Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions. (English) Zbl 1270.80002
Summary: This paper presents two exact explicit solutions for the three dimensional dual-phase lag (DLP) heat conduction equation, during the derivation of which the method of trial and error and the authors’ previous experiences are utilized. To the authors’ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.
MSC:
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 35K05 | Heat equation |
References:
| [1] | Qiu T., Juhasz T., Suarez C., Bron W., Tien C.: Femtosecond laser heating of multi-layer metals-II experiments. Int. J. Heat Mass Transf. 37(17), 2799–2808 (1994) · doi:10.1016/0017-9310(94)90397-2 |
| [2] | Guillemet P., Bardon J., Rauch C.: Experimental route to heat conduction beyond the Fourier equation. Int. J. Heat Mass Transf. 40(17), 4043–4053 (1997) · doi:10.1016/S0017-9310(97)00051-3 |
| [3] | Ozisik M., Tzou D.: On the wave theory in heat conduction. Trans. AMSE J. Heat Transf. 116, 526–535 (1994) · doi:10.1115/1.2910903 |
| [4] | Tzou D.: A unified field approach for heat conduction from macro-to micro-scale. ASME J. Heat Transf. 117, 8–16 (1995) · doi:10.1115/1.2822329 |
| [5] | McCartin B., Causley M.: Angled derivative approximation of the hyperbolic heat conduction equations. Appl. Math. Computat. 182(2), 1581–1607 (2006) · Zbl 1107.65083 · doi:10.1016/j.amc.2006.05.045 |
| [6] | Barletta A., Zanchini E.: Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady-periodic electric field. Int. J. Heat Mass Transf. 39(6), 1307–1315 (1996) · doi:10.1016/0017-9310(95)00202-2 |
| [7] | Barletta A.: Hyperbolic propagation of an axisymmetric thermal signal in an infinite solid medium. Int. J. Heat Mass Transf. 39(15), 3261–3271 (1996) · doi:10.1016/0017-9310(95)00391-6 |
| [8] | Barletta A., Zanchini E.: Three-dimensional propagation of hyperbolic thermal waves in a solid bar with rectangular cross-section. Int. J. Heat Mass Transf. 42, 219–229 (1999) · Zbl 0966.74017 · doi:10.1016/S0017-9310(98)00190-2 |
| [9] | Lewandowska M., Malinowski L.: An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides. Int. Commun. Heat and Mass Transf. 33, 61–69 (2006) · doi:10.1016/j.icheatmasstransfer.2005.08.004 |
| [10] | Cai R., Zhang N.: Explicit analytical solutions of non-Fourier heat conduction equation for IC chip. Chin. Sci. Bull. 43, 1080–1084 (1998) · Zbl 0984.80002 · doi:10.1007/BF02883077 |
| [11] | Cai R., Zhang N., He Y.: One-dimensional algebraic explicit solutions of unsteady non-Fourier heat conduction in a sphere. Prog. Nat. Sci. 9, 34–38 (1999) |
| [12] | Wang L., Xu M.: Well-posedness and solution structure of dual-phase-lagging heat conduction. Int. J. Heat Mass Transf. 44, 1659–1669 (2001) · Zbl 1016.80004 · doi:10.1016/S0017-9310(00)00229-5 |
| [13] | Xu M., Wang L.: Thermal oscillation and resonance in dual-phase-lagging heat conduction. Int. J. Heat Mass Transf. 45, 1055–1061 (2002) · Zbl 0995.80005 · doi:10.1016/S0017-9310(01)00199-5 |
| [14] | Wang L., Xu M.: Well-posedness of dual-phase-lagging heat conduction equation: higher dimensions. Int. J. Heat Mass Transf. 45, 1165–1171 (2002) · Zbl 0995.80004 · doi:10.1016/S0017-9310(01)00188-0 |
| [15] | Lamb H.: Hydrodynamics, 6th edn. Cambridge Univ. Press, London (1932) · JFM 58.1298.04 |
| [16] | Jacob M.: Heat Transfer. John Wiley & Sons, Inc., New York (1949) |
| [17] | Cai, R., Jiang, H., Sun, C.: Some analytical solutions applicable to verify 3D numerical methods in turbomachines. In: ImechE Conference Publication, vol. C80/84, pp. 255–263 (1984) |
| [18] | Xu, J., Shi, J., Ni, W.: Three dimensional incompressible flow solution of an axial compressor using pseudostream-functions formulation. ASME paper 89-GT-319 (1989) |
| [19] | Gong, Y., Cai, R.: 3D MSLM-A new engineering approach to the inverse problem of 3D cascade. ASME paper 89-GT-48 (1989) |
| [20] | Shen M., Liu Q., Zhang Z.: Calculation of 3-D transonic flow in turbomachinery with the generalized von Mises coordinates system. Sci. China Ser. A 39, 1084–1095 (1996) |
| [21] | Hao T.: Exact solution of plane isolated crack normal to a bimaterial interface of infinite extent. Acta Mech. Sin. 22, 455–468 (2006) · Zbl 1202.74143 · doi:10.1007/s10409-006-0024-7 |
| [22] | Chang T.: Explicit solution of the radial breathing mode frequency of single-walled carbon nanotubes. Acta Mech. Sin. 23, 159–162 (2007) · Zbl 1202.74074 · doi:10.1007/s10409-007-0059-4 |
| [23] | Bian X., Chen Y.: An explicit time domain solution for ground stratum response due to harmonic moving load. Acta Mech. Sin. 22, 469–478 (2006) · Zbl 1202.74110 · doi:10.1007/s10409-006-0022-9 |
| [24] | Cai R.: Some explicit analytical solutions of unsteady compressible flow. ASME Trans. J. Fluids Eng. 120, 760–764 (1998) · doi:10.1115/1.2820735 |
| [25] | Cai R., Zhang N.: Some algebraically explicit analytical solutions of unsteady nonlinear heat conduction. ASME Trans. J. Heat Transf. 123, 1189–1191 (2001) · doi:10.1115/1.1392990 |
| [26] | Cai R., Gou C., Li H.: Algebraically explicit analytical solutions of unsteady 3-D nonlinear non-Fourier (hyperbolic) heat conduction. Int. J. Thermal Sci. 45, 893–896 (2006) · doi:10.1016/j.ijthermalsci.2005.12.007 |
| [27] | Cai R., Zhang N., Gou C.: Explicit analytical solutions of axisymmetric laminar natural convection along vertical tubes. Prog. Nat. Sci. 15, 258–261 (2005) · Zbl 1115.76324 · doi:10.1080/10020070512331342080 |
| [28] | Gou C., Cai R., Zhang N.: Explicit analytical solutions of transport equations considering non-Fourier and non-Fick effects in porous media. Prog. Nat. Sci. 15, 545–549 (2005) · Zbl 1086.76600 · doi:10.1080/10020070512331342530 |
| [29] | Cai R., Gou C., Zhang N.: Explicit analytical solutions of the anisotropic Brinkman model for the natural convection in porous media (II). Sci China Ser G 48, 422–430 (2005) · doi:10.1360/142004-0118 |
| [30] | Gou C., Cai R.: An analytical solution of non-Fourier Chen- Holmes bio-heat transfer equation. Chin. Sci. Bull. 50, 2791–2792 (2005) · doi:10.1360/982005-871 |
| [31] | Gou C., Cai R.: One-dimensional unsteady general solution for Wang bioheat transfer equation with heat source proportional to temperature. Prog. Nat. Sci. 15, 545–549 (2005) · Zbl 1086.76600 · doi:10.1080/10020070512331342530 |
| [32] | Cai R., Gou C.: Exact solutions of double diffusive convection in cylindrical coordinates with Le = 1. Int. J. Heat Mass Transf. 49, 3997–4002 (2006) · Zbl 1108.80305 · doi:10.1016/j.ijheatmasstransfer.2006.03.040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
