Analytical solution of the hyperbolic heat conduction equation for moving semi-infinite medium under the effect of time-dependent laser heat source. (English) Zbl 1179.35352

Al-Khairy, R. T.; Al-Ofey, Z. M.

J. Appl. Math. 2009, Article ID 604695, 18 p. (2009).

Summary: This paper presents an analytical solution of the hyperbolic heat conduction equation for moving semi-infinite medium under the effect of time dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given, while the semi-infinite body has insulated boundary. The solution is obtained by Laplace transforms method, and the discussion of solutions for different time characteristics of heat sources capacity (constant, instantaneous, and exponential) is presented. The effect of absorption coefficients on the temperature profiles is examined in detail. It is found that the closed form solution derived from the present study reduces to the previously obtained analytical solution when the medium velocity is set to zero in the closed form solution.

MSC:

35R37	Moving boundary problems for PDEs
35Q79	PDEs in connection with classical thermodynamics and heat transfer
80A20	Heat and mass transfer, heat flow (MSC2010)
35C05	Solutions to PDEs in closed form
35A22	Transform methods (e.g., integral transforms) applied to PDEs
35B33	Critical exponents in context of PDEs

Keywords:

hyperbolic heat conduction equation; laser heat source; Laplace transforms

References:

[1]	J. C. Jaeger, “Moving source of heat and the temperature at sliding contacts,” Proceedings of the Royal Society of NSW, vol. 76, pp. 203-224, 1942.
[2]	J. C. Jaeger and H. S. Carslaw, Conduction of Heat in Solids, Oxford University Press, Oxford, UK, 1959. · Zbl 0029.37801
[3]	M. Kalyon and B. S. Yilbas, “Exact solution for time exponentially varying pulsed laser heating: convective boundary condition case,” Proceedings of the Institution of Mechanical Engineers, Part C, vol. 215, no. 5, pp. 591-602, 2001. · doi:10.1243/0954406011520977
[4]	M. N. Özisik, Heat Conduction, Wiley, New York, NY, USA, 2nd edition, 1993.
[5]	D. Rosenthal, “The theory of moving sources of heat and its application to metal treatments,” Transaction of the American Society of Mechanical Engineers, vol. 68, pp. 849-866, 1946.
[6]	B. S. Yilbas and M. Kalyon, “Parametric variation of the maximum surface temperature during laser heating with convective boundary conditions,” Journal of Mechanical Engineering Science, vol. 216, no. 6, pp. 691-699, 2002. · Zbl 0993.80002 · doi:10.1243/095440602320192337
[7]	M. A. Al-Nimr and V. S. Arpaci, “Picosecond thermal pulses in thin metal films,” Journal of Applied Physics, vol. 85, no. 5, pp. 2517-2521, 1999. · doi:10.1063/1.369568
[8]	M. A. Al-Nimr, “Heat transfer mechanisms during short-duration laser heating of thin metal films,” International Journal of Thermophysics, vol. 18, no. 5, pp. 1257-1268, 1997. · doi:10.1007/BF02575260
[9]	T. Q. Qiu and C. L. Tien, “Short-pulse laser heating on metals,” International Journal of Heat and Mass Transfer, vol. 35, no. 3, pp. 719-726, 1992. · doi:10.1016/0017-9310(92)90131-B
[10]	C. L. Tien and T. Q. Qiu, “Heat transfer mechanism during short pulse laser heating of metals,” American Society of Mechanical Engineers Journal of Heat Transfer, vol. 115, pp. 835-841, 1993.
[11]	C. Catteneo, “A form of heat conduction equation which eliminates the paradox of instantaneous propagation,” Compte Rendus, vol. 247, pp. 431-433, 1958.
[12]	P. Vernotte, “Some possible complications in the phenomenon of thermal conduction,” Compte Rendus, vol. 252, pp. 2190-2191, 1961.
[13]	M. A. Al-Nimr and V. S. Arpaci, “The thermal behavior of thin metal films in the hyperbolic two-step model,” International Journal of Heat and Mass Transfer, vol. 43, no. 11, pp. 2021-2028, 2000. · Zbl 0967.80506 · doi:10.1016/S0017-9310(99)00160-X
[14]	M. A. Al-Nimr, O. M. Haddad, and V. S. Arpaci, “Thermal behavior of metal films-A hyperbolic two-step model,” Heat and Mass Transfer, vol. 35, no. 6, pp. 459-464, 1999. · Zbl 0967.80506 · doi:10.1007/s002310050348
[15]	M. A. Al-Nimr, B. A. Abu-Hijleh, and M. A. Hader, “Effect of thermal losses on the microscopic hyperbolic heat conduction model,” Heat and Mass Transfer, vol. 39, no. 3, pp. 201-207, 2003.
[16]	M. Naji, M.A. Al-Nimr, and M. Hader, “The validity of using the microscopic hyperbolic heat conduction model under a harmonic fluctuating boundary heating source,” International Journal of Thermophysics, vol. 24, no. 2, pp. 545-557, 2003. · Zbl 1027.80004 · doi:10.1023/A:1022984324606
[17]	M. A. Al-Nimr and M. K. Alkam, “Overshooting phenomenon in the hyperbolic microscopic heat conduction model,” International Journal of Thermophysics, vol. 24, no. 2, pp. 577-583, 2003. · doi:10.1023/A:1022988425515
[18]	D. Y. Tzou, Macro-to-Microscale Heat Transfers-The Lagging Behavior, Taylor & Francis, New York, NY, USA, 1997.
[19]	C. I. Christov and P. M. Jordan, “Heat conduction paradox involving second-sound propagation in moving media,” Physical Review Letters, vol. 94, no. 15, Article ID 154301, 4 pages, 2005. · doi:10.1103/PhysRevLett.94.154301
[20]	S. M. Zubair and M. A. Chaudhry, “Heat conduction in a semi-infinite solid due to time-dependent laser source,” International Journal of Heat and Mass Transfer, vol. 39, no. 14, pp. 3067-3074, 1996. · Zbl 0964.74502 · doi:10.1016/0017-9310(95)00388-6
[21]	M. Lewandowska, “Hyperbolic heat conduction in the semi-infinite body with a time-dependent laser heat source,” Heat and Mass Transfer, vol. 37, no. 4-5, pp. 333-342, 2001. · doi:10.1007/s002310000176
[22]	S. H. Chan, J. D. Low, and W. K. Mueller, “Hyperbolic heat conduction in catalytic supported crystallites,” AIChE Journal, vol. 17, pp. 1499-1501, 1971. · doi:10.1002/aic.690170636
[23]	L. G. Hector Jr., W. S. Kim, and M. N. Özisik, “Propagation and reflection of thermal waves in a finite medium due to axisymmetric surface sources,” International Journal of Heat and Mass Transfer, vol. 35, no. 4, pp. 897-912, 1992. · Zbl 0834.35057 · doi:10.1016/0017-9310(92)90256-R
[24]	D. Y. Tzou, “The thermal shock phenomena induced by a rapidly propagating crack tip: experimental evidence,” International Journal of Heat and Mass Transfer, vol. 35, no. 10, pp. 2347-2356, 1992. · doi:10.1016/0017-9310(92)90077-6
[25]	A. Vedavarz, S. Kumar, and M. K. Moallemi, “Significance of non-Fourier heat waves in conduction,” Journal of Heat Transfer, vol. 116, no. 1, pp. 221-226, 1994. · doi:10.1115/1.2910859
[26]	R. V. Churchill, Operational Mathematics, McGraw-Hill, New York, NY, USA, 1958. · Zbl 0083.33102