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 and 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, New York, 1965.

[2]

J. Angulo, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Mathematical Surveys and Monographs, Vol. 156, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/surv/156.

[3]

E. Becker, Gas Dynamics, Academic Press, New York, NY, 1968.

[4]

R. Becker, Stoßbwelle und detonation (in German), Z. Phys. 8 (1922), 321-362. [English transl.: Impact waves and detonation, Part I, N.A.C.A. Technical Memo. No. 505 (N.A.C.A., Washington, DC, 1929); Part II, N.A.C.A. Technical Memo. No. 506 (N.A.C.A., Washington, DC, 1929)].

[5]

R. T. Beyer, The parameter $B/A$, in: Nonlinear Acoustics (eds. M. F. Hamilton and D. T. Blackstock), Academic Press, San Diego, CA, (1997), 25-39.

[6]

D. T. Blackstock, Approximate Equations Governing Finite-Amplitude Sound in Thermoviscous Fluids, Technical Report GD/E Report GD-1463-52, General Dynamics Corp., Rochester, NY, 1963, Chap. IV.

[7]

W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech. 18 (1964), 44-64. doi: 10.1017/S0022112064000040.

[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), 2-18. doi: 10.1016/j.matcom.2013.03.011.

[9]

H. Grad, Principles of the kinetic theory of gases, in: Handbuch der Physik (ed. S. Flügge), Vol. XII, Springer-Verlag, Berlin, 12 (1958), 205-294.

[10]

W. D. Hayes, Gasdynamic Discontinuities, Princeton University Press, Princeton, NJ, 1960, SD,5. doi: 10.1515/9781400879939.

[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), 124-130. doi: 10.1016/j.physleta.2014.10.033.

[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), 382-388. doi: 10.1016/j.wavemoti.2013.08.009.

[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), 56-63. doi: 10.1016/j.euromechflu.2012.01.016.

[14]

B. Kaltenbacher, Well-posedness of a general higher order model in nonlinear acoustics, Appl. Math. Lett., 63 (2017), 21-27. doi: 10.1016/j.aml.2016.07.008.

[15]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1959, p87.

[16]

G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics, Wave Motion, 11 (1989), 23-32. doi: 10.1016/0165-2125(89)90010-3.

[17]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, U.K., 2002. doi: 10.1017/CBO9780511791253.

[18]

M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in: Surveys in Mechanics (eds. G. K. Batchelor and R. M. Davies), Cambridge University Press, London, 1956, pp. 250-351.

[19]

M. J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, U.K., 2001, p 1.2.

[20]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acustica-Acta Acustica, 82 (1996), 579-606.

[21]

L. G. Margolin, J. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles (in preparation).

[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), 674-684, 704. doi: 10.2514/8.11882.

[23]

A. Morro, Shock waves in thermo-viscous fluids with hidden variables, Arch. Mech., 32 (1980), 193-199.

[24]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy, Vol. 37, 2nd edn., Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-2210-1.

[25]

A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Woodbury, NY, 1989.

[26]

Lord Rayleigh, Aerial plane waves of finite amplitude}, Proc. R. Soc. Lond. A, 84 (1910), 247-284. doi: 10.1098/rspa.1910.0075.

[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), 813-816.

[28]

J. Serrin and Y. C. Whang, On the entropy change through a shock layer, J. Aero/Space Sci., 28 (1961), 990-991. doi: 10.2514/8.9282.

[29]

B. Straughan, Heat Waves, Applied Mathematical Sciences, Vol. 177, Springer, New York, NY, 2011. doi: 10.1007/978-1-4614-0493-4.

[30]

P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, NY, 1972.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, New York, 1965.

[2]

J. Angulo, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Mathematical Surveys and Monographs, Vol. 156, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/surv/156.

[3]

E. Becker, Gas Dynamics, Academic Press, New York, NY, 1968.

[4]

R. Becker, Stoßbwelle und detonation (in German), Z. Phys. 8 (1922), 321-362. [English transl.: Impact waves and detonation, Part I, N.A.C.A. Technical Memo. No. 505 (N.A.C.A., Washington, DC, 1929); Part II, N.A.C.A. Technical Memo. No. 506 (N.A.C.A., Washington, DC, 1929)].

[5]

R. T. Beyer, The parameter $B/A$, in: Nonlinear Acoustics (eds. M. F. Hamilton and D. T. Blackstock), Academic Press, San Diego, CA, (1997), 25-39.

[6]

D. T. Blackstock, Approximate Equations Governing Finite-Amplitude Sound in Thermoviscous Fluids, Technical Report GD/E Report GD-1463-52, General Dynamics Corp., Rochester, NY, 1963, Chap. IV.

[7]

W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech. 18 (1964), 44-64. doi: 10.1017/S0022112064000040.

[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), 2-18. doi: 10.1016/j.matcom.2013.03.011.

[9]

H. Grad, Principles of the kinetic theory of gases, in: Handbuch der Physik (ed. S. Flügge), Vol. XII, Springer-Verlag, Berlin, 12 (1958), 205-294.

[10]

W. D. Hayes, Gasdynamic Discontinuities, Princeton University Press, Princeton, NJ, 1960, SD,5. doi: 10.1515/9781400879939.

[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), 124-130. doi: 10.1016/j.physleta.2014.10.033.

[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), 382-388. doi: 10.1016/j.wavemoti.2013.08.009.

[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), 56-63. doi: 10.1016/j.euromechflu.2012.01.016.

[14]

B. Kaltenbacher, Well-posedness of a general higher order model in nonlinear acoustics, Appl. Math. Lett., 63 (2017), 21-27. doi: 10.1016/j.aml.2016.07.008.

[15]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1959, p87.

[16]

G. Lebon and A. Cloot, Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics, Wave Motion, 11 (1989), 23-32. doi: 10.1016/0165-2125(89)90010-3.

[17]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, U.K., 2002. doi: 10.1017/CBO9780511791253.

[18]

M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in: Surveys in Mechanics (eds. G. K. Batchelor and R. M. Davies), Cambridge University Press, London, 1956, pp. 250-351.

[19]

M. J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, U.K., 2001, p 1.2.

[20]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acustica-Acta Acustica, 82 (1996), 579-606.

[21]

L. G. Margolin, J. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles (in preparation).

[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), 674-684, 704. doi: 10.2514/8.11882.

[23]

A. Morro, Shock waves in thermo-viscous fluids with hidden variables, Arch. Mech., 32 (1980), 193-199.

[24]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy, Vol. 37, 2nd edn., Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-2210-1.

[25]

A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Woodbury, NY, 1989.

[26]

Lord Rayleigh, Aerial plane waves of finite amplitude}, Proc. R. Soc. Lond. A, 84 (1910), 247-284. doi: 10.1098/rspa.1910.0075.

[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), 813-816.

[28]

J. Serrin and Y. C. Whang, On the entropy change through a shock layer, J. Aero/Space Sci., 28 (1961), 990-991. doi: 10.2514/8.9282.

[29]

B. Straughan, Heat Waves, Applied Mathematical Sciences, Vol. 177, Springer, New York, NY, 2011. doi: 10.1007/978-1-4614-0493-4.

[30]

P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, NY, 1972.

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