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Introduction to the special volume ``Mathematics of nonlinear acoustics: New approaches in analysis and modeling''
Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids
1. | School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, United States |
References:
[1] |
W. F. Ames, Discontinuity formation in solutions of homogeneous non-linear hyperbolic equations possessing smooth initial data,, Int. J. Non-Linear Mech., 5 (1970), 605.
doi: 10.1016/0020-7462(70)90050-8. |
[2] |
S. Bargmann, Remarks on the Green-Naghdi theory of heat conduction,, J. Non-Equilib. Thermodyn., 38 (2013), 101.
doi: 10.1515/jnetdy-2012-0015. |
[3] |
S. Bargmann and P. Steinmann, Modeling and simulation of first and second sound in solids,, Int. J. Solids Structures, 45 (2008), 6067.
doi: 10.1016/j.ijsolstr.2008.07.026. |
[4] |
S. Bargmann, P. Steinmann and P. M. Jordan, On the propagation of second-sound in linear and nonlinear media: Results from Green-Naghdi theory,, Phys. Lett. A, 372 (2008), 4418.
doi: 10.1016/j.physleta.2008.04.010. |
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J. Bissell and B. Straughan, Discontinuity waves as tipping points: Applications to biological & sociological systems,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1911.
doi: 10.3934/dcdsb.2014.19.1911. |
[7] |
D. T. Blackstock, Approximate equations governing finite-amplitude sound in thermoviscous fluids,, GD/E Report GD-1463-52, (1963), 1463. Google Scholar |
[8] |
D. T. Blackstock, Propagation of plane sound waves of finite amplitude in nondissipative fluids,, J. Acoust. Soc. Am., 34 (1962), 9.
doi: 10.1121/1.1909033. |
[9] |
B. Brunnhuber and B. Kaltenbacher, Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation,, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 4515.
doi: 10.3934/dcds.2014.34.4515. |
[10] |
B. Brunnhuber, B. Kaltenbacher and P. Radu, Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling,, Evol. Equ. Control Theory, 3 (2014), 595.
doi: 10.3934/eect.2014.3.595. |
[11] |
B. Brunnhuber, Well-posedness and exponential decay of solutions for the Blackstock-Crighton-Kuznetsov equation,, J. Math. Anal. Appl., 433 (2016), 1037.
doi: 10.1016/j.jmaa.2015.07.046. |
[12] |
B. Brunnhuber and P. M. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption,, Int. J. Non-Linear Mech., 78 (2016), 131.
doi: 10.1016/j.ijnonlinmec.2015.10.008. |
[13] |
P. J. Chen, Growth and decay of waves in solids,, in Handbuch der Physik, (1973), 303. Google Scholar |
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P. J. Chen, On the growth and decay of one-dimensional temperature rate waves,, Arch. Ration. Mech. Anal., 35 (1969), 1.
doi: 10.1007/BF00248491. |
[15] |
M. Chen, M. Torres and T. Walsh, Existence of travelling wave solutions of a high-order nonlinear acoustic wave equation,, Phys. Lett. A, 373 (2009), 1037.
doi: 10.1016/j.physleta.2009.01.042. |
[16] |
W. Chester, Resonant oscillations in closed tubes,, J. Fluid Mech., 18 (1964), 44.
doi: 10.1017/S0022112064000040. |
[17] |
I. Christov, C. I. Christov and P. M. Jordan, Modeling weakly nonlinear acoustic wave propagation,, Q. J. Mech. Appl. Math., 60 (2007), 473.
doi: 10.1093/qjmam/hbm017. |
[18] |
I. Christov, C. I. Christov and P. M. Jordan, Corrigendum and addendum: Modeling weakly nonlinear acoustic wave propagation,, Q. J. Mech. Appl. Math., 68 (2015), 231.
doi: 10.1093/qjmam/hbu023. |
[19] |
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. Simulat., 127 (2016), 2.
doi: 10.1016/j.matcom.2013.03.011. |
[20] |
I. Christov, P. M. Jordan and C. I. Christov, Nonlinear acoustic propagation in homentropic perfect gases: A numerical study,, Phys. Lett. A, 353 (2006), 273.
doi: 10.1016/j.physleta.2005.12.101. |
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A. M. J. Davis and H. Brenner, Thermal and viscous effects on sound waves: Revised classical theory,, J. Acoust. Soc. Am., 132 (2012), 2963.
doi: 10.1121/1.4757971. |
[24] |
A. R. Elcrat, On the propagation of sonic discontinuities in the unsteady flow of a perfect gas,, Int. J. Eng. Sci., 15 (1977), 29.
doi: 10.1016/0020-7225(77)90066-0. |
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Y. B. Fu and N. H. Scott, The transition from acceleration wave to shock wave,, Int. J. Eng. Sci., 29 (1991), 617.
doi: 10.1016/0020-7225(91)90066-C. |
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A. E. Green and P. M. Naghdi, A new thermoviscous theory for fluids,, J. Non-Newtonian Fluid Mech., 56 (1995), 289.
doi: 10.1016/0377-0257(94)01288-S. |
[27] |
A. E. Green and P. M. Naghdi, An extended theory for incompressible viscous fluid flow,, J. Non-Newtonian Fluid Mech., 66 (1996), 233.
doi: 10.1016/S0377-0257(96)01478-4. |
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A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics,, Proc. R. Soc. Lond. A, 448 (1995), 335.
doi: 10.1098/rspa.1995.0020. |
[29] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua,, Proc. R. Soc. Lond. A, 448 (1995), 357.
doi: 10.1098/rspa.1995.0021. |
[30] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua,, Proc. R. Soc. Lond. A, 448 (1995), 379.
doi: 10.1098/rspa.1995.0022. |
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M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds,, Arch. Rational Mech. Anal., 31 (1968), 113.
doi: 10.1007/BF00281373. |
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M. F. Hamilton and C. L. Morfey, Model equations,, in Nonlinear Acoustics: Theory and Applications (eds. M. F. Hamilton and D. T. Blackstock), (1997), 41. Google Scholar |
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B. M. Johnson, Analytical shock solutions at large and small Prandtl number,, J. Fluid Mech., 726 (2013).
doi: 10.1017/jfm.2013.262. |
[34] |
B. M. Johnson, Closed-form shock solutions,, J. Fluid Mech., 745 (2014).
doi: 10.1017/jfm.2014.107. |
[35] |
P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009,, Mech. Res. Commun., 73 (2016), 127.
doi: 10.1016/j.mechrescom.2016.02.014. |
[36] |
P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation,, Phys. Lett. A, 326 (2004), 77.
doi: 10.1016/j.physleta.2004.03.067. |
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P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189.
doi: 10.3934/dcdsb.2014.19.2189. |
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P. M. Jordan, A note on the Lambert $W$-function: Applications in the mathematical and physical sciences,, in Mathematics of Continuous and Discrete Dynamical Systems (ed. A. B. Gumel), 618 (2014), 247.
doi: 10.1090/conm/618. |
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P. M. Jordan and C. I. Christov, A simple finite difference scheme for modeling the finite-time blow-up of acoustic acceleration waves,, J. Sound Vib., 281 (2005), 1207.
doi: 10.1016/j.jsv.2004.03.067. |
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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.
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show all references
References:
[1] |
W. F. Ames, Discontinuity formation in solutions of homogeneous non-linear hyperbolic equations possessing smooth initial data,, Int. J. Non-Linear Mech., 5 (1970), 605.
doi: 10.1016/0020-7462(70)90050-8. |
[2] |
S. Bargmann, Remarks on the Green-Naghdi theory of heat conduction,, J. Non-Equilib. Thermodyn., 38 (2013), 101.
doi: 10.1515/jnetdy-2012-0015. |
[3] |
S. Bargmann and P. Steinmann, Modeling and simulation of first and second sound in solids,, Int. J. Solids Structures, 45 (2008), 6067.
doi: 10.1016/j.ijsolstr.2008.07.026. |
[4] |
S. Bargmann, P. Steinmann and P. M. Jordan, On the propagation of second-sound in linear and nonlinear media: Results from Green-Naghdi theory,, Phys. Lett. A, 372 (2008), 4418.
doi: 10.1016/j.physleta.2008.04.010. |
[5] |
R. T. Beyer, The parameter $B/A$,, in Nonlinear Acoustics: Theory and Applications (eds. M. F. Hamilton and D. T. Blackstock), (1997), 25. Google Scholar |
[6] |
J. Bissell and B. Straughan, Discontinuity waves as tipping points: Applications to biological & sociological systems,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1911.
doi: 10.3934/dcdsb.2014.19.1911. |
[7] |
D. T. Blackstock, Approximate equations governing finite-amplitude sound in thermoviscous fluids,, GD/E Report GD-1463-52, (1963), 1463. Google Scholar |
[8] |
D. T. Blackstock, Propagation of plane sound waves of finite amplitude in nondissipative fluids,, J. Acoust. Soc. Am., 34 (1962), 9.
doi: 10.1121/1.1909033. |
[9] |
B. Brunnhuber and B. Kaltenbacher, Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation,, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 4515.
doi: 10.3934/dcds.2014.34.4515. |
[10] |
B. Brunnhuber, B. Kaltenbacher and P. Radu, Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling,, Evol. Equ. Control Theory, 3 (2014), 595.
doi: 10.3934/eect.2014.3.595. |
[11] |
B. Brunnhuber, Well-posedness and exponential decay of solutions for the Blackstock-Crighton-Kuznetsov equation,, J. Math. Anal. Appl., 433 (2016), 1037.
doi: 10.1016/j.jmaa.2015.07.046. |
[12] |
B. Brunnhuber and P. M. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption,, Int. J. Non-Linear Mech., 78 (2016), 131.
doi: 10.1016/j.ijnonlinmec.2015.10.008. |
[13] |
P. J. Chen, Growth and decay of waves in solids,, in Handbuch der Physik, (1973), 303. Google Scholar |
[14] |
P. J. Chen, On the growth and decay of one-dimensional temperature rate waves,, Arch. Ration. Mech. Anal., 35 (1969), 1.
doi: 10.1007/BF00248491. |
[15] |
M. Chen, M. Torres and T. Walsh, Existence of travelling wave solutions of a high-order nonlinear acoustic wave equation,, Phys. Lett. A, 373 (2009), 1037.
doi: 10.1016/j.physleta.2009.01.042. |
[16] |
W. Chester, Resonant oscillations in closed tubes,, J. Fluid Mech., 18 (1964), 44.
doi: 10.1017/S0022112064000040. |
[17] |
I. Christov, C. I. Christov and P. M. Jordan, Modeling weakly nonlinear acoustic wave propagation,, Q. J. Mech. Appl. Math., 60 (2007), 473.
doi: 10.1093/qjmam/hbm017. |
[18] |
I. Christov, C. I. Christov and P. M. Jordan, Corrigendum and addendum: Modeling weakly nonlinear acoustic wave propagation,, Q. J. Mech. Appl. Math., 68 (2015), 231.
doi: 10.1093/qjmam/hbu023. |
[19] |
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. Simulat., 127 (2016), 2.
doi: 10.1016/j.matcom.2013.03.011. |
[20] |
I. Christov, P. M. Jordan and C. I. Christov, Nonlinear acoustic propagation in homentropic perfect gases: A numerical study,, Phys. Lett. A, 353 (2006), 273.
doi: 10.1016/j.physleta.2005.12.101. |
[21] |
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function,, Adv. Comput. Math., 5 (1996), 329.
doi: 10.1007/BF02124750. |
[22] |
D. G. Crighton, Model equations of nonlinear acoustics,, Annu. Rev. Fluid Mech., 11 (1979), 11.
doi: 10.1146/annurev.fl.11.010179.000303. |
[23] |
A. M. J. Davis and H. Brenner, Thermal and viscous effects on sound waves: Revised classical theory,, J. Acoust. Soc. Am., 132 (2012), 2963.
doi: 10.1121/1.4757971. |
[24] |
A. R. Elcrat, On the propagation of sonic discontinuities in the unsteady flow of a perfect gas,, Int. J. Eng. Sci., 15 (1977), 29.
doi: 10.1016/0020-7225(77)90066-0. |
[25] |
Y. B. Fu and N. H. Scott, The transition from acceleration wave to shock wave,, Int. J. Eng. Sci., 29 (1991), 617.
doi: 10.1016/0020-7225(91)90066-C. |
[26] |
A. E. Green and P. M. Naghdi, A new thermoviscous theory for fluids,, J. Non-Newtonian Fluid Mech., 56 (1995), 289.
doi: 10.1016/0377-0257(94)01288-S. |
[27] |
A. E. Green and P. M. Naghdi, An extended theory for incompressible viscous fluid flow,, J. Non-Newtonian Fluid Mech., 66 (1996), 233.
doi: 10.1016/S0377-0257(96)01478-4. |
[28] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics,, Proc. R. Soc. Lond. A, 448 (1995), 335.
doi: 10.1098/rspa.1995.0020. |
[29] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua,, Proc. R. Soc. Lond. A, 448 (1995), 357.
doi: 10.1098/rspa.1995.0021. |
[30] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua,, Proc. R. Soc. Lond. A, 448 (1995), 379.
doi: 10.1098/rspa.1995.0022. |
[31] |
M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds,, Arch. Rational Mech. Anal., 31 (1968), 113.
doi: 10.1007/BF00281373. |
[32] |
M. F. Hamilton and C. L. Morfey, Model equations,, in Nonlinear Acoustics: Theory and Applications (eds. M. F. Hamilton and D. T. Blackstock), (1997), 41. Google Scholar |
[33] |
B. M. Johnson, Analytical shock solutions at large and small Prandtl number,, J. Fluid Mech., 726 (2013).
doi: 10.1017/jfm.2013.262. |
[34] |
B. M. Johnson, Closed-form shock solutions,, J. Fluid Mech., 745 (2014).
doi: 10.1017/jfm.2014.107. |
[35] |
P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009,, Mech. Res. Commun., 73 (2016), 127.
doi: 10.1016/j.mechrescom.2016.02.014. |
[36] |
P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation,, Phys. Lett. A, 326 (2004), 77.
doi: 10.1016/j.physleta.2004.03.067. |
[37] |
P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189.
doi: 10.3934/dcdsb.2014.19.2189. |
[38] |
P. M. Jordan, A note on the Lambert $W$-function: Applications in the mathematical and physical sciences,, in Mathematics of Continuous and Discrete Dynamical Systems (ed. A. B. Gumel), 618 (2014), 247.
doi: 10.1090/conm/618. |
[39] |
P. M. Jordan and C. I. Christov, A simple finite difference scheme for modeling the finite-time blow-up of acoustic acceleration waves,, J. Sound Vib., 281 (2005), 1207.
doi: 10.1016/j.jsv.2004.03.067. |
[40] |
P. M. Jordan and B. Straughan, Acoustic acceleration waves in homentropic Green and Naghdi gases,, Proc. R. Soc. A, 462 (2006), 3601.
doi: 10.1098/rspa.2006.1739. |
[41] |
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.
doi: 10.1016/j.euromechflu.2012.01.016. |
[42] |
B. Kaltenbacher, Mathematics of nonlinear acoustics,, Evol. Equ. Control Theory, 4 (2015), 447.
doi: 10.3934/eect.2015.4.447. |
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