers Within Slab Heat Transfer, In: Proceedings of International Conference on Modeling and Simulation (Konia, 2006), pp. 895–899Google Scholar
  211. 211.
    Z. Shomali, J. Ghazanfarian, A. Abbassi, Superlattices Microstruct. (2015). doi:10.1016/j.spmi.2015.03.060
  212. 212.
    B. Vermeersch, G.D. Mey, Analog Integr. Circuits Signal Process. 55, 197 (2008)CrossRefGoogle Scholar
  213. 213.
    D. Donadio, G. Galli, Nano Lett. 10, 847 (2010)ADSCrossRefGoogle Scholar
  214. 214.
    M. Moghaddam, J. Ghazanfarian, A. Abbassi, IEEE Trans. Electron Devices. 61, 3131 (2014)ADSCrossRefGoogle Scholar
  215. 215.
    C.W. Chang, D. Okawa, H. Garcia, A. Majumdar, A. Zettl, Phys. Rev. Lett. 101, 075903 (2008)ADSCrossRefGoogle Scholar
  216. 216.
    J. Shiomi, S. Maruyama, Therm. Sci. Eng. 13, 89 (2005)Google Scholar
  217. 217.
    P. Kim, L. Shi, A. Majumdar, P.L. McEuen, Phys. Rev. Lett. 87, 215502 (2001)ADSCrossRefGoogle Scholar
  218. 218.
    M.A. Osman, D. Srivastava, Nanotechnology 12, 21 (2001)ADSCrossRefGoogle Scholar
  219. 219.
    D. Donadio, G. Galli, Phys. Rev. Lett. 99, 255502 (2007)ADSCrossRefGoogle Scholar
  220. 220.
    C. Yu, L. Shi, Z. Yao, D. Li, A. Majumdar, Nano Lett. 5, 1842 (2005)ADSCrossRefGoogle Scholar
  221. 221.
    H.-D. Wang, B.-Y. Cao, Z.-Y. Guo, Int. J. Heat Mass Transf. 53, 1796 (2010)CrossRefGoogle Scholar
  222. 222.
    H.-D. Wang, B.Y. Cao, Z.Y. Guo, J. Heat Transf.-Trans. ASME 134, 051004 (2012)CrossRefGoogle Scholar
  223. 223.
    X. Zhang, M. Hu, D. Poulikakos, Nano Lett. 12, 3410 (2012)ADSCrossRefGoogle Scholar
  224. 224.
    J.H. Seol, I. Jo, A.L. Moore, L. Lindsay, Z.H. Aitken, M.T. Pettes, M. Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R.S. Ruoff, L. Shi, Science 328, 213 (2010)ADSCrossRefGoogle Scholar
  225. 225.
    L. Lindsay, D.A. Broido, N. Mingo, Phys. Rev. B 82, 161402 (2010)ADSCrossRefGoogle Scholar
  226. 226.
    S. Ghosh, W. Bao, D.L. Nika, S. Subrina, E.P. Pokatilov, C.N. Lau, A.A. Balandin, Nat. Mat. 9, 555 (2010)CrossRefGoogle Scholar
  227. 227.
    A. Sellitto, F.X. Alvarez, Nanoscale Sys. MMTA 1, 38 (2012)Google Scholar
  228. 228.
    S. Volz, J.-B. Saulnier, M. Lallemand, B. Perrin, P. Depondt, M. Mareschal, Phys. Rev. B 45, 340 (1996)ADSCrossRefGoogle Scholar
  229. 229.
    M.A. Al-Nimr, A.F. Khadrawi, Int. J. Thermophys. 25, 1953 (2004)ADSCrossRefGoogle Scholar
  230. 230.
    Q. Liu, P. Jiang, H. Xiang, Prog. Nat. Sci. 18, 999 (2008)CrossRefGoogle Scholar
  231. 231.
    T.-W. Tsai, Y.-M. Lee, Int. J. Heat Mass Transf. 62, 87 (2013)CrossRefGoogle Scholar
  232. 232.
    Zahra Shomalia, Abbas Abbassi, Int. J. Therm. Sci. 83, 56 (2014)CrossRefGoogle Scholar
  233. 233.
    M.A. Al-Nimr, M. Naji, R.I. Abdallah, Int. J. Thermophys. 25, 949 (2004)ADSCrossRefGoogle Scholar
  234. 234.
    K.-C. Liu, P.J. Cheng, Numer. Heat Transf. A 49, 589 (2006)ADSCrossRefGoogle Scholar
  235. 235.
    K. Ramadan, M.A. Al-Nimr, J. Heat Transf.-Trans. ASME 130, 074501 (2008)CrossRefGoogle Scholar
  236. 236.
    K. Ramadan, Int. J. Therm. Sci. 48, 14 (2009)CrossRefGoogle Scholar
  237. 237.
    M.A. Al-Nimr, K. Ramadan, Heat Transf. Eng. 30, 677 (2009)ADSCrossRefGoogle Scholar
  238. 238.
    K.-C. Liu, J. Phys. D Appl. Phys. 38, 3722 (2005)ADSCrossRefGoogle Scholar
  239. 239.
    K. Ramadan, M.A. Al-Nimr, Int. J. Therm. Sci. 48, 1718 (2009)CrossRefGoogle Scholar
  240. 240.
    G. Espinosa-Paredes, E.-G. Espinosa-Martnez, Ann. Nucl. Energy 36, 680 (2009)CrossRefGoogle Scholar
  241. 241.
    S.P. Filopoulos, Th.K. Papathanassiou, G.J. Tsamasphyros, J. Therm. Stress. 32, 905 (2009)Google Scholar
  242. 242.
    H.W. Lord, Y. Shulman, J. Mech. Phys. Solids 15, 299 (1967)ADSCrossRefGoogle Scholar
  243. 243.
    E. Green, K.A. Lindsay, J. Elast. 2, 1 (1972)CrossRefGoogle Scholar
  244. 244.
    R.B. Hetnarski, J. Ignaczak, J. Therm. Stress. 17, 377 (1994)MathSciNetCrossRefGoogle Scholar
  245. 245.
    E. Green, P.M. Naghdi, J. Elast. 31, 189 (1993)MathSciNetCrossRefGoogle Scholar
  246. 246.
    S.K. Roychoudhuri, J. Therm. Stress. 30, 231 (2007)CrossRefGoogle Scholar
  247. 247.
    A.S. El-Karamany, Int. J. Eng. Sci. 40, 2097 (2002)MathSciNetCrossRefGoogle Scholar
  248. 248.
    A.S. El-Karamany, J. Appl. Mech. 70, 661 (2003)ADSCrossRefGoogle Scholar
  249. 249.
    A.S. El-Karamany, M.A. Ezzat, Int. J. Mech. Sci. 46, 389 (2004)CrossRefGoogle Scholar
  250. 250.
    A.S. El-Karamany, Int. J. Eng. Sci. 42, 157 (2004)MathSciNetCrossRefGoogle Scholar
  251. 251.
    M.K. Ghosh, M. Kanoria, J. Therm. Stress. 31, 665 (2008)CrossRefGoogle Scholar
  252. 252.
    M. Islam, M. Kanoria, Math. Mech. Sol. 0, 1 (2013)Google Scholar
  253. 253.
    S. Mukhopadhyay, R. Prasad, R. Kumar, J. Therm. Stress. 34, 352 (2011)CrossRefGoogle Scholar
  254. 254.
    M.A. Ezzat, A.S. El-Karamany, S.M. Ezzat, Nuc. Eng. Des. 252, 267 (2012)CrossRefGoogle Scholar
  255. 255.
    K.K. Tamma, S.B. Railkar, Comput. Struct. 34, 5 (1990)CrossRefGoogle Scholar
  256. 256.
    B. Said-Houari, R. Rahali, Evol. Equ. Control Theory 2, 423 (2013)MathSciNetCrossRefGoogle Scholar
  257. 257.
    B. Wang, Appl. Mech, Mater. 249–250, 962 (2013)CrossRefGoogle Scholar
  258. 258.
    M.A. Al-Nimr, N.S. Al-Huniti, J. Therm. Stress. 23, 731 (2000)CrossRefGoogle Scholar
  259. 259.
    A.S. El-Karamany, M.A. Ezzat, Meccanica 49, 79 (2014)MathSciNetCrossRefGoogle Scholar
  260. 260.
    R. Quintanilla, J. Therm. Stress. 26, 713 (2003)MathSciNetCrossRefGoogle Scholar
  261. 261.
    N.S. Al-Huniti, M.A. Al-Nimr, J. Therm. Stress. 27, 607 (2004)CrossRefGoogle Scholar
  262. 262.
    N.S. AL-Huniti, M.A. Al-Nimr, Heat. Transf. Eng. 26, 41 (2005)ADSCrossRefGoogle Scholar
  263. 263.
    S.K.R. RoyChoudhuri, J. Mech. Mater. Struct. 2, 239 (2007)CrossRefGoogle Scholar
  264. 264.
    R. Prasad, R. Kumar, S. Mukhopadhyay, Int. J. Eng. Sci. 48, 2028 (2010)MathSciNetCrossRefGoogle Scholar
  265. 265.
    S. Mukhopadhyay, S. Kothari, R. Kumar, J. Therm. Stress. 34, 923 (2011)CrossRefGoogle Scholar
  266. 266.
    A.E. Abouelregal, Int. J. Comput. Methods Eng. Sci. Mech. 12, 96 (2011)MathSciNetCrossRefGoogle Scholar
  267. 267.
    A.E. Abouelregal, Int. J. Eng. Sci. 49, 781 (2011)MathSciNetCrossRefGoogle Scholar
  268. 268.
    A.E. Abouelregal, S.M. Abo-Dahab, J. Therm. Stress. 35, 820 (2012)CrossRefGoogle Scholar
  269. 269.
    Y. Yu, X. Tian, T.J. Lu, Eur. J. Mech. A 42, 188 (2013)MathSciNetCrossRefGoogle Scholar
  270. 270.
    A. Miranville, R. Quintanilla, Appl. Math. Optim. 63, 133 (2011)MathSciNetCrossRefGoogle Scholar
  271. 271.
    M.A. Ezzat, A.S. El Karamany, M.A. Fayik, Arch. Appl. Mech. 82, 557 (2012)ADSCrossRefGoogle Scholar
  272. 272.
    S. Mukhopadhyay, R. Kumar, Acta Mech. 210, 331 (2010)CrossRefGoogle Scholar
  273. 273.
    R. Kumar, S. Mukhopadhyay, J. Therm. Stress. 32, 1149 (2009)CrossRefGoogle Scholar
  274. 274.
    S. Mukhopadhyay, S. Kothari, R. Kumar, Acta Mech. 214, 305 (2010)CrossRefGoogle Scholar
  275. 275.
    R. Kumar, V. Chawla, Int. Commun. Heat Mass Transf. 38, 1262 (2011)CrossRefGoogle Scholar
  276. 276.
    A. Kar, M. Kanoriab, Eur. J. Mech. A Sol. 28, 757 (2009)CrossRefGoogle Scholar
  277. 277.
    A. Kar, M. Kanoria, Appl. Math. Model. 33, 3287 (2009)MathSciNetCrossRefGoogle Scholar
  278. 278.
    M. Kanoria, S.H. Mallik, Eur. J. Mech. A Sol. 29, 695 (2010)CrossRefGoogle Scholar
  279. 279.
    A. Kar, M. Kanoria, Eur. J. Pure Appl. Math. 4, 304 (2011)MathSciNetGoogle Scholar
  280. 280.
    M.H. Babaei, Z. Chen, J. Thermophys. Heat Transf. 24, 325 (2010)CrossRefGoogle Scholar
  281. 281.
    A.H. Akbarzadeh, Z.T. Chen, Int. J. Thermophys. 33, 1100 (2012)ADSCrossRefGoogle Scholar
  282. 282.
    D.Y. Tzou, Int. J. Heat Mass Transf. 35, 481 (1992)CrossRefGoogle Scholar
  283. 283.
    Y. Zhou, X. Li, D. Yu, Int. J. Sol. Struct. 47, 768 (2010)CrossRefGoogle Scholar
  284. 284.
    B.L. Wang, J.C. Han, Int. J. Heat Mass Transf. 55, 4631 (2012)MathSciNetCrossRefGoogle Scholar
  285. 285.
    W. BaoLin, H. JieCai, Sci. China Phys. Mech. Ast. 55, 493 (2012)CrossRefGoogle Scholar
  286. 286.
    B. Wang, Adv. Mater. Res. 716, 402 (2013)CrossRefGoogle Scholar
  287. 287.
    Z.T. Chen, K.Q. Hu, Int. J. Thermophys. 33, 895 (2012)ADSCrossRefGoogle Scholar
  288. 288.
    K. Hu, Z. Chen, Int. J. Eng. Sci. 51, 144 (2012)CrossRefGoogle Scholar
  289. 289.
    B.L. Wang, J.C. Han, Int. J. Eng. Sci. 55, 66 (2012)MathSciNetCrossRefGoogle Scholar
  290. 290.
    B. Wang, Appl. Mech. Mater. 271–272, 1312 (2013)Google Scholar
  291. 291.
    K.Q. Hu, Z.T. Chen, Int. J. Heat Mass Transf. 62, 445 (2013)CrossRefGoogle Scholar
  292. 292.
    G.Y. Li, B.L. Chen, IEEE Trans. Comp. Pack. Technol. 26, 651 (2003)CrossRefGoogle Scholar
  293. 293.
    A.D. Polyanin, A.V. Vyazmin, Theor. Found. Chem. Eng. 47, 217 (2013)CrossRefGoogle Scholar
  294. 294.
    J.K. Chen, J.E. Beraun, D.Y. Tzou, J. Mater. Sci. 34, 6183 (1999)ADSCrossRefGoogle Scholar
  295. 295.
    J.K. Chen, J.E. Beraun, D.Y. Tzou, Semicond. Sci. Technol. 15, 235 (2000)ADSCrossRefGoogle Scholar
  296. 296.
    Z. Huang, P.P. Conway, R. Qin, Microsyst. Technol. 15, 101 (2009)CrossRefGoogle Scholar
  297. 297.
    J.K. Chen, J.E. Beraun, D.Y. Tzou, Microsyst. Technol. 123, 52 (2001)Google Scholar
  298. 298.
    P. Vadasz, Int. J. Heat Mass Transf. 48, 2822 (2005)CrossRefGoogle Scholar
  299. 299.
    P.J. Antaki, J. Thermophys. Heat Transf. 14, 276 (2000)CrossRefGoogle Scholar
  300. 300.
    S. Andarwa, H. Basirat Tabrizi, Int. J. Heat Mass Transf. 53, 3080 (2010)CrossRefGoogle Scholar
  301. 301.
    M. Al-Odat, M.A. Al-Nimr, M. Hamdan, Int. J. Thermophys. 23, 855 (2002)CrossRefGoogle Scholar
  302. 302.
    M.Q. Al-Odat, M.A. Al-Nimr, Heat Mass Transf. 40, 211 (2004)ADSCrossRefGoogle Scholar
  303. 303.
    M.Q. Al-Odat, F.M. Al-Hussien, Int. J. Thermophys. 29, 1523 (2008)ADSCrossRefGoogle Scholar
  304. 304.
    J. Ordóñez-Miranda, J.J. Alvarado-Gil, Granul. Matter 12, 569 (2010)CrossRefGoogle Scholar
  305. 305.
    D.Y. Tzou, J.K. Chen, J. Thermophys. Heat Transf. 12, 567 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jafar Ghazanfarian
    • 1
  • Zahra Shomali
    • 2
  • Abbas Abbassi
    • 3
  1. 1.Department of Mechanical Engineering, Faculty of EngineeringUniversity of ZanjanZanjanIran
  2. 2.Department of PhysicsInstitute for Advanced Studies in Basic Sciences (IASBS)ZanjanIran
  3. 3.Mechanical Engineering DepartmentAmirkabir University of TechnologyTehranIran

Personalised recommendations