Skip to Main Content
Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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
Full-text PDF Free Access

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 ${\vartheta {_\lambda }}$. 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.


References [Enhancements On Off] (What's this?)

    C. C. Ackerman, B. Bertman, H. A. Fairbank, and R. A. Guyer, Second sound in solid helium, Phys. Rev. Letters 16, 789–791 (1966) Cz. Bajer and W. Kosiński, Numerical modelling of thermal waves via internal state variable approach, Comp. Assisted Mech. and Engrg. Sciences 2 (4), (1995) V. A. Cimmelli and W. Kosiński, Nonequilibrium semi-empirical temperature in materials with thermal relaxation, Arch. Mech. 43 (6), 753–767 (1991)
  • Vito Antonio Cimmelli and Witold Kosiński, Evolution hyperbolic equations for heat conduction, Thermodynamics and kinetic theory (Mądralin, 1990) Ser. Adv. Math. Appl. Sci., vol. 12, World Sci. Publ., River Edge, NJ, 1992, pp. 11–22. MR 1170162
  • V. A. Cimmelli, W. Kosiński, and K. Saxton, Modified Fourier law—comparison of two approaches, Arch. Mech. 44 (4), 409–415 (1992) V. A. Cimmelli and W. Kosiński, Well-posedness results for a nonlinear hyperbolic heat equation, Ricerche di Matematica XLII (1), 49–68 (1993) 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)
  • Bernard D. Coleman and Po-Hsien Lai, Waves of discontinuity and sinusoidal waves in the theory of second sound in solids, Arch. Rational Mech. Anal. 126 (1994), no. 1, 1–20. MR 1268046, DOI https://doi.org/10.1007/BF00375693
  • C. M. Dafermos, Hyperbolic systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 25–70. MR 725517
  • W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Contin. Mech. Thermodyn. 5 (1993), no. 1, 3–50. MR 1207459, DOI https://doi.org/10.1007/BF01135371
  • K. Frischmuth and V. A. Cimmelli, Numerical reconstruction of heat pulse experiments, Internat. J. Engrg. Sci. 33 (2), 209–215 (1995) K. Frischmuth and V. A. Cimmelli, Hyperbolic heat conduction with variable relaxation time, J. Theor. Appl. Mechanics 34 (1), 67–76 (1996) H. E. Jackson, C. T. Walker, and T. F. McNelly, Second sound in NaF, Phys. Rev. Letters 25 (1), 26–28 (1970) 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)
  • D. D. Joseph and Luigi Preziosi, Heat waves, Rev. Modern Phys. 61 (1989), no. 1, 41–73. MR 977943, DOI https://doi.org/10.1103/RevModPhys.61.41
  • D. D. Joseph and Luigi Preziosi, Addendum to the paper: “Heat waves” [Rev. Modern Phys. 61 (1989), no. 1, 41–73; MR0977943 (89k:80001)], Rev. Modern Phys. 62 (1990), no. 2, 375–391. MR 1056235, DOI https://doi.org/10.1103/RevModPhys.62.375
  • 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
  • W. Kosiński and K. Saxton, The effect on finite time breakdown due to modified Fourier laws, Quart. Appl. Math. 51 (1993), no. 1, 55–68. MR 1205936, DOI https://doi.org/10.1090/qam/1205936
  • W. Kosiński and W. Wojno, Gradient generalization to internal state variable approach, Arch. Mech. (Arch. Mech. Stos.) 47 (1995), no. 3, 523–536. MR 1364476
  • W. Kosiński and W. Wojno, On parabolic regularization of hyperbolic heat conductivity in rigid body, J. Theor. Appl. Mechanics 34 (1) (1996) 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) V. Narayanamurti and R. C. Dynes, Observation of second sound in bismuth, Phys. Rev. Letters 28 (22), 1461–1465 (1972) 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) 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)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35Q99, 35L60, 74A15, 80A20

Retrieve articles in all journals with MSC: 35Q99, 35L60, 74A15, 80A20


Additional Information

Article copyright: © Copyright 1999 American Mathematical Society