On second sound at the critical temperature

Authors:
K. Saxton, R. Saxton and W. Kosinski

Journal:
Quart. Appl. Math. **57** (1999), 723-740

MSC:
Primary 35Q99; Secondary 35L60, 74A15, 80A20

DOI:
https://doi.org/10.1090/qam/1724302

MathSciNet review:
MR1724302

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Abstract | References | Similar Articles | Additional Information

Abstract: Based on low-temperature experimental data in solid dielectric crystals, we derive a model of heat conduction for rigid materials using the theory of thermo-dynamic internal state variables. The model is intended to admit wavelike propagation of heat below--and diffusive conduction above--a particular temperature value . A rapid decay of the speed of thermal waves occurs just below this temperature, coincident with the conductivity of the material reaching a peak. An analysis of weak and strong discontinuity waves is given in order to exhibit several main features of the proposed model.

**[1]**C. C. Ackerman, B. Bertman, H. A. Fairbank, and R. A. Guyer,*Second sound in solid helium*, Phys. Rev. Letters**16**, 789-791 (1966)**[2]**Cz. Bajer and W. Kosiński,*Numerical modelling of thermal waves via internal state variable approach*, Comp. Assisted Mech. and Engrg. Sciences**2**(4), (1995)**[3]**V. A. Cimmelli and W. Kosiński,*Nonequilibrium semi-empirical temperature in materials with thermal relaxation*, Arch. Mech.**43**(6), 753-767 (1991)**[4]**V. A. Cimmelli and W. Kosiński,*Evolution hyperbolic equations for heat conduction*, Proc. 5th Bilateral Polish-Italian Meeting, Thermodynamics and Kinetic Theory, W. Kosiński et al, eds., World Scientific Series on Advances in Math. Appl. Sci., Vol. 12, Singapore, New Jersey, London, Hong Kong, 1992, pp. 11-22 MR**1170162****[5]**V. A. Cimmelli, W. Kosiński, and K. Saxton,*Modified Fourier law--comparison of two approaches*, Arch. Mech.**44**(4), 409-415 (1992)**[6]**V. A. Cimmelli and W. Kosiński,*Well-posedness results for a nonlinear hyperbolic heat equation*, Ricerche di Matematica**XLII**(1), 49-68 (1993)**[7]**B. D. Coleman and D. C. Neumann,*Implication of a nonlinearity in the theory of second sound in solids*, Phys. Rev. B**32**, 1492-1498 (1988)**[8]**B. D. Coleman and Po-Sien Lai,*Waves of discontinuity and sinusoidal waves in the theory of second sound in solids*, Arch. Rational Mech. Anal.**126**, 1-20 (1994) MR**1268046****[9]**C. M. Dafermos,*Hyperbolic systems of conservation laws*, Systems of Nonlinear Partial Differential Equations, J. M. Ball, ed., NATO ASI series (C),**111**, D. Reidel Publishing Company, 1983, pp. 25-70 MR**725517****[10]**W. Dreyer and H. Struchtrup,*Heat pulse experiments revisited*, Continuum Mech. Thermodyn.**5**, 3-50 (1993) MR**1207459****[11]**K. Frischmuth and V. A. Cimmelli,*Numerical reconstruction of heat pulse experiments*, Internat. J. Engrg. Sci.**33**(2), 209-215 (1995)**[12]**K. Frischmuth and V. A. Cimmelli,*Hyperbolic heat conduction with variable relaxation time*, J. Theor. Appl. Mechanics**34**(1), 67-76 (1996)**[13]**H. E. Jackson, C. T. Walker, and T. F. McNelly,*Second sound in NaF*, Phys. Rev. Letters**25**(1), 26-28 (1970)**[14]**H. E. Jackson and C. T. Walker,*Thermal conductivity, second sound, and phonon-phonon interactions in NaF*, Physical Review B**3**(4), 1428-1439 (1971)**[15]**D. D. Joseph and L. Preziosi,*Heat waves*, Rev. Mod. Phys.**61**(41), 1-62 (1989) MR**977943****[16]**D. D. Joseph and L. Preziosi,*Addendum to the paper ``Heat Waves"*, (Rev. Mod. Phys.**61**, (41) (1989)), Rev. Mod. Phys.**62**(2), 375-392 (1990) MR**1056235****[17]**W. Kosiński,*Elastic waves in the presence of a new temperature scale*, in Elastic Wave Propagation, M. F. McCarthy and M. Hayes, eds., Elsevier Science, North Holland, 1989, pp. 629-634**[18]**W. Kosiński and K. Saxton,*The effect on finite time breakdown due to modified Fourier laws*, Quart. Appl. Math.**51**(1), 55-68 (1993) MR**1205936****[19]**W. Kosiński and W. Wojno,*Gradient generalization to internal state variable approach*, Arch. Mech.**47**(3), 523-536 (1995) MR**1364476****[20]**W. Kosiński and W. Wojno,*On parabolic regularization of hyperbolic heat conductivity in rigid body*, J. Theor. Appl. Mechanics**34**(1) (1996)**[21]**T. F. McNelly, S. J. Rogers, D. J. Chamin, R. J. Rollefson, W. M. Goubau, G. E. Schmidt, J. A. Krumhansi, and R. O. Pohl,*Heat pulses in NaF: Onset of second sound*, Phys. Rev. Letters**24**(3), 100-102 (1970)**[22]**V. Narayanamurti and R. C. Dynes,*Observation of second sound in bismuth*, Phys. Rev. Letters**28**(22), 1461-1465 (1972)**[23]**T. Ruggeri, A. Muracchini, and L. Seccia,*Shock waves and second sound in a rigid heat conductor: A critical temperature for NaF and Bi*, Phys. Rev. Letters**64**(22), 2640-2643 (1990)**[24]**T. Ruggeri, A. Muracchini, and L. Seccia,*Continuum approach to phonon gas and shape changes of second sound via shock waves theory*, Il Nuovo Cimento**16**D (1), 15-44 (1994)

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DOI:
https://doi.org/10.1090/qam/1724302

Article copyright:
© Copyright 1999
American Mathematical Society