Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids

Previous Article
Oscillating nonlinear acoustic shock waves
EECT Home
This Issue
Next Article
Introduction to the special volume ``Mathematics of nonlinear acoustics: New approaches in analysis and modeling''

September 2016, 5(3): 349-365. doi: 10.3934/eect.2016008

Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids

Ivan C. Christov ^1,

1.	School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, United States

Received January 2016 Revised January 2016 Published August 2016

Abstract
Related Papers

The equations of motion of lossless compressible nonclassical fluids under the so-called Green--Naghdi theory are considered for two classes of barotropic fluids: (i) perfect gases and (ii) liquids obeying a quadratic equation of state. An exact reduction in terms of a scalar acoustic potential and the (scalar) thermal displacement is achieved. Properties and simplifications of these model nonlinear acoustic equations for unidirectional flows are noted. Specifically, the requirement that the governing system of equations for such flows remain hyperbolic is shown to lead to restrictions on the physical parameters and/or applicability of the model. A weakly nonlinear model is proposed on the basis of neglecting only terms proportional to the square of the Mach number in the governing equations, without any further approximation or modification of the nonlinear terms. Shock formation via acceleration wave blow up is studied numerically in a one-dimensional context using a high-resolution Godunov-type finite-volume scheme, thereby verifying prior analytical results on the blow up time and contrasting these results with the corresponding ones for classical (Euler, i.e., lossless compressible) fluids.

Keywords: barotropic fluids, Evolution equations, Green--Naghdi theory, shock formation., nonlinear acoustics.

Mathematics Subject Classification: Primary: 35Q35, 76N15; Secondary: 76L05, 35L6.

Citation: Ivan C. Christov. Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids. Evolution Equations and Control Theory, 2016, 5 (3) : 349-365. doi: 10.3934/eect.2016008

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), 605-615. 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), 101-118. 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), 6067-6073. 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), 4418-4424. 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), Academic Press, (1997), 25-39.
[6]	J. Bissell and B. Straughan, Discontinuity waves as tipping points: Applications to biological & sociological systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1911-1934. 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.
[8]	D. T. Blackstock, Propagation of plane sound waves of finite amplitude in nondissipative fluids, J. Acoust. Soc. Am., 34 (1962), 9-30. 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), 4515-4535, arXiv:1311.1692. 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), 595-626, arXiv:1410.0797. 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), 1037-1054, arXiv:1405.6494. 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), 131-132. doi: 10.1016/j.ijnonlinmec.2015.10.008.
[13]	P. J. Chen, Growth and decay of waves in solids, in Handbuch der Physik, vol. VIa/3 (eds. S. Flügge and C. Truesdell), Springer, Berlin, (1973), 303-402.
[14]	P. J. Chen, On the growth and decay of one-dimensional temperature rate waves, Arch. Ration. Mech. Anal., 35 (1969), 1-15. 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), 1037-1043. doi: 10.1016/j.physleta.2009.01.042.
[16]	W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64. 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), 473-495. 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), 231-233. 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), 2-18. 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), 273-280. 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), 329-359. doi: 10.1007/BF02124750.
[22]	D. G. Crighton, Model equations of nonlinear acoustics, Annu. Rev. Fluid Mech., 11 (1979), 11-33. 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), 2963-2969. 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), 29-34. 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), 617-624. 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), 289-306. 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), 233-255. 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), 335-356. 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), 357-377. 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), 379-388. 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), 113-126. 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), Academic Press, (1997), 41-63.
[33]	B. M. Johnson, Analytical shock solutions at large and small Prandtl number, J. Fluid Mech., 726 (2013), R4, 12pp. doi: 10.1017/jfm.2013.262.
[34]	B. M. Johnson, Closed-form shock solutions, J. Fluid Mech., 745 (2014), R1, 11pp. doi: 10.1017/jfm.2014.107.
[35]	P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009, Mech. Res. Commun., 73 (2016), 127-139. 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), 77-84. 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), 2189-2205. 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), American Mathematical Society, 618 (2014), 247-263. 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), 1207-1216. 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), 3601-3611. 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), 56-63. doi: 10.1016/j.euromechflu.2012.01.016.
[42]	B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447.
[43]	B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 503-523. doi: 10.3934/dcdss.2009.2.503.
[44]	B. Kaltenbacher, Boundary observability and stabilization for Westervelt type wave equations without interior damping, Appl. Math. Optim., 62 (2010), 381-410. doi: 10.1007/s00245-010-9108-7.
[45]	B. Kaltenbacher, I. Lasiecka and S. Veljović, Well-posedness and exponential decay for the Westervelt equation with inhomogeneous Dirichlet boundary data, in Parabolic Problems: Herbert Amann Festschrift (eds. J. Escher et al.), Springer, 80 (2011), 357-387. doi: 10.1007/978-3-0348-0075-4_19.
[46]	B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321. doi: 10.1002/mana.201000007.
[47]	R. S. Keiffer, R. McNorton, P. M. Jordan and I. C. Christov, Dissipative acoustic solitons under a weakly-nonlinear, Lagrangian-averaged Euler-$\alpha$ model of single-phase lossless fluids, Wave Motion, 48 (2011), 782-790. doi: 10.1016/j.wavemoti.2011.04.013.
[48]	W. Lauterborn, T. Kurz and I. Akhatov, Nonlinear acoustics in fluids, in Springer Handbook of Acoustics (ed. T. D. Rossing), Springer, (2007), 257-297. doi: 10.1007/978-0-387-30425-0_8.
[49]	P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973. doi: 10.1137/1.9781611970562.
[50]	M. B. Lesser and 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.
[51]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, New York, NY, 2002. doi: 10.1017/CBO9780511791253.
[52]	H. Lin and A. J. Szeri, Shock formation in the presence of entropy gradients, J. Fluid Mech., 431 (2001), 161-188. doi: 10.1017/S0022112000003104.
[53]	K. A. Lindsay and B. Straughan, Acceleration waves and second sound in a perfect fluid, Arch. Rational Mech. Anal., 68 (1978), 53-87. doi: 10.1007/BF00276179.
[54]	S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acta Acoust. united Ac., 82 (1996), 579-606.
[55]	S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part II, Acta Acust. united Ac., 83 (1997), 197-222.
[56]	S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part III, Acta Acust. united Ac., 83 (1997), 827-846.
[57]	A. Morro, Jump relations and discontinuity waves in conductors with memory, Math. Comput. Modell., 43 (2006), 138-149. doi: 10.1016/j.mcm.2005.04.016.
[58]	K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics, Cambridge University Press, New York, NY, 1998. doi: 10.2277/052139984X.
[59]	H. Ockendon and J. R. Ockendon, Waves and Compressible Flow, Springer, Berlin, 2004. doi: 10.1007/b97537.
[60]	H. Ockendon and J. R. Ockendon, Nonlinearity in fluid resonances, Meccanica, 36 (2001), 297-321. doi: 10.1023/A:1013911407811.
[61]	M. Ostoja-Starzewski and J. Trębicki, On the growth and decay of acceleration waves in random media, Proc. R. Soc. Lond. A, 455 (1999), 2577-2614. doi: 10.1098/rspa.1999.0418.
[62]	R. Quintanilla and B. Straughan, Nonlinear waves in a Green-Naghdi dissipationless fluid, J. Non-Newtonian Fluid Mech., 154 (2008), 207-210. doi: 10.1016/j.jnnfm.2008.04.006.
[63]	R. Quintanilla and B. Straughan, Green-Naghdi type III viscous fluids, Int. J. Heat Mass Transf., 55 (2012), 710-714. doi: 10.1016/j.ijheatmasstransfer.2011.10.039.
[64]	A. R. Rassmusen, M. P. Sørensen, Yu. B. Gaididei and P. L. Christiansen, Interacting wave fronts and rarefaction waves in a second order model of nonlinear thermoviscous fluids, Acta Appl. Math., 115 (2011), 43-61. doi: 10.1007/s10440-010-9581-7.
[65]	A. R. Rasmussen, M. P. Sørensen, Yu. B. Gaididei and P. L. Christiansen, Compound waves in a higher order nonlinear model of thermoviscous fluids, Math. Comput. Simulat., 127 (2016), 236-251. doi: 10.1016/j.matcom.2014.01.009.
[66]	R. A. Saenger and G. E. Hudson, Periodic shock waves in resonating gas columns, J. Acoust. Soc. Am., 32 (1960), 961-970. doi: 10.1121/1.1908343.
[67]	L. I. Sedov, Mechanics of Continuous Media, World Scientific, River Edge, NJ, 1997. doi: 10.1142/0712.
[68]	R. L. Seliger and G. B. Whitham, Variational principles in continuum mechanics, Proc. R. Soc. A, 305 (1968), 1-25. doi: 10.1098/rspa.1968.0103.
[69]	L. H. Söderholm, A higher order acoustic equation for the slightly viscous case, Acta Acust. united Ac., 87 (2000), 29-33.
[70]	B. Straughan, Green-Naghdi fluid with non-thermal equilibrium effects, Proc. R. Soc. A, 466 (2010), 2021-2032. doi: 10.1098/rspa.2009.0523.
[71]	B. Straughan, Heat Waves, Springer, New York, NY, 2011. doi: 10.1007/978-1-4614-0493-4.
[72]	B. Straughan, Shocks and acceleration waves in modern continuum mechanics and in social systems, Evol. Equ. Control Theory, 3 (2014), 541-555. doi: 10.3934/eect.2014.3.541.
[73]	P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, NY, 1972.
[74]	T. Y. Thomas, The growth and decay of sonic discontinuities in ideal gases, J. Math. Mech., 6 (1957), 455-469. doi: 10.1512/iumj.1957.6.56022.
[75]	E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, $3^{rd}$ edition, Springer-Verlag, Berlin/Heidelberg, 2009. doi: 10.1007/b79761.
[76]	G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, NY, 1974. doi: 10.1002/9781118032954.
[77]	T. W. Wright, An intrinsic description of unsteady shock waves, Q. J. Mech. Appl. Math., 29 (1976), 311-324. doi: 10.1093/qjmam/29.3.311.

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), 605-615. 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), 101-118. 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), 6067-6073. 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), 4418-4424. 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), Academic Press, (1997), 25-39.
[6]	J. Bissell and B. Straughan, Discontinuity waves as tipping points: Applications to biological & sociological systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1911-1934. 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.
[8]	D. T. Blackstock, Propagation of plane sound waves of finite amplitude in nondissipative fluids, J. Acoust. Soc. Am., 34 (1962), 9-30. 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), 4515-4535, arXiv:1311.1692. 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), 595-626, arXiv:1410.0797. 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), 1037-1054, arXiv:1405.6494. 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), 131-132. doi: 10.1016/j.ijnonlinmec.2015.10.008.
[13]	P. J. Chen, Growth and decay of waves in solids, in Handbuch der Physik, vol. VIa/3 (eds. S. Flügge and C. Truesdell), Springer, Berlin, (1973), 303-402.
[14]	P. J. Chen, On the growth and decay of one-dimensional temperature rate waves, Arch. Ration. Mech. Anal., 35 (1969), 1-15. 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), 1037-1043. doi: 10.1016/j.physleta.2009.01.042.
[16]	W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64. 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), 473-495. 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), 231-233. 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), 2-18. 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), 273-280. 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), 329-359. doi: 10.1007/BF02124750.
[22]	D. G. Crighton, Model equations of nonlinear acoustics, Annu. Rev. Fluid Mech., 11 (1979), 11-33. 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), 2963-2969. 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), 29-34. 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), 617-624. 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), 289-306. 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), 233-255. 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), 335-356. 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), 357-377. 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), 379-388. 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), 113-126. 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), Academic Press, (1997), 41-63.
[33]	B. M. Johnson, Analytical shock solutions at large and small Prandtl number, J. Fluid Mech., 726 (2013), R4, 12pp. doi: 10.1017/jfm.2013.262.
[34]	B. M. Johnson, Closed-form shock solutions, J. Fluid Mech., 745 (2014), R1, 11pp. doi: 10.1017/jfm.2014.107.
[35]	P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009, Mech. Res. Commun., 73 (2016), 127-139. 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), 77-84. 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), 2189-2205. 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), American Mathematical Society, 618 (2014), 247-263. 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), 1207-1216. 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), 3601-3611. 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), 56-63. doi: 10.1016/j.euromechflu.2012.01.016.
[42]	B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447.
[43]	B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 503-523. doi: 10.3934/dcdss.2009.2.503.
[44]	B. Kaltenbacher, Boundary observability and stabilization for Westervelt type wave equations without interior damping, Appl. Math. Optim., 62 (2010), 381-410. doi: 10.1007/s00245-010-9108-7.
[45]	B. Kaltenbacher, I. Lasiecka and S. Veljović, Well-posedness and exponential decay for the Westervelt equation with inhomogeneous Dirichlet boundary data, in Parabolic Problems: Herbert Amann Festschrift (eds. J. Escher et al.), Springer, 80 (2011), 357-387. doi: 10.1007/978-3-0348-0075-4_19.
[46]	B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321. doi: 10.1002/mana.201000007.
[47]	R. S. Keiffer, R. McNorton, P. M. Jordan and I. C. Christov, Dissipative acoustic solitons under a weakly-nonlinear, Lagrangian-averaged Euler-$\alpha$ model of single-phase lossless fluids, Wave Motion, 48 (2011), 782-790. doi: 10.1016/j.wavemoti.2011.04.013.
[48]	W. Lauterborn, T. Kurz and I. Akhatov, Nonlinear acoustics in fluids, in Springer Handbook of Acoustics (ed. T. D. Rossing), Springer, (2007), 257-297. doi: 10.1007/978-0-387-30425-0_8.
[49]	P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973. doi: 10.1137/1.9781611970562.
[50]	M. B. Lesser and 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.
[51]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, New York, NY, 2002. doi: 10.1017/CBO9780511791253.
[52]	H. Lin and A. J. Szeri, Shock formation in the presence of entropy gradients, J. Fluid Mech., 431 (2001), 161-188. doi: 10.1017/S0022112000003104.
[53]	K. A. Lindsay and B. Straughan, Acceleration waves and second sound in a perfect fluid, Arch. Rational Mech. Anal., 68 (1978), 53-87. doi: 10.1007/BF00276179.
[54]	S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acta Acoust. united Ac., 82 (1996), 579-606.
[55]	S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part II, Acta Acust. united Ac., 83 (1997), 197-222.
[56]	S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part III, Acta Acust. united Ac., 83 (1997), 827-846.
[57]	A. Morro, Jump relations and discontinuity waves in conductors with memory, Math. Comput. Modell., 43 (2006), 138-149. doi: 10.1016/j.mcm.2005.04.016.
[58]	K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics, Cambridge University Press, New York, NY, 1998. doi: 10.2277/052139984X.
[59]	H. Ockendon and J. R. Ockendon, Waves and Compressible Flow, Springer, Berlin, 2004. doi: 10.1007/b97537.
[60]	H. Ockendon and J. R. Ockendon, Nonlinearity in fluid resonances, Meccanica, 36 (2001), 297-321. doi: 10.1023/A:1013911407811.
[61]	M. Ostoja-Starzewski and J. Trębicki, On the growth and decay of acceleration waves in random media, Proc. R. Soc. Lond. A, 455 (1999), 2577-2614. doi: 10.1098/rspa.1999.0418.
[62]	R. Quintanilla and B. Straughan, Nonlinear waves in a Green-Naghdi dissipationless fluid, J. Non-Newtonian Fluid Mech., 154 (2008), 207-210. doi: 10.1016/j.jnnfm.2008.04.006.
[63]	R. Quintanilla and B. Straughan, Green-Naghdi type III viscous fluids, Int. J. Heat Mass Transf., 55 (2012), 710-714. doi: 10.1016/j.ijheatmasstransfer.2011.10.039.
[64]	A. R. Rassmusen, M. P. Sørensen, Yu. B. Gaididei and P. L. Christiansen, Interacting wave fronts and rarefaction waves in a second order model of nonlinear thermoviscous fluids, Acta Appl. Math., 115 (2011), 43-61. doi: 10.1007/s10440-010-9581-7.
[65]	A. R. Rasmussen, M. P. Sørensen, Yu. B. Gaididei and P. L. Christiansen, Compound waves in a higher order nonlinear model of thermoviscous fluids, Math. Comput. Simulat., 127 (2016), 236-251. doi: 10.1016/j.matcom.2014.01.009.
[66]	R. A. Saenger and G. E. Hudson, Periodic shock waves in resonating gas columns, J. Acoust. Soc. Am., 32 (1960), 961-970. doi: 10.1121/1.1908343.
[67]	L. I. Sedov, Mechanics of Continuous Media, World Scientific, River Edge, NJ, 1997. doi: 10.1142/0712.
[68]	R. L. Seliger and G. B. Whitham, Variational principles in continuum mechanics, Proc. R. Soc. A, 305 (1968), 1-25. doi: 10.1098/rspa.1968.0103.
[69]	L. H. Söderholm, A higher order acoustic equation for the slightly viscous case, Acta Acust. united Ac., 87 (2000), 29-33.
[70]	B. Straughan, Green-Naghdi fluid with non-thermal equilibrium effects, Proc. R. Soc. A, 466 (2010), 2021-2032. doi: 10.1098/rspa.2009.0523.
[71]	B. Straughan, Heat Waves, Springer, New York, NY, 2011. doi: 10.1007/978-1-4614-0493-4.
[72]	B. Straughan, Shocks and acceleration waves in modern continuum mechanics and in social systems, Evol. Equ. Control Theory, 3 (2014), 541-555. doi: 10.3934/eect.2014.3.541.
[73]	P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, NY, 1972.
[74]	T. Y. Thomas, The growth and decay of sonic discontinuities in ideal gases, J. Math. Mech., 6 (1957), 455-469. doi: 10.1512/iumj.1957.6.56022.
[75]	E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, $3^{rd}$ edition, Springer-Verlag, Berlin/Heidelberg, 2009. doi: 10.1007/b79761.
[76]	G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, NY, 1974. doi: 10.1002/9781118032954.
[77]	T. W. Wright, An intrinsic description of unsteady shock waves, Q. J. Mech. Appl. Math., 29 (1976), 311-324. doi: 10.1093/qjmam/29.3.311.

[1]	Lijun Zhang, Yixia Shi, Maoan Han. Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2917-2926. doi: 10.3934/dcdss.2020217
[2]	Vincent Duchêne, Christian Klein. Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021300
[3]	Colette Guillopé, Samer Israwi, Raafat Talhouk. Large-time existence for one-dimensional Green-Naghdi equations with vorticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2947-2974. doi: 10.3934/dcdss.2021040
[4]	Barbara Kaltenbacher. Mathematics of nonlinear acoustics. Evolution Equations and Control Theory, 2015, 4 (4) : 447-491. doi: 10.3934/eect.2015.4.447
[5]	Michael Shearer, Nicholas Giffen. Shock formation and breaking in granular avalanches. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 693-714. doi: 10.3934/dcds.2010.27.693
[6]	Cristóbal Rodero, J. Alberto Conejero, Ignacio García-Fernández. Shock wave formation in compliant arteries. Evolution Equations and Control Theory, 2019, 8 (1) : 221-230. doi: 10.3934/eect.2019012
[7]	Xin Zhong. Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1083-1096. doi: 10.3934/dcdsb.2019209
[8]	Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133
[9]	Bernard Ducomet, Eduard Feireisl, Hana Petzeltová, Ivan Straškraba. Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 113-130. doi: 10.3934/dcds.2004.11.113
[10]	Adrien Dekkers, Anna Rozanova-Pierrat, Vladimir Khodygo. Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4231-4258. doi: 10.3934/dcds.2020179
[11]	Yuri Gaididei, Anders Rønne Rasmussen, Peter Leth Christiansen, Mads Peter Sørensen. Oscillating nonlinear acoustic shock waves. Evolution Equations and Control Theory, 2016, 5 (3) : 367-381. doi: 10.3934/eect.2016009
[12]	Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681
[13]	Risei Kano, Yusuke Murase. Solvability of nonlinear evolution equations generated by subdifferentials and perturbations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 75-93. doi: 10.3934/dcdss.2014.7.75
[14]	Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046
[15]	Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331
[16]	Akisato Kubo. Nonlinear evolution equations associated with mathematical models. Conference Publications, 2011, 2011 (Special) : 881-890. doi: 10.3934/proc.2011.2011.881
[17]	Pedro M. Jordan, Barbara Kaltenbacher. Introduction to the special volume ``Mathematics of nonlinear acoustics: New approaches in analysis and modeling''. Evolution Equations and Control Theory, 2016, 5 (3) : i-ii. doi: 10.3934/eect.201603i
[18]	H. T. Banks, R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 119-149. doi: 10.3934/dcds.1995.1.119
[19]	Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[20]	Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139

2021 Impact Factor: 1.169

Tools

Metrics

Other articles
by authors

on AIMS
Ivan C. Christov
on Google Scholar
Ivan C. Christov

[Back to Top]