Second-sound phenomena in inviscid, thermally relaxing gases

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September 2014, 19(7): 2189-2205. doi: 10.3934/dcdsb.2014.19.2189

Second-sound phenomena in inviscid, thermally relaxing gases

Pedro M. Jordan ^1,

1.	Acoustics Div., U.S. Naval Research Laboratory, Stennis Space Ctr., MS 39529, United States

Received April 2013 Revised August 2013 Published August 2014

Abstract
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We consider the propagation of acoustic and thermal waves in a class of inviscid, thermally relaxing gases wherein the flow of heat is described by the Maxwell--Cattaneo law, i.e., in Cattaneo--Christov gases. After first considering the start-up piston problem under the linear theory, we then investigate traveling wave phenomena under the weakly-nonlinear approximation. In particular, a shock analysis is carried out, comparisons with predictions from classical gases dynamics theory are performed, and critical values of the parameters are derived. Special case results are also presented and connections to other fields are noted.

Keywords: Cattaneo--Christov gas, hyperbolic Burgers equation, traveling waves., Nonlinear acoustics, shock waves.

Mathematics Subject Classification: Primary: 76L05, 76N15, 80A20; Secondary: 35L70, 44A1.

Citation: Pedro M. Jordan. Second-sound phenomena in inviscid, thermally relaxing gases. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2189-2205. doi: 10.3934/dcdsb.2014.19.2189

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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