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

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

1. 

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

Received  January 2016 Revised  January 2016 Published  August 2016

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.
Citation: Ivan C. Christov. Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids. Evolution Equations & 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. 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.

[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.

[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.

[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.

[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.

[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. 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. 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.), 80 (2011), 357. 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. 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. 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), (2007), 257. 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, (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. doi: 10.1017/S0022112068000303.

[51]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (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. 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. 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.

[55]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part II,, Acta Acust. united Ac., 83 (1997), 197.

[56]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part III,, Acta Acust. united Ac., 83 (1997), 827.

[57]

A. Morro, Jump relations and discontinuity waves in conductors with memory,, Math. Comput. Modell., 43 (2006), 138. doi: 10.1016/j.mcm.2005.04.016.

[58]

K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics,, Cambridge University Press, (1998). doi: 10.2277/052139984X.

[59]

H. Ockendon and J. R. Ockendon, Waves and Compressible Flow,, Springer, (2004). doi: 10.1007/b97537.

[60]

H. Ockendon and J. R. Ockendon, Nonlinearity in fluid resonances,, Meccanica, 36 (2001), 297. 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. 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. 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. 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. 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. 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. doi: 10.1121/1.1908343.

[67]

L. I. Sedov, Mechanics of Continuous Media,, World Scientific, (1997). doi: 10.1142/0712.

[68]

R. L. Seliger and G. B. Whitham, Variational principles in continuum mechanics,, \emph{Proc. R. Soc. A}, 305 (1968), 1. 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.

[70]

B. Straughan, Green-Naghdi fluid with non-thermal equilibrium effects,, Proc. R. Soc. A, 466 (2010), 2021. doi: 10.1098/rspa.2009.0523.

[71]

B. Straughan, Heat Waves,, Springer, (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. doi: 10.3934/eect.2014.3.541.

[73]

P. A. Thompson, Compressible-Fluid Dynamics,, McGraw-Hill, (1972).

[74]

T. Y. Thomas, The growth and decay of sonic discontinuities in ideal gases,, J. Math. Mech., 6 (1957), 455. doi: 10.1512/iumj.1957.6.56022.

[75]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics,, $3^{rd}$ edition, (2009). doi: 10.1007/b79761.

[76]

G. B. Whitham, Linear and Nonlinear Waves,, Wiley-Interscience, (1974). doi: 10.1002/9781118032954.

[77]

T. W. Wright, An intrinsic description of unsteady shock waves,, Q. J. Mech. Appl. Math., 29 (1976), 311. 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. 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.

[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.

[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.

[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.

[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.

[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. 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. 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.), 80 (2011), 357. 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. 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. 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), (2007), 257. 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, (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. doi: 10.1017/S0022112068000303.

[51]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (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. 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. 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.

[55]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part II,, Acta Acust. united Ac., 83 (1997), 197.

[56]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part III,, Acta Acust. united Ac., 83 (1997), 827.

[57]

A. Morro, Jump relations and discontinuity waves in conductors with memory,, Math. Comput. Modell., 43 (2006), 138. doi: 10.1016/j.mcm.2005.04.016.

[58]

K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics,, Cambridge University Press, (1998). doi: 10.2277/052139984X.

[59]

H. Ockendon and J. R. Ockendon, Waves and Compressible Flow,, Springer, (2004). doi: 10.1007/b97537.

[60]

H. Ockendon and J. R. Ockendon, Nonlinearity in fluid resonances,, Meccanica, 36 (2001), 297. 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. 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. 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. 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. 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. 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. doi: 10.1121/1.1908343.

[67]

L. I. Sedov, Mechanics of Continuous Media,, World Scientific, (1997). doi: 10.1142/0712.

[68]

R. L. Seliger and G. B. Whitham, Variational principles in continuum mechanics,, \emph{Proc. R. Soc. A}, 305 (1968), 1. 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.

[70]

B. Straughan, Green-Naghdi fluid with non-thermal equilibrium effects,, Proc. R. Soc. A, 466 (2010), 2021. doi: 10.1098/rspa.2009.0523.

[71]

B. Straughan, Heat Waves,, Springer, (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. doi: 10.3934/eect.2014.3.541.

[73]

P. A. Thompson, Compressible-Fluid Dynamics,, McGraw-Hill, (1972).

[74]

T. Y. Thomas, The growth and decay of sonic discontinuities in ideal gases,, J. Math. Mech., 6 (1957), 455. doi: 10.1512/iumj.1957.6.56022.

[75]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics,, $3^{rd}$ edition, (2009). doi: 10.1007/b79761.

[76]

G. B. Whitham, Linear and Nonlinear Waves,, Wiley-Interscience, (1974). doi: 10.1002/9781118032954.

[77]

T. W. Wright, An intrinsic description of unsteady shock waves,, Q. J. Mech. Appl. Math., 29 (1976), 311. doi: 10.1093/qjmam/29.3.311.

[1]

Barbara Kaltenbacher. Mathematics of nonlinear acoustics. Evolution Equations & Control Theory, 2015, 4 (4) : 447-491. doi: 10.3934/eect.2015.4.447

[2]

Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133

[3]

Michael Shearer, Nicholas Giffen. Shock formation and breaking in granular avalanches. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 693-714. doi: 10.3934/dcds.2010.27.693

[4]

Bernard Ducomet, Eduard Feireisl, Hana Petzeltová, Ivan Straškraba. Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 113-130. doi: 10.3934/dcds.2004.11.113

[5]

Pedro M. Jordan, Barbara Kaltenbacher. Introduction to the special volume ``Mathematics of nonlinear acoustics: New approaches in analysis and modeling''. Evolution Equations & Control Theory, 2016, 5 (3) : i-ii. doi: 10.3934/eect.201603i

[6]

H. T. Banks, R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 119-149. doi: 10.3934/dcds.1995.1.119

[7]

Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681

[8]

Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331

[9]

Akisato Kubo. Nonlinear evolution equations associated with mathematical models. Conference Publications, 2011, 2011 (Special) : 881-890. doi: 10.3934/proc.2011.2011.881

[10]

Risei Kano, Yusuke Murase. Solvability of nonlinear evolution equations generated by subdifferentials and perturbations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 75-93. doi: 10.3934/dcdss.2014.7.75

[11]

Yuri Gaididei, Anders Rønne Rasmussen, Peter Leth Christiansen, Mads Peter Sørensen. Oscillating nonlinear acoustic shock waves. Evolution Equations & Control Theory, 2016, 5 (3) : 367-381. doi: 10.3934/eect.2016009

[12]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[13]

Eduard Feireisl. Mathematical theory of viscous fluids: Retrospective and future perspectives. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 533-555. doi: 10.3934/dcds.2010.27.533

[14]

Chjan C. Lim. Extremal free energy in a simple mean field theory for a coupled Barotropic fluid - rotating sphere system. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 361-386. doi: 10.3934/dcds.2007.19.361

[15]

Marco Caponigro, Anna Chiara Lai, Benedetto Piccoli. A nonlinear model of opinion formation on the sphere. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4241-4268. doi: 10.3934/dcds.2015.35.4241

[16]

Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801

[17]

Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841

[18]

Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11

[19]

Laura Levaggi. Existence of sliding motions for nonlinear evolution equations in Banach spaces. Conference Publications, 2013, 2013 (special) : 477-487. doi: 10.3934/proc.2013.2013.477

[20]

Mustapha Mokhtar-Kharroubi. On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory. Kinetic & Related Models, 2010, 3 (3) : 473-499. doi: 10.3934/krm.2010.3.473

2017 Impact Factor: 1.049

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]