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Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids
September  2016, 5(3): 367-381. doi: 10.3934/eect.2016009

## Oscillating nonlinear acoustic shock waves

 1 Bogolyubov Institute for Theoretical Physics, 03143 Kiev, Ukraine 2 GreenHydrogen, DK-6000 Kolding, Denmark 3 Department of Physics and Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 4 Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

Received  January 2016 Revised  March 2016 Published  August 2016

We investigate oscillating shock waves in a tube using a higher order weakly nonlinear acoustic model. The model includes thermoviscous effects and is non isentropic. The oscillating shock waves are generated at one end of the tube by a sinusoidal driver. Numerical simulations show that at resonance a stationary state arise consisting of multiple oscillating shock waves. Off resonance driving leads to a nearly linear oscillating ground state but superimposed by bursts of a fast oscillating shock wave. Based on a travelling wave ansatz for the fluid velocity potential with an added 2'nd order polynomial in the space and time variables, we find analytical approximations to the observed single shock waves in an infinitely long tube. Using perturbation theory for the driven acoustic system approximative analytical solutions for the off resonant case are determined.
Citation: 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
##### References:
 [1] B. O. Enflo and C. M. Hedberg, Theory of Nonlinear Acoustics in Fluids, $1^{st}$ edition, Kluwer Academic, Dordrecht, 2002. [2] W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64. doi: 10.1017/S0022112064000040. [3] I. Christov, C. I. Christov and P. M. Jordan, Modeling weakly nonlinear acoustic wave propagation, Q. Jl Mech. Appl. Math., 60 (2007), 473-495. doi: 10.1093/qjmam/hbm017. [4] I. Christov, C. I. Christov and P. M. Jordan, Corrigendum and addendum: Modeling weakly nonlinear acoustic wave propagation, Q. Jl Mech. Appl. Math., 68 (2015), 231-233. doi: 10.1093/qjmam/hbu023. [5] S. M. Hagsäter, T. G. Jensen, H. Bruus and J. P. Kutter, Acoustic resonances in piezo-actuated microfluidic chips: Full-image micro-piv experiments and numerical simulations, Lab Chip, 7 (2007), 1336-1344. [6] S. M. Hagsäter, A. Lenshof, P. Skafte-Pedersen, J. P. Kutter, T. Laurell and H. Bruus, Acoustic resonances in straight micro channels: Beyond the 1d-approximation, Lab Chip, 8 (2008), 1178-1184. [7] M. F. Hamilton and C. L. Morfey, In: M.F. Hamilton and D.T. Blackstock, (eds.), Nonlinear Acoustics, Chap. 3, Academic Press, San Diego, (1998), 41-64. [8] P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Physics Letters A, 326 (2004), 77-84. doi: 10.1016/j.physleta.2004.03.067. [9] 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, European Journal of Mechanics B-Fluids, 34 (2012), 56-63. doi: 10.1016/j.euromechflu.2012.01.016. [10] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evolutiuon equations and control theory, 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447. [11] 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. [12] V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470. [13] S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part I, Acustica, 82 (1996), 579-606. [14] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07. Online companion to [OLBC10]. [15] W. L. Nyborg, Acoustic streaming, Physical Acoustics, 2 (1965), 265-331. doi: 10.1016/B978-0-12-395662-0.50015-1. [16] A. R. Rasmussen, 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. [17] Anders Rønne Rasmussen, Thermoviscous Model Equations in Nonlinear Acoustics, Ph.D Thesis, Department of Mathematics, Technical University of Denmark in Kongens Lyngby, Denmark, 2009.

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##### References:
 [1] B. O. Enflo and C. M. Hedberg, Theory of Nonlinear Acoustics in Fluids, $1^{st}$ edition, Kluwer Academic, Dordrecht, 2002. [2] W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), 44-64. doi: 10.1017/S0022112064000040. [3] I. Christov, C. I. Christov and P. M. Jordan, Modeling weakly nonlinear acoustic wave propagation, Q. Jl Mech. Appl. Math., 60 (2007), 473-495. doi: 10.1093/qjmam/hbm017. [4] I. Christov, C. I. Christov and P. M. Jordan, Corrigendum and addendum: Modeling weakly nonlinear acoustic wave propagation, Q. Jl Mech. Appl. Math., 68 (2015), 231-233. doi: 10.1093/qjmam/hbu023. [5] S. M. Hagsäter, T. G. Jensen, H. Bruus and J. P. Kutter, Acoustic resonances in piezo-actuated microfluidic chips: Full-image micro-piv experiments and numerical simulations, Lab Chip, 7 (2007), 1336-1344. [6] S. M. Hagsäter, A. Lenshof, P. Skafte-Pedersen, J. P. Kutter, T. Laurell and H. Bruus, Acoustic resonances in straight micro channels: Beyond the 1d-approximation, Lab Chip, 8 (2008), 1178-1184. [7] M. F. Hamilton and C. L. Morfey, In: M.F. Hamilton and D.T. Blackstock, (eds.), Nonlinear Acoustics, Chap. 3, Academic Press, San Diego, (1998), 41-64. [8] P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Physics Letters A, 326 (2004), 77-84. doi: 10.1016/j.physleta.2004.03.067. [9] 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, European Journal of Mechanics B-Fluids, 34 (2012), 56-63. doi: 10.1016/j.euromechflu.2012.01.016. [10] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evolutiuon equations and control theory, 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447. [11] 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. [12] V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470. [13] S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part I, Acustica, 82 (1996), 579-606. [14] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07. Online companion to [OLBC10]. [15] W. L. Nyborg, Acoustic streaming, Physical Acoustics, 2 (1965), 265-331. doi: 10.1016/B978-0-12-395662-0.50015-1. [16] A. R. Rasmussen, 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. [17] Anders Rønne Rasmussen, Thermoviscous Model Equations in Nonlinear Acoustics, Ph.D Thesis, Department of Mathematics, Technical University of Denmark in Kongens Lyngby, Denmark, 2009.
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