Green’s Function Solution for the Dual-Phase-Lag Heat Equation

Reem Alkhairy

Dammam University, Dammam, KSA.

DOI: 10.4236/am.2012.310171                

Abstract

The present work is devoted to define a generalized Green’s function solution for the dual-phase-lag model in homogeneous materials in a unified manner .The high-order mixed derivative with respect to space and time which reflect the lagging behavior is treated in special manner in the dual-phase-lag heat equation in order to construct a general solution applicable to wide range of dual-phase-lag heat transfer problems of general initial-boundary conditions using Green’s function solution method. Also, the Green’s function for a finite medium subjected to arbitrary heat source and arbitrary initial and boundary conditions is constructed. Finally, four examples of different physical situations are analyzed in order to illustrate the accuracy and potentialities of the proposed unified method. The obtained results show good agreement with works of [1-4].

Keywords

Dual-Phase-Lag Heat Conduction; Green’s Function; Integral Transforms

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	M. Xu, J. Guo, L. Wang and L. Cheng, “Thermal Wave Interference as the Origin of the Overshooting Phenomenon in Dual-Phase-Lagging Heat Conduction,” International Journal of Thermal Sciences, Vol. 50, No. 5, 2011, pp. 825-830. doi:10.1016/j.ijthermalsci.2010.12.006
[2]	R. Al-Khairy, “Analytical Solution of Non-Fourier Temperature Response in a Finite Medium Symmetrically Heated on Both Sides,” Physics of Wave Phenomena, Vol. 17, No. 4, 2009, pp. 277-285. doi:10.3103/S1541308X09040049
[3]	M. Lewandowska and L. Malinowski, “An Analytical Solution of the Hyperbolic Heat Conduction Equation for the Case of a Finite Medium Symmetrically Heated on Both Sides,” International Communications in Heat and Mass Transfer, Vol. 33, No. 1, 2006, pp. 61-69. doi:10.1016/j.icheatmasstransfer.2005.08.004
[4]	P. Han, D. W. Tang and L. Zhou, “Numerical Analysis of Two Dimensional Lagging Thermal Behavior under Short-Pulse-Laser Heating on Surface,” International Journal of Engineering Science, Vol. 44, No. 20, 2006, pp. 1510-1519. doi:10.1016/j.ijengsci.2006.08.012
[5]	B. Shen and P. Zhang, “Notable Physical Anomalies Manifested in Non-Fourier Heat Conduction under the Dual-Phase-Lag Model,” International Journal of Heat and Mass Transfer, Vol. 51, No. 7-8, 2008, pp. 1713-1727. doi:10.1016/j.ijheatmasstransfer.2007.07.039
[6]	A. Haji-Sheikh and J. V. Beck, “Green’s Function Solution for Thermal Wave Equation in Finite Bodies,” International Journal of Heat and Mass Transfer, Vol. 37, No. 17, 1994, pp. 2615-2626. doi:10.1016/0017-9310(94)90379-4
[7]	F. Loureiroa, P. Oyarzúna, J. Santosa, W. Mansura and C. Vasconcellosb, “A Hybrid Time/Laplace Domain Method Base on Numerical Green’s Functions Applied to Parabolic and Hyperbolic Bioheat Transfer Problems,” Mecnica Computational, Vol. 29, 2010, pp. 5599-5611.
[8]	D. W. Tang and N. Araki, “Wavy, Wavelike, Diffusive Thermal Responses of Finite Rigid Slabs to High-Speed Heating of Laser-Pulses,” International Journal of Heat and Mass Transfer, Vol. 42, No. 5, 1999, pp. 855-860. doi:10.1016/S0017-9310(98)00244-0
[9]	D. W. Tang and N. Araki, “Non-Fourier Heat Conduction Behavior in a Finite Mediums under Pulse Surface Heating,” Materials Science and Engineering, Vol. A292, No. 2, 2000, pp. 173-178.
[10]	G. Duffy, “Green’s Functions with Applications,” Chapman & Hall, New York, 2001. doi:10.1201/9781420034790
[11]	D. Y. Tzou, “Macro-to Microscale Heat Transfer: The Lagging Behavior,” Taylor and Francis, Bristol, 1990.