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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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O. G. Bakunin, Mysteries of diffusion and labyrinths of destiny, Phys.-Usp., 46 (2003), 309-313. doi: 10.1070/PU2003v046n03ABEH001289. Google Scholar

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S. BargmannP. 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. Google Scholar

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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.Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, The Clarendon Press, Oxford University Press, New York, 1988. Google Scholar

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C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1949), 83-101. Google Scholar

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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. Google Scholar

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M. Chester, Second sound in solids, Phys. Rev., 131 (1963), 2013-2015. doi: 10.1103/PhysRev.131.2013. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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M. CiarlettaB. 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. Google Scholar

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[21]

R. M. CorlessG. H. GonnetD. E. G. HareD. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. Google Scholar

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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. Google Scholar

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G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, Berlin, 1974. Google Scholar

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W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Continuum Mech. Thermodyn., 5 (1993), 3-50. doi: 10.1007/BF01135371. Google Scholar

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T. S. Fisher, Thermal Energy at the Nanoscale, vol. 3 of Lessons from Nanoscience, World Scientific Publishing Co., Singapore, 2013. doi: 10.1142/8716. Google Scholar

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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. Google Scholar

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C. S. GardnerJ. M. GreeneM. 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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice Hall, Englewood Hills, NJ, 1988. Google Scholar

[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. Google Scholar

[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. Google Scholar
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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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P. M. Jordan and A. Puri, Digital signal propagation in dispersive media, J. Appl. Phys., 85 (1999), 1273-1282. doi: 10.1063/1.369258. Google Scholar

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D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys, 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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M. F. Modest, Radiative Heat Transfer, 3rd edition, Elsevier, Oxford, UK, 2013. doi: 10.1016/B978-0-12-386944-9.50025-X. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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show all references

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

O. G. Bakunin, Mysteries of diffusion and labyrinths of destiny, Phys.-Usp., 46 (2003), 309-313. doi: 10.1070/PU2003v046n03ABEH001289. Google Scholar

[5]

S. BargmannP. 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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[8] D. R. Bland, Wave Theory and Applications, Oxford University Press, London, 1988. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, The Clarendon Press, Oxford University Press, New York, 1988. Google Scholar

[13]

C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1949), 83-101. Google Scholar

[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. Google Scholar

[15]

M. Chester, Second sound in solids, Phys. Rev., 131 (1963), 2013-2015. doi: 10.1103/PhysRev.131.2013. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

M. CiarlettaB. 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. Google Scholar

[20]

B. D. ColemanM. 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. Google Scholar

[21]

R. M. CorlessG. H. GonnetD. E. G. HareD. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. Google Scholar

[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. Google Scholar

[23]

G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, Berlin, 1974. Google Scholar

[24]

W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Continuum Mech. Thermodyn., 5 (1993), 3-50. doi: 10.1007/BF01135371. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

C. S. GardnerJ. M. GreeneM. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice Hall, Englewood Hills, NJ, 1988. Google Scholar

[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. Google Scholar

[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. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys, 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

M. F. Modest, Radiative Heat Transfer, 3rd edition, Elsevier, Oxford, UK, 2013. doi: 10.1016/B978-0-12-386944-9.50025-X. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[56]

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Figure 1.  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 $
Figure 2.  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
Figure 3.  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
Figure 4.  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
Figure 5.  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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