Approximate analyses of Fourier and non-Fourier heat conduction models by the variational principles based on Laplace transforms

References

• A. N. Smith, J. L. Hostetler, and P. M. Norris, Nonequilibrium Heating in Metal Films: An Analytical and Numerical Analysis, Numer. Heat Transfer A, vol. 35, pp. 859–873, 1999.
   Web of Science ®Google Scholar
• A. Horvat, Y. Sinai, and P. Tofilo, Semi-Analytical Treatment of Wall Heat Transfer Coupled to a Numerical Simulation Model of Fire, Numer. Heat Transfer A, vol. 55, pp. 517–533, 2008.
   Web of Science ®Google Scholar
• A. Yevtushenko and S. J. Matysiak, On Approximate Solutions of Temperature and Thermal Stresses in an Elastic Semi-Space Due to Laser Heating, Numer. Heat Transfer A, vol. 47, pp. 899–915, 2005.
   Web of Science ®Google Scholar
• H. A. Machado and R. M. Cotta, Analysis of Internal Convection with Variable Physical Properties via Integral Transformation, Numer. Heat Transfer A, vol. 36, pp. 699–724, 1999.
   Web of Science ®Google Scholar
• J. W. Yang and S. G. Bankoff, Solidification Effects on the Fragmentation of Molten Metal Drops behind a Pressure Shock Wave, J. Heat Transfer, vol. 109, pp. 226–231, 1987.
   Web of Science ®Google Scholar
• K. T. Yang and A. Szewczyk, An Approximate Treatment of Unsteady Heat Conduction in Semi-Infinite Solids with Variable Thermal Properties, J. Heat Transfer, vol. 81, pp. 251–252, 1959.
   Google Scholar
• B. T. F. Chung and J. S. Hsiao, Heat Transfer with Ablation in a Finite Slab Subjected to Time-Variant Heat Fluxes, Aiaa J., vol. 23, pp. 145–150, 1985.
   Google Scholar
• T. R. Goodman, Application of Integral Methods to Transient Non-Linear Heat Transfer, Adv. Heat Transfer, vol. 1, pp. 51–122, 1964.
   Google Scholar
• P. V´An and W. Muschik, Structure of Variational Principles in Nonequilibrium Thermodynamics, Phys. Rev. E., vol. 52, pp. 3584–3590, 1995.
   Web of Science ®Google Scholar
• G. Lebon, P. Perzyna, and K. Wilmanski, Recent Developments in Thermomechanics of Solids, pp. 221–366, Springer, Berlin, 1980.
   Google Scholar
• L. Onsager, Reciprocal Relations in Irreversible Processes. I, Phys. Rev., vol. 37, pp. 405–426, 1931.
   Google Scholar
• L. Onsager, Reciprocal Relations in Irreversible Processes. II, Phys. Rev., vol. 38, pp. 2265–2279, 1931.
   Google Scholar
• L. Onsager and S. Maclup, Fluctuations and Irreversible Processes, Phys. Rev., vol. 91, pp. 1505–1512, 1953.
   Web of Science ®Google Scholar
• P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, pp. 126–219, Wiley, New York, 1971.
   Google Scholar
• M. A. Biot, Variational Principles in Irreversible Thermodynamics with Application to Viscoelasticity, Phys. Rev., vol. 97, pp. 1463–1469, 1955.
   Web of Science ®Google Scholar
• M. A. Biot, Variational Principles in Heat Transfer, pp. 1–50, Clarendon Press, Oxford University Press, London, 1970.
   Google Scholar
• I. Gyarmati, On the “Governing Principle of Dissipative Processes” and Its Extension to Non-Linear Problems, Annalen Der Physik, vol. 23, pp. 353–378, 1969.
   Web of Science ®Google Scholar
• J. Verh´as, Gyarmati’s Variational Principle of Dissipative Processes, Entropy, vol. 16, pp. 2362–2383, 2014.
   Web of Science ®Google Scholar
• V. A. Cimmelli, Weakly Nonlocal Thermodynamics of Anisotropic Rigid Heat Conductors Revisited, J. Non-Equilib. Thermodyn., vol. 36, pp. 285–309, 2011.
   Web of Science ®Google Scholar
• W. Muschik, P. V´an, and C. Papenfuss, Variational Principles in Thermodynamics, Technische Mechanik, vol. 20, pp. 105–112, 2005.
   Google Scholar
• J. Merker and M. Krueger, On a Variational Principle in Thermodynamics, Contin. Mech. Thermodyn., vol. 25, pp. 779–793, 2013.
   Web of Science ®Google Scholar
• B. Vujanovic and D. Djukic, On One Variational Principle of Hamilton’s Type for Nonlinear Heat Transfer Problem, Int. J. Heat Mass Transfer, vol. 15, pp. 1111–1123, 1972.
   Web of Science ®Google Scholar
• P. Rosen, On Variational Principles for Irreversible Processes, J. Chem. Phys., vol. 21, pp. 1220–1221, 1953.
   Web of Science ®Google Scholar
• A. Prasad and H. C. Agrawal, Biot’s Variational Principle for a Stefan Problem, AIAA J., vol. 10, pp. 325–327, 1972.
   Google Scholar
• A. Prasad and S. N. Sinha, Radiative Ablation of Melting Solids, AIAA J., vol. 14, pp. 1494–1497, 1975.
   Google Scholar
• V. D. Rao, P. K. Sarma, and G. J. V. J. Raju, Biot’s Variational Method to Fluidized-Bed Coating on Thin Plates, J. Heat Transfer, vol. 107, pp. 258–260, 1985.
   Web of Science ®Google Scholar
• M. A. Biot, New Methods in Heat Flow Analysis with Application to Flight Structures, J. Aeronaut. Sci., vol. 24, pp. 857–873, 1957.
   Google Scholar
• A. Stark, Approximation Methods for the Solution of Heat Conduction Problems Using Gyarmati’s Principle, Ann. Phys., vol. 31, pp. 53–75, 1974.
   Google Scholar
• P. Singh, The Application of the Governing Principle of Dissipative Processes to Benard Convection, Int. J. Heat Mass Transfer, vol. 19, pp. 581–588, 1976.
   Web of Science ®Google Scholar
• N. H. Tung, An Application of the Governing Principle of Dissipative Processes to Eigenvalue Problem of Hydro-Thermodynamics Fluids, Acta Phys. Hung., vol. 65, pp. 69–78, 1989.
   Google Scholar
• P. Singh, Lagrangian Thermodynamics as a Particular Case of Governing Principle of Dissipative Processes, Acta Phys. Acad. Sci. Hung., vol. 43, pp. 133–140, 1977.
   Google Scholar
• D. D. Joseph and L. Preziosi, Heat Waves, Rev. Modern Phys., vol. 61, pp. 41–73, 1989.
   Google Scholar
• D. S. Tang, Y. C. Hua, and B. Y. Cao, Thermal Wave Propagation Through Nanofilms in Ballistic-Diffusive Regime by Monte Carlo Simulations, Int. J. Therm. Sci., vol. 109, pp. 81–89, 2016.
   Web of Science ®Google Scholar
• M. K. Zhang, B. Y. Cao, and Y. C. Guo, Numerical Studies on Damping of Thermal Waves, Int. J. Therm. Sci., vol. 84, pp. 9–20, 2014.
   Web of Science ®Google Scholar
• W. J. Yao and B. Y. Cao, Thermal Wave Propagation in Graphene by Molecular Dynamics Simulations, Chin. Sci. Bulletin, vol. 59, pp. 3495–3503, 2014.
   Google Scholar
• Y. Dong, B. Y. Cao, and Z. Y. Guo, Generalized Heat Conduction Laws Based on Thermomass Theory and Phonon Hydrodynamics, J. Appl. Phys., vol. 110, pp. 063504, 2011.
   Web of Science ®Google Scholar
• B. Y. Cao and Z. Y. Guo, Equation of Motion of Phonon Gas and Non-Fourier Heat Conduction, J. Appl. Phys., vol. 102, pp. 053503, 2007.
   Web of Science ®Google Scholar
• J. T. O’Toole, Variational Principles for Time-Dependent Transport Problems, Chem. Eng. Sci., vol. 22, pp. 313–318, 1967.
   Web of Science ®Google Scholar
• L. Davies and D. Brian, Integral Transforms and Their Applications, Chap. 7, Springer, New York, 2002.
   Google Scholar
• S. N. Li and B. Y. Cao, Generalized Variational Principles for Heat Conduction Models Based on Laplace Transforms, Int. J. Heat Mass Transfer, vol. 103, pp. 1176–1180, 2016.
   Web of Science ®Google Scholar
• C. Cattaneo, Sur une forme de léquation de lachaleur éliminant le paradoxe d’une propagation instantanée, Comptes Rendus, vol. 247, pp. 431–433, 1958.
   Google Scholar
• P. Vernotte, Les Paradoxes De La Théorie Continue De L’équation De La Chaleur, Comptes Rendus, vol. 246, pp. 3154–3155, 1958.
   Google Scholar
• S. I. Anisinov, B. L. Kapeliovich, and T. L. Perelman, Electron Emission from Metal Surfaces Exposed to Ultrashort Laser Pulses, Soviet Phys. JETP, vol. 39, pp. 375–377, 1974.
   Google Scholar
• R. A. Guyer and J. A. Krumhansl, Solution of the Linearized Phonon Boltzmann Equation, Phys. Rev., vol. 14, pp. 766–778, 1996.
   Google Scholar