# American Institute of Mathematical Sciences

September  2016, 5(3): 383-397. doi: 10.3934/eect.2016010

## The effects of coupling on finite-amplitude acoustic traveling waves in thermoviscous gases: Blackstock's models

 1 Acoustics Division, U.S. Naval Research Laboratory, Stennis Space Center, MS 39529, United States

Received  May 2016 Revised  May 2016 Published  August 2016

We consider the propagation of acoustic and thermal waves in classical perfect gases under a coupled, weakly-nonlinear system first derived by Blackstock. Our primary aim is to ascertain the usefulness of Blackstock's system as an approximate model of nonlinear acoustic phenomena. Working in the context of the piston problem, and using a solvable special case of the Navier--Stokes--Fourier system as our benchmark, we compare Blackstock's system against a simpler weakly-nonlinear model whose constitute equations are not coupled. In particular, traveling wave solutions (TWS)s are determined, the structure of the solution profiles is analyzed, numerical comparisons are presented, and follow-on studies are suggested.
Citation: P. M. Jordan. The effects of coupling on finite-amplitude acoustic traveling waves in thermoviscous gases: Blackstock's models. Evolution Equations & Control Theory, 2016, 5 (3) : 383-397. doi: 10.3934/eect.2016010
##### References:
 [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions,, Dover, (1965). Google Scholar [2] J. Angulo, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions,, Mathematical Surveys and Monographs, (2009). doi: 10.1090/surv/156. Google Scholar [3] E. Becker, Gas Dynamics,, Academic Press, (1968). Google Scholar [4] R. Becker, Stoßbwelle und detonation (in German),, Z. Phys. 8 (1922), 8 (1922), 321. Google Scholar [5] R. T. Beyer, The parameter $B/A$,, in: Nonlinear Acoustics (eds. M. F. Hamilton and D. T. Blackstock), (1997), 25. Google Scholar [6] D. T. Blackstock, Approximate Equations Governing Finite-Amplitude Sound in Thermoviscous Fluids,, Technical Report GD/E Report GD-1463-52, (1963), 1463. Google Scholar [7] W. Chester, Resonant oscillations in closed tubes,, J. Fluid Mech. 18 (1964), 18 (1964), 44. doi: 10.1017/S0022112064000040. Google Scholar [8] 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. Simul. 127 (2016), 127 (2016), 2. doi: 10.1016/j.matcom.2013.03.011. Google Scholar [9] H. Grad, Principles of the kinetic theory of gases,, in: Handbuch der Physik (ed. S. Flügge), 12 (1958), 205. Google Scholar [10] W. D. Hayes, Gasdynamic Discontinuities,, Princeton University Press, (1960). doi: 10.1515/9781400879939. Google Scholar [11] P. M. Jordan and R. S. Keiffer, A note on finite-scale Navier-Stokes theory: The case of constant viscosity, strictly adiabatic flow,, Phys. Lett. A, 379 (2015), 124. doi: 10.1016/j.physleta.2014.10.033. Google Scholar [12] P. M. Jordan, R. S. Keiffer and G. Saccomandi, Anomalous propagation of acoustic traveling waves in thermoviscous fluids under the Rubin-Rosenau-Gottlieb theory of dispersive media,, Wave Motion, 51 (2014), 382. doi: 10.1016/j.wavemoti.2013.08.009. Google Scholar [13] 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. Google Scholar [14] B. Kaltenbacher, Well-posedness of a general higher order model in nonlinear acoustics,, Appl. Math. Lett., 63 (2017), 21. doi: 10.1016/j.aml.2016.07.008. Google Scholar [15] L. D. Landau and E. M. Lifshitz, Fluid Mechanics,, Pergamon Press, (1959). Google Scholar [16] G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics,, Wave Motion, 11 (1989), 23. doi: 10.1016/0165-2125(89)90010-3. Google Scholar [17] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511791253. Google Scholar [18] M. J. Lighthill, Viscosity effects in sound waves of finite amplitude,, in: Surveys in Mechanics (eds. G. K. Batchelor and R. M. Davies), (1956), 250. Google Scholar [19] M. J. Lighthill, Waves in Fluids,, Cambridge University Press, (2001). Google Scholar [20] S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I,, Acustica-Acta Acustica, 82 (1996), 579. Google Scholar [21] L. G. Margolin, J. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles, (in preparation)., (). Google Scholar [22] M. Morduchow and P. A. Libby, On a complete solution of the one-dimensional flow equations of a viscous, heat-conducting, compressible gas,, J. Aeronaut. Sci., 16 (1949), 674. doi: 10.2514/8.11882. Google Scholar [23] A. Morro, Shock waves in thermo-viscous fluids with hidden variables,, Arch. Mech., 32 (1980), 193. Google Scholar [24] I. Müller and T. Ruggeri, Rational Extended Thermodynamics,, Springer Tracts in Natural Philosophy, (1998). doi: 10.1007/978-1-4612-2210-1. Google Scholar [25] A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications,, Acoustical Society of America, (1989). Google Scholar [26] Lord Rayleigh, Aerial plane waves of finite amplitude},, Proc. R. Soc. Lond. A, 84 (1910), 247. doi: 10.1098/rspa.1910.0075. Google Scholar [27] M. Roy, Sur la structure de l'onde de choc, limite d'une quasi-onde de choc dans un fluide compressible et visqueux (in French),, C. R. Acad. Sci., 218 (1944), 813. Google Scholar [28] J. Serrin and Y. C. Whang, On the entropy change through a shock layer,, J. Aero/Space Sci., 28 (1961), 990. doi: 10.2514/8.9282. Google Scholar [29] B. Straughan, Heat Waves,, Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4614-0493-4. Google Scholar [30] P. A. Thompson, Compressible-Fluid Dynamics,, McGraw-Hill, (1972). Google Scholar

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##### References:
 [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions,, Dover, (1965). Google Scholar [2] J. Angulo, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions,, Mathematical Surveys and Monographs, (2009). doi: 10.1090/surv/156. Google Scholar [3] E. Becker, Gas Dynamics,, Academic Press, (1968). Google Scholar [4] R. Becker, Stoßbwelle und detonation (in German),, Z. Phys. 8 (1922), 8 (1922), 321. Google Scholar [5] R. T. Beyer, The parameter $B/A$,, in: Nonlinear Acoustics (eds. M. F. Hamilton and D. T. Blackstock), (1997), 25. Google Scholar [6] D. T. Blackstock, Approximate Equations Governing Finite-Amplitude Sound in Thermoviscous Fluids,, Technical Report GD/E Report GD-1463-52, (1963), 1463. Google Scholar [7] W. Chester, Resonant oscillations in closed tubes,, J. Fluid Mech. 18 (1964), 18 (1964), 44. doi: 10.1017/S0022112064000040. Google Scholar [8] 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. Simul. 127 (2016), 127 (2016), 2. doi: 10.1016/j.matcom.2013.03.011. Google Scholar [9] H. Grad, Principles of the kinetic theory of gases,, in: Handbuch der Physik (ed. S. Flügge), 12 (1958), 205. Google Scholar [10] W. D. Hayes, Gasdynamic Discontinuities,, Princeton University Press, (1960). doi: 10.1515/9781400879939. Google Scholar [11] P. M. Jordan and R. S. Keiffer, A note on finite-scale Navier-Stokes theory: The case of constant viscosity, strictly adiabatic flow,, Phys. Lett. A, 379 (2015), 124. doi: 10.1016/j.physleta.2014.10.033. Google Scholar [12] P. M. Jordan, R. S. Keiffer and G. Saccomandi, Anomalous propagation of acoustic traveling waves in thermoviscous fluids under the Rubin-Rosenau-Gottlieb theory of dispersive media,, Wave Motion, 51 (2014), 382. doi: 10.1016/j.wavemoti.2013.08.009. Google Scholar [13] 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. Google Scholar [14] B. Kaltenbacher, Well-posedness of a general higher order model in nonlinear acoustics,, Appl. Math. Lett., 63 (2017), 21. doi: 10.1016/j.aml.2016.07.008. Google Scholar [15] L. D. Landau and E. M. Lifshitz, Fluid Mechanics,, Pergamon Press, (1959). Google Scholar [16] G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics,, Wave Motion, 11 (1989), 23. doi: 10.1016/0165-2125(89)90010-3. Google Scholar [17] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511791253. Google Scholar [18] M. J. Lighthill, Viscosity effects in sound waves of finite amplitude,, in: Surveys in Mechanics (eds. G. K. Batchelor and R. M. Davies), (1956), 250. Google Scholar [19] M. J. Lighthill, Waves in Fluids,, Cambridge University Press, (2001). Google Scholar [20] S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I,, Acustica-Acta Acustica, 82 (1996), 579. Google Scholar [21] L. G. Margolin, J. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles, (in preparation)., (). Google Scholar [22] M. Morduchow and P. A. Libby, On a complete solution of the one-dimensional flow equations of a viscous, heat-conducting, compressible gas,, J. Aeronaut. Sci., 16 (1949), 674. doi: 10.2514/8.11882. Google Scholar [23] A. Morro, Shock waves in thermo-viscous fluids with hidden variables,, Arch. Mech., 32 (1980), 193. Google Scholar [24] I. Müller and T. Ruggeri, Rational Extended Thermodynamics,, Springer Tracts in Natural Philosophy, (1998). doi: 10.1007/978-1-4612-2210-1. Google Scholar [25] A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications,, Acoustical Society of America, (1989). Google Scholar [26] Lord Rayleigh, Aerial plane waves of finite amplitude},, Proc. R. Soc. Lond. A, 84 (1910), 247. doi: 10.1098/rspa.1910.0075. Google Scholar [27] M. Roy, Sur la structure de l'onde de choc, limite d'une quasi-onde de choc dans un fluide compressible et visqueux (in French),, C. R. Acad. Sci., 218 (1944), 813. Google Scholar [28] J. Serrin and Y. C. Whang, On the entropy change through a shock layer,, J. Aero/Space Sci., 28 (1961), 990. doi: 10.2514/8.9282. Google Scholar [29] B. Straughan, Heat Waves,, Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4614-0493-4. Google Scholar [30] P. A. Thompson, Compressible-Fluid Dynamics,, McGraw-Hill, (1972). Google Scholar
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