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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

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

Received  April 2013 Revised  August 2013 Published  August 2014

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.
Citation: Pedro M. Jordan. Second-sound phenomena in inviscid, thermally relaxing gases. Discrete & 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. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [6] H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Dover, New York, NY, 1963.  Google Scholar [7] C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. Fis. Della Univ. Modena, 3 (1949), 83-101.  Google Scholar [8] D. S. Chandrasekharaiah, Thermoelasticity with second sound: A Review, Appl. Mech. Rev., 39 (1986), 355-376. Google Scholar [9] W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64. doi: 10.1017/S0022112064000040.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] D. G. Crighton, Basic theoretical nonlinear acoustics, in Frontiers in Physical Acoustics, (ed. D. Sette), North-Holland, Amsterdam, (1986), 1-52. Google Scholar [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.  Google Scholar [17] W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Cont. Mech. Thermodyn, 5 (1993), 3-50. doi: 10.1007/BF01135371.  Google Scholar [18] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, 2nd edn., Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9781420035148.  Google Scholar [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.  Google Scholar [20] H. Grad, Thermodynamics of gases, in Handbuch der Physik (ed. S. Flügge), Springer, Berlin, XII (1960), 205-294.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] P. M. Jordan and P. Puri, Revisiting the Danilovskaya problem, J. Thermal Stresses, 29 (2006), 865-878. doi: 10.1080/01495730600705505.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acustica-Acta Acustica, 82 (1996), 579-606. Google Scholar [30] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. London, Ser. A, 157 (1867), 49-88. Google Scholar [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. Google Scholar [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.  Google Scholar [33] A. Morro, Wave propagation in thermo-viscous materials with hidden variables, Arch. Mech., 32 (1980), 145-161.  Google Scholar [34] A. Morro, Shock waves in thermo-viscous fluids with hidden variables, Arch. Mech., 32 (1980), 193-199.  Google Scholar [35] I. Müller, Zum Paradoxon der Wärmeleitungstheorie, Z. Phys., 198 (1967), 329-344. Google Scholar [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.  Google Scholar [37] V. Peshkov, "Second sound'' in helium II, J. Phys., (USSR) 8 (1944), 381. Google Scholar [38] A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Woodbury, NY, 1989. Google Scholar [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.  Google Scholar [40] T. Ruggeri, Galilean invariance and entropy principle for systems of balance laws, Cont. Mech. Thermodyn, 1 (1989), 3-20. doi: 10.1007/BF01125883.  Google Scholar [41] J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, (ed. S. Flügge), Vol. VIII/1, Springer, Berlin, (1959), 125-263.  Google Scholar [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.  Google Scholar [43] B. Straughan, Nonlinear acceleration waves in porous media, Math. Comput. Simul., 80 (2009), 763-769. doi: 10.1016/j.matcom.2009.08.013.  Google Scholar [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.  Google Scholar [45] B. Straughan, Heat Waves, Applied Mathematical Sciences, Vol. 177, Springer, New York, NY, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar [46] P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, NY, 1972. Google Scholar [47] J. S. Toll, Causality and the dispersion relation: Logical foundations, Phys. Rev., 104 (1956), 1760-1770. doi: 10.1103/PhysRev.104.1760.  Google Scholar [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.  Google Scholar [49] D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, Washington, DC, 1997. Google Scholar [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.  Google Scholar

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. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [6] H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Dover, New York, NY, 1963.  Google Scholar [7] C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. Fis. Della Univ. Modena, 3 (1949), 83-101.  Google Scholar [8] D. S. Chandrasekharaiah, Thermoelasticity with second sound: A Review, Appl. Mech. Rev., 39 (1986), 355-376. Google Scholar [9] W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64. doi: 10.1017/S0022112064000040.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] D. G. Crighton, Basic theoretical nonlinear acoustics, in Frontiers in Physical Acoustics, (ed. D. Sette), North-Holland, Amsterdam, (1986), 1-52. Google Scholar [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.  Google Scholar [17] W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Cont. Mech. Thermodyn, 5 (1993), 3-50. doi: 10.1007/BF01135371.  Google Scholar [18] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, 2nd edn., Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9781420035148.  Google Scholar [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.  Google Scholar [20] H. Grad, Thermodynamics of gases, in Handbuch der Physik (ed. S. Flügge), Springer, Berlin, XII (1960), 205-294.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] P. M. Jordan and P. Puri, Revisiting the Danilovskaya problem, J. Thermal Stresses, 29 (2006), 865-878. doi: 10.1080/01495730600705505.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acustica-Acta Acustica, 82 (1996), 579-606. Google Scholar [30] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. London, Ser. A, 157 (1867), 49-88. Google Scholar [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. Google Scholar [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.  Google Scholar [33] A. Morro, Wave propagation in thermo-viscous materials with hidden variables, Arch. Mech., 32 (1980), 145-161.  Google Scholar [34] A. Morro, Shock waves in thermo-viscous fluids with hidden variables, Arch. Mech., 32 (1980), 193-199.  Google Scholar [35] I. Müller, Zum Paradoxon der Wärmeleitungstheorie, Z. Phys., 198 (1967), 329-344. Google Scholar [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.  Google Scholar [37] V. Peshkov, "Second sound'' in helium II, J. Phys., (USSR) 8 (1944), 381. Google Scholar [38] A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Woodbury, NY, 1989. Google Scholar [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.  Google Scholar [40] T. Ruggeri, Galilean invariance and entropy principle for systems of balance laws, Cont. Mech. Thermodyn, 1 (1989), 3-20. doi: 10.1007/BF01125883.  Google Scholar [41] J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, (ed. S. Flügge), Vol. VIII/1, Springer, Berlin, (1959), 125-263.  Google Scholar [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.  Google Scholar [43] B. Straughan, Nonlinear acceleration waves in porous media, Math. Comput. Simul., 80 (2009), 763-769. doi: 10.1016/j.matcom.2009.08.013.  Google Scholar [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.  Google Scholar [45] B. Straughan, Heat Waves, Applied Mathematical Sciences, Vol. 177, Springer, New York, NY, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar [46] P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, NY, 1972. Google Scholar [47] J. S. Toll, Causality and the dispersion relation: Logical foundations, Phys. Rev., 104 (1956), 1760-1770. doi: 10.1103/PhysRev.104.1760.  Google Scholar [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.  Google Scholar [49] D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, Washington, DC, 1997. Google Scholar [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.  Google Scholar
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