# American Institute of Mathematical Sciences

March  2019, 8(1): 57-72. doi: 10.3934/eect.2019004

## On a C-integrable equation for second sound propagation in heated dielectrics

 School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA

Received  January 2018 Revised  April 2018 Published  January 2019

An exactly solvable model in heat conduction is considered. The $C$-integrable (i.e., change-of-variables-integrable) equation for second sound (i.e., heat wave) propagation in a thin, rigid dielectric heat conductor uniformly heated on its lateral side by a surrounding medium under the Stefan-Boltzmann law is derived. A simple change-of-variables transformation is shown to exactly map the nonlinear governing partial differential equation to the classical linear telegrapher's equation. In a one-dimensional context, known integral-transform solutions of the latter are adapted to construct exact solutions relevant to heat transfer applications: (ⅰ) the initial-value problem on an infinite domain (the real line), and (ⅱ) the initial-boundary-value problem on a semi-infinite domain (the half-line). Possible "second law violations" and restrictions on the $C$-transformation are noted for some sets of parameters.

Citation: Ivan C. Christov. On a C-integrable equation for second sound propagation in heated dielectrics. Evolution Equations & Control Theory, 2019, 8 (1) : 57-72. doi: 10.3934/eect.2019004
##### References:
 [1] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Studies in Applied and Numerical Mathematics, SIAM, Philadelphia, PA, 1981. doi: 10.1137/1.9781611970883. [2] P. J. Antaki, Importance of nonFourier heat conduction in solid-phase reactions, Combust. Flame, 112 (1998), 329-341. doi: 10.1016/S0010-2180(97)00131-4. [3] C. Bai and A. S. Lavine, On hyperbolic heat conduction and the second law of thermodynamics, ASME J. Heat Transfer, 117 (1995), 256-263. doi: 10.1115/1.2822514. [4] O. G. Bakunin, Mysteries of diffusion and labyrinths of destiny, Phys.-Usp., 46 (2003), 309-313. doi: 10.1070/PU2003v046n03ABEH001289. [5] S. Bargmann, P. Steinmann and P. M. Jordan, On the propagation of second-sound in linear and nonlinear media: Results from Green-Naghdi theory, Phys. Lett. A, 372 (2008), 4418-4424. doi: 10.1016/j.physleta.2008.04.010. [6] T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, Introduction to Heat Transfer, 6th ed., John Wiley & Sons, Hoboken, NJ, 2011. [7] J. J. Bissell, Thermal convection in a magnetized conducting fluid with the Cattaneo-Christov heat-flow model, Proc. R. Soc. A, 472 (2016), 20160649, 20pp. doi: 10.1098/rspa.2016.0649. [8] D. R. Bland, Wave Theory and Applications, Oxford University Press, London, 1988. [9] F. Calogero, Why are certain nonlinear PDEs both widely applicable and integrable?, in What Is Integrability? (ed. V. E. Zakharov), Springer Series in Nonlinear Dynamics, Springer, Berlin/Heidelberg, 1991, 1-62. doi: 10.1007/978-3-642-88703-1_1. [10] F. Calogero, New C-integrable and S-integrable systems of nonlinear partial differential equations, J. Nonlinear Math. Phys., 24 (2017), 142-148. doi: 10.1080/14029251.2017.1287387. [11] D. Campos and V. Méndez, Different microscopic interpretations of the reaction-telegrapher equation, J. Phys. A: Math. Theor., 42 (2009), 075003, 13pp. doi: 10.1088/1751-8113/42/7/075003. [12] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, The Clarendon Press, Oxford University Press, New York, 1988. [13] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1949), 83-101. [14] C. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée, C. R. Acad. Sci. Paris, 247 (1958), 431-433. [15] M. Chester, Second sound in solids, Phys. Rev., 131 (1963), 2013-2015. doi: 10.1103/PhysRev.131.2013. [16] C. I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mech. Res. Commun., 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003. [17] I. C. Christov, Wave solutions, in Encyclopedia of Thermal Stresses (ed. R. B. Hetnarski), Springer, Netherlands, 2014, 6495-6506. doi: 10.1007/978-94-007-2739-7_33. [18] I. C. Christov and P. M. Jordan, On the propagation of second-sound in nonlinear media: Shock, acceleration and traveling wave results, J. Thermal Stresses, 33 (2010), 1109-1135. doi: 10.1080/01495739.2010.517674. [19] M. Ciarletta, B. Straughan and V. Tibullo, Christov–Morro theory for non-isothermal diffusion, Nonlinear Anal. RWA, 13 (2012), 1224-1228. doi: 10.1016/j.nonrwa.2011.10.014. [20] B. D. Coleman, M. Fabrizio and D. R. Owen, On the thermodynamics of second sound in dielectric crystals, Arch. Rational Mech. Anal., 80 (1982), 135-158. doi: 10.1007/BF00250739. [21] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. [22] R. B. Dingle, The velocity of second sound in various media, Proc. Phys. Soc. A, 65 (1952), 1044-1050. doi: 10.1088/0370-1298/65/12/313. [23] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, Berlin, 1974. [24] W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Continuum Mech. Thermodyn., 5 (1993), 3-50. doi: 10.1007/BF01135371. [25] T. S. Fisher, Thermal Energy at the Nanoscale, vol. 3 of Lessons from Nanoscience, World Scientific Publishing Co., Singapore, 2013. doi: 10.1142/8716. [26] P. H. Francis, Thermo-mechanical effects in elastic wave propagation: A survey, J. Sound Vib., 21 (1972), 181-192. doi: 10.1016/0022-460X(72)90905-4. [27] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett., 18 (1967), 1095-1097. doi: 10.1137/1018076. [28] R. Garra, On the generalized Hardy–Hardy–Maurer model with memory effects, Nonlinear Dyn., 86 (2016), 861-868. doi: 10.1007/s11071-016-2928-5. [29] M. Gentile and B. Straughan, Hyperbolic diffusion with Christov–Morro theory, Math. Comput. Simulat., 127 (2016), 94-100. doi: 10.1016/j.matcom.2012.07.010. [30] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅰ. Classical continuum physics, Proc. R. Soc. Lond. A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020. [31] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅱ. Generalized continua, Proc. R. Soc. Lond. A, 448 (1995), 357-377. doi: 10.1098/rspa.1995.0021. [32] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅲ. Mixtures of interacting continua, Proc. R. Soc. Lond. A, 448 (1995), 379-388. doi: 10.1098/rspa.1995.0022. [33] R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice Hall, Englewood Hills, NJ, 1988. [34] L. Guo, S. L. Hodson, T. S. Fisher and X. Xu, Heat transfer across metal-dielectric interfaces during ultrafast-laser heating, ASME J. Heat Transfer, 134 (2012), 042402. doi: 10.1115/IMECE2011-64165. [35] J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford Mathematical Monographs, Oxford University Press, New York, 2010. doi: 10.1093/acprof:oso/9780199541645.001.000. [36] J. Jaisaardsuetrong and B. Straughan, Thermal waves in a rigid heat conductor, Phys. Lett. A, 366 (2007), 433-436. doi: 10.1016/j.physleta.2007.02.058. [37] P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189. [38] P. M. Jordan, A note on the Lambert $W$-function: Applications in the mathematical and physical sciences, in Mathematics of Continuous and Discrete Dynamical Systems (ed. A. B. Gumel), American Mathematical Society, 618 (2014), 247-263. doi: 10.1090/conm/618. [39] P. M. Jordan, A nonstandard finite difference scheme for nonlinear heat transfer in a thin finite rod, J. Diff. Eq. Appl., 9 (2003), 1015-1021. doi: 10.1080/1023619031000146922. [40] P. M. Jordan, Second-sound propagation in rigid, nonlinear conductors, Mech. Res. Commun., 68 (2015), 52-59. doi: 10.1016/j.mechrescom.2015.04.005. [41] P. M. Jordan and A. Puri, Digital signal propagation in dispersive media, J. Appl. Phys., 85 (1999), 1273-1282. doi: 10.1063/1.369258. [42] D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys, 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41. [43] D. D. Joseph and L. Preziosi, Addendum to the paper "Heat Waves" [Rev. Mod. Phys. 61, 41 (1989)], Rev. Mod. Phys, 62 (1990), 375-391. doi: 10.1103/RevModPhys.62.375. [44] D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics, 4th edition, Springer Science+Business Media, Dordrecht, 2010. doi: 10.1007/978-90-481-3074-0. [45] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. Ser. 5, 39 (1895), 422-443. doi: 10.1080/14786449508620739. [46] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. A, 157 (1867), 49-88. doi: 10.1098/rstl.1867.0004. [47] R. E. Mickens and P. M. Jordan, A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Eq., 20 (2004), 639-649. doi: 10.1002/num.20003. [48] M. F. Modest, Radiative Heat Transfer, 3rd edition, Elsevier, Oxford, UK, 2013. doi: 10.1016/B978-0-12-386944-9.50025-X. [49] A. Morro, Evolution equations and thermodynamic restrictions for dissipative solids, Math. Comput. Modell., 52 (2010), 1869-1876. doi: 10.1016/j.mcm.2010.07.021. [50] A. Morro, Governing equations in non-isothermal diffusion, Int. J. Non-Linear Mech., 55 (2013), 90-97. doi: 10.1016/j.ijnonlinmec.2013.04.010. [51] I. Müller and T. Ruggeri, Extended Thermodynamics, no. 37 in Tracts in Natural Philosophy, Springer-Verlag, Berlin/Heidelberg, 1993. doi: 10.1007/978-1-4684-0447-0. [52] M. Ostoja-Starzewski, A derivation of the Maxwell–Cattaneo equation from the free energy and dissipation potentials, Int. J. Eng. Sci., 47 (2009), 807-810. doi: 10.1016/j.ijengsci.2009.03.002. [53] M. Ostoja-Starzewski and A. Malyarenko, Continuum mechanics beyond the second law of thermodynamics, Proc. R. Soc. A, 470 (2014), 20140531. doi: 10.1098/rspa.2014.0531. [54] M. Ostoja-Starzewski and B. V. Raghavan, Continuum mechanics versus violations of the second law of thermodynamics, J. Thermal Stresses, 39 (2016), 734-749. doi: 10.1080/01495739.2016.1169140. [55] A. Pantokratoras, Comment on the paper "On Cattaneo–Christov heat flux model for Carreau fluid flow over a slendering sheet, Hashim, Masood Khan, Results in Physics, 7 (2017), 310–319", Res. Phys., 7 (2017), 1504-1505. doi: 10.1016/j.rinp.2017.04.008. [56] M. B. Rubin, Hyperbolic heat conduction and the second law, Int. J. Eng. Sci., 30 (1992), 1665-1676. doi: 10.1016/0020-7225(92)90134-3. [57] S. L. Sobolev, On hyperbolic heat-mass transfer equation, Int. J. Heat Mass Transfer, 122 (2018), 629-630. doi: 10.1016/j.ijheatmasstransfer.2018.02.022. [58] B. Straughan, Thermal convection with the Cattaneo–Christov model, Int. J. Heat Mass Transfer, 53 (2010), 95-98. doi: 10.1016/j.ijheatmasstransfer.2009.10.001. [59] B. Straughan, Heat Waves, vol. 117 of Applied Mathematical Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4. [60] P. Vernotte, Les paradoxes de la théorie continue de l'équation de la chaleur, C. R. Acad. Sci. Paris, 246 (1958), 3154-3155. [61] A. G. Webster, Partial Differential Equations of Mathematical Physics, Dover Publications, Mineola, NY, 1955. [62] J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids, Oxford University Press, New York, 2001. doi: 10.1093/acprof:oso/9780198507796.001.0001.

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##### References:
 [1] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Studies in Applied and Numerical Mathematics, SIAM, Philadelphia, PA, 1981. doi: 10.1137/1.9781611970883. [2] P. J. Antaki, Importance of nonFourier heat conduction in solid-phase reactions, Combust. Flame, 112 (1998), 329-341. doi: 10.1016/S0010-2180(97)00131-4. [3] C. Bai and A. S. Lavine, On hyperbolic heat conduction and the second law of thermodynamics, ASME J. Heat Transfer, 117 (1995), 256-263. doi: 10.1115/1.2822514. [4] O. G. Bakunin, Mysteries of diffusion and labyrinths of destiny, Phys.-Usp., 46 (2003), 309-313. doi: 10.1070/PU2003v046n03ABEH001289. [5] S. Bargmann, P. Steinmann and P. M. Jordan, On the propagation of second-sound in linear and nonlinear media: Results from Green-Naghdi theory, Phys. Lett. A, 372 (2008), 4418-4424. doi: 10.1016/j.physleta.2008.04.010. [6] T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, Introduction to Heat Transfer, 6th ed., John Wiley & Sons, Hoboken, NJ, 2011. [7] J. J. Bissell, Thermal convection in a magnetized conducting fluid with the Cattaneo-Christov heat-flow model, Proc. R. Soc. A, 472 (2016), 20160649, 20pp. doi: 10.1098/rspa.2016.0649. [8] D. R. Bland, Wave Theory and Applications, Oxford University Press, London, 1988. [9] F. Calogero, Why are certain nonlinear PDEs both widely applicable and integrable?, in What Is Integrability? (ed. V. E. Zakharov), Springer Series in Nonlinear Dynamics, Springer, Berlin/Heidelberg, 1991, 1-62. doi: 10.1007/978-3-642-88703-1_1. [10] F. Calogero, New C-integrable and S-integrable systems of nonlinear partial differential equations, J. Nonlinear Math. Phys., 24 (2017), 142-148. doi: 10.1080/14029251.2017.1287387. [11] D. Campos and V. Méndez, Different microscopic interpretations of the reaction-telegrapher equation, J. Phys. A: Math. Theor., 42 (2009), 075003, 13pp. doi: 10.1088/1751-8113/42/7/075003. [12] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, The Clarendon Press, Oxford University Press, New York, 1988. [13] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1949), 83-101. [14] C. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée, C. R. Acad. Sci. Paris, 247 (1958), 431-433. [15] M. Chester, Second sound in solids, Phys. Rev., 131 (1963), 2013-2015. doi: 10.1103/PhysRev.131.2013. [16] C. I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mech. Res. Commun., 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003. [17] I. C. Christov, Wave solutions, in Encyclopedia of Thermal Stresses (ed. R. B. Hetnarski), Springer, Netherlands, 2014, 6495-6506. doi: 10.1007/978-94-007-2739-7_33. [18] I. C. Christov and P. M. Jordan, On the propagation of second-sound in nonlinear media: Shock, acceleration and traveling wave results, J. Thermal Stresses, 33 (2010), 1109-1135. doi: 10.1080/01495739.2010.517674. [19] M. Ciarletta, B. Straughan and V. Tibullo, Christov–Morro theory for non-isothermal diffusion, Nonlinear Anal. RWA, 13 (2012), 1224-1228. doi: 10.1016/j.nonrwa.2011.10.014. [20] B. D. Coleman, M. Fabrizio and D. R. Owen, On the thermodynamics of second sound in dielectric crystals, Arch. Rational Mech. Anal., 80 (1982), 135-158. doi: 10.1007/BF00250739. [21] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. [22] R. B. Dingle, The velocity of second sound in various media, Proc. Phys. Soc. A, 65 (1952), 1044-1050. doi: 10.1088/0370-1298/65/12/313. [23] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, Berlin, 1974. [24] W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Continuum Mech. Thermodyn., 5 (1993), 3-50. doi: 10.1007/BF01135371. [25] T. S. Fisher, Thermal Energy at the Nanoscale, vol. 3 of Lessons from Nanoscience, World Scientific Publishing Co., Singapore, 2013. doi: 10.1142/8716. [26] P. H. Francis, Thermo-mechanical effects in elastic wave propagation: A survey, J. Sound Vib., 21 (1972), 181-192. doi: 10.1016/0022-460X(72)90905-4. [27] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett., 18 (1967), 1095-1097. doi: 10.1137/1018076. [28] R. Garra, On the generalized Hardy–Hardy–Maurer model with memory effects, Nonlinear Dyn., 86 (2016), 861-868. doi: 10.1007/s11071-016-2928-5. [29] M. Gentile and B. Straughan, Hyperbolic diffusion with Christov–Morro theory, Math. Comput. Simulat., 127 (2016), 94-100. doi: 10.1016/j.matcom.2012.07.010. [30] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅰ. Classical continuum physics, Proc. R. Soc. Lond. A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020. [31] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅱ. Generalized continua, Proc. R. Soc. Lond. A, 448 (1995), 357-377. doi: 10.1098/rspa.1995.0021. [32] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅲ. Mixtures of interacting continua, Proc. R. Soc. Lond. A, 448 (1995), 379-388. doi: 10.1098/rspa.1995.0022. [33] R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice Hall, Englewood Hills, NJ, 1988. [34] L. Guo, S. L. Hodson, T. S. Fisher and X. Xu, Heat transfer across metal-dielectric interfaces during ultrafast-laser heating, ASME J. Heat Transfer, 134 (2012), 042402. doi: 10.1115/IMECE2011-64165. [35] J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford Mathematical Monographs, Oxford University Press, New York, 2010. doi: 10.1093/acprof:oso/9780199541645.001.000. [36] J. Jaisaardsuetrong and B. Straughan, Thermal waves in a rigid heat conductor, Phys. Lett. A, 366 (2007), 433-436. doi: 10.1016/j.physleta.2007.02.058. [37] P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189. [38] P. M. Jordan, A note on the Lambert $W$-function: Applications in the mathematical and physical sciences, in Mathematics of Continuous and Discrete Dynamical Systems (ed. A. B. Gumel), American Mathematical Society, 618 (2014), 247-263. doi: 10.1090/conm/618. [39] P. M. Jordan, A nonstandard finite difference scheme for nonlinear heat transfer in a thin finite rod, J. Diff. Eq. Appl., 9 (2003), 1015-1021. doi: 10.1080/1023619031000146922. [40] P. M. Jordan, Second-sound propagation in rigid, nonlinear conductors, Mech. Res. Commun., 68 (2015), 52-59. doi: 10.1016/j.mechrescom.2015.04.005. [41] P. M. Jordan and A. Puri, Digital signal propagation in dispersive media, J. Appl. Phys., 85 (1999), 1273-1282. doi: 10.1063/1.369258. [42] D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys, 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41. [43] D. D. Joseph and L. Preziosi, Addendum to the paper "Heat Waves" [Rev. Mod. Phys. 61, 41 (1989)], Rev. Mod. Phys, 62 (1990), 375-391. doi: 10.1103/RevModPhys.62.375. [44] D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics, 4th edition, Springer Science+Business Media, Dordrecht, 2010. doi: 10.1007/978-90-481-3074-0. [45] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. Ser. 5, 39 (1895), 422-443. doi: 10.1080/14786449508620739. [46] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. A, 157 (1867), 49-88. doi: 10.1098/rstl.1867.0004. [47] R. E. Mickens and P. M. Jordan, A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Eq., 20 (2004), 639-649. doi: 10.1002/num.20003. [48] M. F. Modest, Radiative Heat Transfer, 3rd edition, Elsevier, Oxford, UK, 2013. doi: 10.1016/B978-0-12-386944-9.50025-X. [49] A. Morro, Evolution equations and thermodynamic restrictions for dissipative solids, Math. Comput. Modell., 52 (2010), 1869-1876. doi: 10.1016/j.mcm.2010.07.021. [50] A. Morro, Governing equations in non-isothermal diffusion, Int. J. Non-Linear Mech., 55 (2013), 90-97. doi: 10.1016/j.ijnonlinmec.2013.04.010. [51] I. Müller and T. Ruggeri, Extended Thermodynamics, no. 37 in Tracts in Natural Philosophy, Springer-Verlag, Berlin/Heidelberg, 1993. doi: 10.1007/978-1-4684-0447-0. [52] M. Ostoja-Starzewski, A derivation of the Maxwell–Cattaneo equation from the free energy and dissipation potentials, Int. J. Eng. Sci., 47 (2009), 807-810. doi: 10.1016/j.ijengsci.2009.03.002. [53] M. Ostoja-Starzewski and A. Malyarenko, Continuum mechanics beyond the second law of thermodynamics, Proc. R. Soc. A, 470 (2014), 20140531. doi: 10.1098/rspa.2014.0531. [54] M. Ostoja-Starzewski and B. V. Raghavan, Continuum mechanics versus violations of the second law of thermodynamics, J. Thermal Stresses, 39 (2016), 734-749. doi: 10.1080/01495739.2016.1169140. [55] A. Pantokratoras, Comment on the paper "On Cattaneo–Christov heat flux model for Carreau fluid flow over a slendering sheet, Hashim, Masood Khan, Results in Physics, 7 (2017), 310–319", Res. Phys., 7 (2017), 1504-1505. doi: 10.1016/j.rinp.2017.04.008. [56] M. B. Rubin, Hyperbolic heat conduction and the second law, Int. J. Eng. Sci., 30 (1992), 1665-1676. doi: 10.1016/0020-7225(92)90134-3. [57] S. L. Sobolev, On hyperbolic heat-mass transfer equation, Int. J. Heat Mass Transfer, 122 (2018), 629-630. doi: 10.1016/j.ijheatmasstransfer.2018.02.022. [58] B. Straughan, Thermal convection with the Cattaneo–Christov model, Int. J. Heat Mass Transfer, 53 (2010), 95-98. doi: 10.1016/j.ijheatmasstransfer.2009.10.001. [59] B. Straughan, Heat Waves, vol. 117 of Applied Mathematical Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4. [60] P. Vernotte, Les paradoxes de la théorie continue de l'équation de la chaleur, C. R. Acad. Sci. Paris, 246 (1958), 3154-3155. [61] A. G. Webster, Partial Differential Equations of Mathematical Physics, Dover Publications, Mineola, NY, 1955. [62] J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids, Oxford University Press, New York, 2001. doi: 10.1093/acprof:oso/9780198507796.001.0001.
Schematic of a thin, rigid rod of a dielectric material (contained within the domain $\Omega$ with boundary $\partial\Omega$ and unit surface normal $\mathit{\boldsymbol{\hat{n}}}$) subject to uniform heating/cooling of its lateral surface by its surroundings. The dielectric is long in the $x$-direction and thin in the cross-sectional $y$- and $z$-directions, so that heat conduction can be assumed to be unidirectional and radiation to be a volumetric source term in the energy equation. In this particular illustration, the temperature $\vartheta$ at one end ($x = 0$) can be prescribed. The temperature in the surrounding medium (i.e., in $\mathbb{R}^3\setminus\Omega$) is the constant $\vartheta_\infty$
Oscillatory behavior of solutions, given in Eq. (27), to the ODE (25) for $\lambda_0 = \epsilon = 1 \Rightarrow \lambda_0^2 < 4\epsilon$. Solid curve correspond to $\Theta_{\rm i} = 1 > \Theta_{\rm R} = 0.1$, dashed curve corresponds to $\Theta_{\rm i} = 1 < \Theta_{\rm R} = 2$. For this choice of parameters, the solid curve's first minimum "dips" below $\Theta^4 = 0$; thus, this solution is not strictly non-negative and $\Theta$ can become imaginary
Dimensionless temperature $\Theta$ profiles versus $X$ at different dimensionless times $T$ showing the relaxation of a unit pulse via the exact solution in Eq. (40). (a) $T = 0.5$, (b) $T = 1$, (c) $T = 2$, (d) $T = 4$. Here, $\epsilon = 0.1$, $\lambda_0 = \tfrac{3}{2}\sqrt{4\epsilon}$ ($k<0$) for solid curves, while $\lambda_0 = \sqrt{4\epsilon}$ ($k = 0$) for dashed curves
Dimensionless temperature $\Theta$ profiles versus $X$ at different dimensionless times $T$ showing the relaxation of a unit pulse via the exact solution in Eq. (40). (a) $T = 0.5$, (b) $T = 1$, (c) $T = 2$, (d) $T = 4$. Here, $\epsilon = 0.5$, $\lambda_0 = \tfrac{3}{2}\sqrt{4\epsilon}$ ($k<0$) for solid curves, while $\lambda_0 = \sqrt{4\epsilon}$ ($k = 0$) for dashed curves
Evolution of a (dimensionless) heat pulse $\Theta$ under the solution from Eq. (44). (a, b) $\epsilon = 0.1$, (c, d) $\epsilon = 0.5$; (a, c) $T = 2$, (b, d) $T = 4$. In all panels $\lambda_0 = \tfrac{3}{2}\sqrt{4\epsilon}$ ($k<0$) for the solid curves, while $\lambda_0 = \sqrt{4\epsilon}$ ($k = 0$) for the dashed curves
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