American Institute of Mathematical Sciences

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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

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 & Control Theory, 2016, 5 (3) : 349-365. doi: 10.3934/eect.2016008
References:
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show all references

References:
[1]

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[2]

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[3]

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[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. Google Scholar

[5]

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[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. Google Scholar

[7]

D. T. Blackstock, Approximate equations governing finite-amplitude sound in thermoviscous fluids,, GD/E Report GD-1463-52, (1963), 1463. Google Scholar

[8]

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[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

W. Chester, Resonant oscillations in closed tubes,, J. Fluid Mech., 18 (1964), 44. doi: 10.1017/S0022112064000040. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

D. G. Crighton, Model equations of nonlinear acoustics,, Annu. Rev. Fluid Mech., 11 (1979), 11. doi: 10.1146/annurev.fl.11.010179.000303. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

B. M. Johnson, Closed-form shock solutions,, J. Fluid Mech., 745 (2014). doi: 10.1017/jfm.2014.107. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

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