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Second-sound phenomena in inviscid, thermally relaxing gases
1. | Acoustics Div., U.S. Naval Research Laboratory, Stennis Space Ctr., MS 39529, United States |
References:
[1] |
R. T. Beyer, The parameter $B/A$, in Nonlinear Acoustics, (eds. M. F. Hamilton and D. T. Blackstock), Academic Press, San Diego, CA, (1997), 25-39. |
[2] |
B. A. Boley and R. B. Hetnarski, Propagation of discontinuities in coupled thermoelastic problems, J. Appl. Mech. (ASME), 35 (1968), 489-494.
doi: 10.1115/1.3601240. |
[3] |
S. Carillo, Bäcklund transformations & heat conduction with memory, in New Trends in Fluid and Solid Models: Proceedings of the International Conference in Honour of Brian Straughan (Supplementary) (eds. M. Ciarletta, M. Fabrizio, A. Morro, and S. Rionero), World Scientific, Hackensack, NJ, (2010), 8-17. |
[4] |
S. Carillo, Nonlinear hyperbolic equations and linear heat conduction with memory, in Mechanics of Microstructured Solids 2, (eds. J.-F. Ganghoffer and F. Pastrone), Lecture Notes in Applied and Computational Mechanics, Vol. 50, Springer, Berlin, (2010), 63-70.
doi: 10.1007/978-3-642-05171-5_7. |
[5] |
M. Carrassi and A. Morro, A modified Navier-Stokes equations and its consequences on sound dispersion, Nuovo Cimento B, 9 (1972), 321-343.
doi: 10.1007/BF02734451. |
[6] |
H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Dover, New York, NY, 1963. |
[7] |
C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. Fis. Della Univ. Modena, 3 (1949), 83-101. |
[8] |
D. S. Chandrasekharaiah, Thermoelasticity with second sound: A Review, Appl. Mech. Rev., 39 (1986), 355-376. |
[9] |
W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64.
doi: 10.1017/S0022112064000040. |
[10] |
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. |
[11] |
I. C. Christov and P. M. Jordan, Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation, New J. Phys., 10 (2008), 043027.
doi: 10.1088/1367-2630/10/4/043027. |
[12] |
I. C. Christov, P. M. Jordan, S. A. Chin-Bing and A. Warn-Varnas, Acoustic traveling waves in thermoviscous perfect gases: Kinks, acceleration waves, and shocks under the Taylor-Lighthill balance, Math. Comput. Simul., 2013, in press (doi: 10.1016/j.matcom.2013.03.011).
doi: 10.1016/j.matcom.2013.03.011. |
[13] |
B. D. Coleman, M. Fabrizio and D. R. Owen, On the thermodynamics of second sound in dielectric crystals, Arch. Rat. Mech. Anal., 80 (1982), 135-158.
doi: 10.1007/BF00250739. |
[14] |
D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.
doi: 10.1146/annurev.fl.11.010179.000303. |
[15] |
D. G. Crighton, Basic theoretical nonlinear acoustics, in Frontiers in Physical Acoustics, (ed. D. Sette), North-Holland, Amsterdam, (1986), 1-52. |
[16] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edn., A Series of Comprehensive Studies in Mathematics, Vol. 325, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-04048-1. |
[17] |
W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Cont. Mech. Thermodyn, 5 (1993), 3-50.
doi: 10.1007/BF01135371. |
[18] |
D. G. Duffy, Transform Methods for Solving Partial Differential Equations, 2nd edn., Chapman & Hall/CRC, Boca Raton, FL, 2004.
doi: 10.1201/9781420035148. |
[19] |
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. |
[20] |
H. Grad, Thermodynamics of gases, in Handbuch der Physik (ed. S. Flügge), Springer, Berlin, XII (1960), 205-294. |
[21] |
P. M. Jordan, On the application of the Cole-Hopf transformation to hyperbolic equations based on second-sound models, Math. Comput. Simul., 81 (2010), 18-25.
doi: 10.1016/j.matcom.2010.06.011. |
[22] |
P. M. Jordan, G. V. Norton, S. A. Chin-Bing and A. Warn-Varnas, On the propagation of nonlinear acoustic waves in viscous and thermoviscous fluids, Eur. J. Mech. B/Fluids, 34 (2012), 56-63.
doi: 10.1016/j.euromechflu.2012.01.016. |
[23] |
P. M. Jordan and P. Puri, Revisiting the Danilovskaya problem, J. Thermal Stresses, 29 (2006), 865-878.
doi: 10.1080/01495730600705505. |
[24] |
D. Jou, C. Cásas-Vazquez and G. Lebon, Extended irreversible thermodynamics revisited (1988-98), Rep. Prog. Phys., 62 (1999), 1035-1142.
doi: 10.1088/0034-4885/62/7/201. |
[25] |
B. Kaltenbacher, I. Lasieck and M. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34pp.
doi: 10.1142/S0218202512500352. |
[26] |
R. E. Khayat and M. Ostoja-Starzewski, On the objective rate of heat and stress fluxes: Connection with micro/nano-scale heat convection, Discrete Cont. Dyn. Sys., Ser. B (DCDS-B), 15 (2011), 991-998.
doi: 10.3934/dcdsb.2011.15.991. |
[27] |
M. B. Lesser, R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, J. Fluid Mech., 31 (1968), 501-528.
doi: 10.1017/S0022112068000303. |
[28] |
K. A. Lindsay, B. Straughan, Acceleration waves and second sound in a perfect fluid, Arch. Rat. Mech. Anal., 68 (1978), 53-87.
doi: 10.1007/BF00276179. |
[29] |
S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acustica-Acta Acustica, 82 (1996), 579-606. |
[30] |
J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. London, Ser. A, 157 (1867), 49-88. |
[31] |
F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aero/Space Sci., 27 (1960), 117-127. |
[32] |
J. P. Moran and S. F. Shen, On the formation of weak plane shock waves by impulsive motion of a piston, J. Fluid Mech., 25 (1966), 705-718.
doi: 10.1017/S0022112066000351. |
[33] |
A. Morro, Wave propagation in thermo-viscous materials with hidden variables, Arch. Mech., 32 (1980), 145-161. |
[34] |
A. Morro, Shock waves in thermo-viscous fluids with hidden variables, Arch. Mech., 32 (1980), 193-199. |
[35] |
I. Müller, Zum Paradoxon der Wärmeleitungstheorie, Z. Phys., 198 (1967), 329-344. |
[36] |
I. Müller and T. Ruggeri, Extended Thermodynamics, Springer Tracts in Natural Philosophy, Vol. 37, Springer, New York, NY, 1993.
doi: 10.1007/978-1-4684-0447-0. |
[37] |
V. Peshkov, "Second sound'' in helium II, J. Phys., (USSR) 8 (1944), 381. |
[38] |
A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Woodbury, NY, 1989. |
[39] |
T. Ruggeri, Symmetric-hyperbolic system of conservative equations for a viscous heat conducting fluid, Acta Mech., 47 (1983), 167-183.
doi: 10.1007/BF01189206. |
[40] |
T. Ruggeri, Galilean invariance and entropy principle for systems of balance laws, Cont. Mech. Thermodyn, 1 (1989), 3-20.
doi: 10.1007/BF01125883. |
[41] |
J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, (ed. S. Flügge), Vol. VIII/1, Springer, Berlin, (1959), 125-263. |
[42] |
G. G. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Phil. Mag. (Ser. 4), 3 (2009), 142-154.
doi: 10.1017/CBO9780511702266.005. |
[43] |
B. Straughan, Nonlinear acceleration waves in porous media, Math. Comput. Simul., 80 (2009), 763-769.
doi: 10.1016/j.matcom.2009.08.013. |
[44] |
B. Straughan, Acoustic waves in a Cattaneo-Christov gas, Phys. Lett. A, 374 (2010), 2667-2669.
doi: 10.1016/j.physleta.2010.04.054. |
[45] |
B. Straughan, Heat Waves, Applied Mathematical Sciences, Vol. 177, Springer, New York, NY, 2011.
doi: 10.1007/978-1-4614-0493-4. |
[46] |
P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, NY, 1972. |
[47] |
J. S. Toll, Causality and the dispersion relation: Logical foundations, Phys. Rev., 104 (1956), 1760-1770.
doi: 10.1103/PhysRev.104.1760. |
[48] |
V. Tibullo and V. Zampoli, A uniqueness result for the Cattaneo-Christov heat conduction model applied to incompressible fluids, Mech. Res. Commun., 38 (2011), 77-79.
doi: 10.1016/j.mechrescom.2010.10.008. |
[49] |
D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, Washington, DC, 1997. |
[50] |
H. D. Weymann, Finite speed of propagation in heat conduction, diffusion, and viscous shear motion, Amer. J. Phys., 35 (1967), 488-496.
doi: 10.1119/1.1974155. |
show all references
References:
[1] |
R. T. Beyer, The parameter $B/A$, in Nonlinear Acoustics, (eds. M. F. Hamilton and D. T. Blackstock), Academic Press, San Diego, CA, (1997), 25-39. |
[2] |
B. A. Boley and R. B. Hetnarski, Propagation of discontinuities in coupled thermoelastic problems, J. Appl. Mech. (ASME), 35 (1968), 489-494.
doi: 10.1115/1.3601240. |
[3] |
S. Carillo, Bäcklund transformations & heat conduction with memory, in New Trends in Fluid and Solid Models: Proceedings of the International Conference in Honour of Brian Straughan (Supplementary) (eds. M. Ciarletta, M. Fabrizio, A. Morro, and S. Rionero), World Scientific, Hackensack, NJ, (2010), 8-17. |
[4] |
S. Carillo, Nonlinear hyperbolic equations and linear heat conduction with memory, in Mechanics of Microstructured Solids 2, (eds. J.-F. Ganghoffer and F. Pastrone), Lecture Notes in Applied and Computational Mechanics, Vol. 50, Springer, Berlin, (2010), 63-70.
doi: 10.1007/978-3-642-05171-5_7. |
[5] |
M. Carrassi and A. Morro, A modified Navier-Stokes equations and its consequences on sound dispersion, Nuovo Cimento B, 9 (1972), 321-343.
doi: 10.1007/BF02734451. |
[6] |
H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Dover, New York, NY, 1963. |
[7] |
C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. Fis. Della Univ. Modena, 3 (1949), 83-101. |
[8] |
D. S. Chandrasekharaiah, Thermoelasticity with second sound: A Review, Appl. Mech. Rev., 39 (1986), 355-376. |
[9] |
W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64.
doi: 10.1017/S0022112064000040. |
[10] |
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. |
[11] |
I. C. Christov and P. M. Jordan, Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation, New J. Phys., 10 (2008), 043027.
doi: 10.1088/1367-2630/10/4/043027. |
[12] |
I. C. Christov, P. M. Jordan, S. A. Chin-Bing and A. Warn-Varnas, Acoustic traveling waves in thermoviscous perfect gases: Kinks, acceleration waves, and shocks under the Taylor-Lighthill balance, Math. Comput. Simul., 2013, in press (doi: 10.1016/j.matcom.2013.03.011).
doi: 10.1016/j.matcom.2013.03.011. |
[13] |
B. D. Coleman, M. Fabrizio and D. R. Owen, On the thermodynamics of second sound in dielectric crystals, Arch. Rat. Mech. Anal., 80 (1982), 135-158.
doi: 10.1007/BF00250739. |
[14] |
D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.
doi: 10.1146/annurev.fl.11.010179.000303. |
[15] |
D. G. Crighton, Basic theoretical nonlinear acoustics, in Frontiers in Physical Acoustics, (ed. D. Sette), North-Holland, Amsterdam, (1986), 1-52. |
[16] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edn., A Series of Comprehensive Studies in Mathematics, Vol. 325, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-04048-1. |
[17] |
W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Cont. Mech. Thermodyn, 5 (1993), 3-50.
doi: 10.1007/BF01135371. |
[18] |
D. G. Duffy, Transform Methods for Solving Partial Differential Equations, 2nd edn., Chapman & Hall/CRC, Boca Raton, FL, 2004.
doi: 10.1201/9781420035148. |
[19] |
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. |
[20] |
H. Grad, Thermodynamics of gases, in Handbuch der Physik (ed. S. Flügge), Springer, Berlin, XII (1960), 205-294. |
[21] |
P. M. Jordan, On the application of the Cole-Hopf transformation to hyperbolic equations based on second-sound models, Math. Comput. Simul., 81 (2010), 18-25.
doi: 10.1016/j.matcom.2010.06.011. |
[22] |
P. M. Jordan, G. V. Norton, S. A. Chin-Bing and A. Warn-Varnas, On the propagation of nonlinear acoustic waves in viscous and thermoviscous fluids, Eur. J. Mech. B/Fluids, 34 (2012), 56-63.
doi: 10.1016/j.euromechflu.2012.01.016. |
[23] |
P. M. Jordan and P. Puri, Revisiting the Danilovskaya problem, J. Thermal Stresses, 29 (2006), 865-878.
doi: 10.1080/01495730600705505. |
[24] |
D. Jou, C. Cásas-Vazquez and G. Lebon, Extended irreversible thermodynamics revisited (1988-98), Rep. Prog. Phys., 62 (1999), 1035-1142.
doi: 10.1088/0034-4885/62/7/201. |
[25] |
B. Kaltenbacher, I. Lasieck and M. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34pp.
doi: 10.1142/S0218202512500352. |
[26] |
R. E. Khayat and M. Ostoja-Starzewski, On the objective rate of heat and stress fluxes: Connection with micro/nano-scale heat convection, Discrete Cont. Dyn. Sys., Ser. B (DCDS-B), 15 (2011), 991-998.
doi: 10.3934/dcdsb.2011.15.991. |
[27] |
M. B. Lesser, R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, J. Fluid Mech., 31 (1968), 501-528.
doi: 10.1017/S0022112068000303. |
[28] |
K. A. Lindsay, B. Straughan, Acceleration waves and second sound in a perfect fluid, Arch. Rat. Mech. Anal., 68 (1978), 53-87.
doi: 10.1007/BF00276179. |
[29] |
S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acustica-Acta Acustica, 82 (1996), 579-606. |
[30] |
J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. London, Ser. A, 157 (1867), 49-88. |
[31] |
F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aero/Space Sci., 27 (1960), 117-127. |
[32] |
J. P. Moran and S. F. Shen, On the formation of weak plane shock waves by impulsive motion of a piston, J. Fluid Mech., 25 (1966), 705-718.
doi: 10.1017/S0022112066000351. |
[33] |
A. Morro, Wave propagation in thermo-viscous materials with hidden variables, Arch. Mech., 32 (1980), 145-161. |
[34] |
A. Morro, Shock waves in thermo-viscous fluids with hidden variables, Arch. Mech., 32 (1980), 193-199. |
[35] |
I. Müller, Zum Paradoxon der Wärmeleitungstheorie, Z. Phys., 198 (1967), 329-344. |
[36] |
I. Müller and T. Ruggeri, Extended Thermodynamics, Springer Tracts in Natural Philosophy, Vol. 37, Springer, New York, NY, 1993.
doi: 10.1007/978-1-4684-0447-0. |
[37] |
V. Peshkov, "Second sound'' in helium II, J. Phys., (USSR) 8 (1944), 381. |
[38] |
A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Woodbury, NY, 1989. |
[39] |
T. Ruggeri, Symmetric-hyperbolic system of conservative equations for a viscous heat conducting fluid, Acta Mech., 47 (1983), 167-183.
doi: 10.1007/BF01189206. |
[40] |
T. Ruggeri, Galilean invariance and entropy principle for systems of balance laws, Cont. Mech. Thermodyn, 1 (1989), 3-20.
doi: 10.1007/BF01125883. |
[41] |
J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, (ed. S. Flügge), Vol. VIII/1, Springer, Berlin, (1959), 125-263. |
[42] |
G. G. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Phil. Mag. (Ser. 4), 3 (2009), 142-154.
doi: 10.1017/CBO9780511702266.005. |
[43] |
B. Straughan, Nonlinear acceleration waves in porous media, Math. Comput. Simul., 80 (2009), 763-769.
doi: 10.1016/j.matcom.2009.08.013. |
[44] |
B. Straughan, Acoustic waves in a Cattaneo-Christov gas, Phys. Lett. A, 374 (2010), 2667-2669.
doi: 10.1016/j.physleta.2010.04.054. |
[45] |
B. Straughan, Heat Waves, Applied Mathematical Sciences, Vol. 177, Springer, New York, NY, 2011.
doi: 10.1007/978-1-4614-0493-4. |
[46] |
P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, NY, 1972. |
[47] |
J. S. Toll, Causality and the dispersion relation: Logical foundations, Phys. Rev., 104 (1956), 1760-1770.
doi: 10.1103/PhysRev.104.1760. |
[48] |
V. Tibullo and V. Zampoli, A uniqueness result for the Cattaneo-Christov heat conduction model applied to incompressible fluids, Mech. Res. Commun., 38 (2011), 77-79.
doi: 10.1016/j.mechrescom.2010.10.008. |
[49] |
D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, Washington, DC, 1997. |
[50] |
H. D. Weymann, Finite speed of propagation in heat conduction, diffusion, and viscous shear motion, Amer. J. Phys., 35 (1967), 488-496.
doi: 10.1119/1.1974155. |
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