Abstract
Reasonable unsteady three-dimensional explicit analytical solutions are derived with different methods for the widely used bio-heat transfer equation-Pennes equation. The condition to decide temperature oscillation is obtained in this paper. In other cases the temperature would vary monotonously along geometric coordinates as time goes by. There have been very few open reports of explicit unsteady multidimensional exact analytical solutions published in literature. Besides its irreplaceable theoretical value, the analytical solution can also serve as standard solution to check numerical calculation, and therefore promote the development of numerical method of computational heat transfer. In addition, some new special methods have been given originally and deserved further attention.
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References
Pennes H H. Analysis of tissue and arterial temperatures in the resting human forearm. J Appl Physiol, 1948, 1: 93–122
Chen D. The study of non-Fourier heat conduction effect and the influence on heat transfer of low-temperature living beings. PhD Dissertation. Shanghai: East China University of Industry, 1994
Wang B, Wang Y. The study of bio-heat equation. J Eng Thermophys, 1993, 14: 166–170
Liu J, Ren Z, Wang C. Thermal wave theory about temperature oscillations effect in living tissues. Chin J Med Phys, 1995, 12(4): 215–218
Roemer R B, Oleson J R, Cetas T C. Oscillatory temperatures response to constant power applied to canine muscle. Amer. J Appl Physiol, 1985, 249: 153–158
Wang Y. The basic study on thermophysics in tumor hyperthermia.: PhD Dissertation. Beijing: Tsinghua University, 1994
Cai R, Zhang N. Unsteady one-dimensional analytical solutions for bio-heat transfer equations. Prog Nat Sci, 1998, 8(3): 331–336
Cai R. Some explicit analytical solutions of unsteady compressible flow. ASME J Fluids Eng, 1998, 120: 760–764
Cai R, Zhang N. Some algebraically explicit analytical solutions of unsteady nonlinear heat conduction. ASME J Heat Transfer, 2001, 123: 1189–1191
Cai R, Zhang D. Some explicit analytical solutions of unsteady two phase flow. Chin J Mech Eng, 2001, 37(9): 1–3
Cai R, Zhang N. Explicit analytical solutions of incompressible unsteady 2-D laminar flow with heat transfer. Int J Heat Mass Transfer, 2002, 45: 2623–2627
Cai R, Gou C. Discussion on the convective heat transfer and field synergy principle. Int J Heat Mass Transfer, 2007, 50: 5168–5176
Cai R. Some explicit analytical solutions for unsteady compressible isentropic flow. J Eng Thermophys, 2001, 22(2): 159–162
Cai R, Liu Q. A new method for deriving analytical solutions of partial differential equations-algebraically explicit analytical solutions of two-buoyancy natural convection in porous media. Sci China Ser G-Phys Mech Astro, 2008, 51(11): 1733–1744
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Li, Y., Cai, R. Unsteady 3D algebraically explicit analytical solutions for bio-heat transfer equations. Sci. China Technol. Sci. 54, 362–368 (2011). https://doi.org/10.1007/s11431-010-4203-1
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DOI: https://doi.org/10.1007/s11431-010-4203-1