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

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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

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

Received January 2016 Revised January 2016 Published August 2016

Abstract
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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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References:

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[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
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[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
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[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
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[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. Google Scholar
[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. Google Scholar
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