March 2019, 8(1): 101-116. doi: 10.3934/eect.2019006

Finite-amplitude acoustics under the classical theory of particle-laden flows

Acoustics Division, U.S. Naval Research Laboratory, Stennis Space Center, MS 39529, USA

Received  April 2018 Revised  July 2018 Published  January 2019

We consider acoustic propagation in a particle-laden fluid, specifically, a perfect gas, under a model system based on the theories of Marble (1970) and Thompson (1972). Our primary aim is to understand, via analytical methods, the impact of the particle phase on the acoustic velocity field. Working under the finite-amplitude approximation, we investigate singular surface and traveling wave phenomena, as admitted by both phases of the flow. We show, among other things, that the particle velocity field admits a singular surface one order higher than that of the gas phase, that the particle-to-gas density ratio plays a number of critical roles, and that traveling wave solutions are only possible for sufficiently small values of the Mach number.

Citation: Pedro M. Jordan. Finite-amplitude acoustics under the classical theory of particle-laden flows. Evolution Equations & Control Theory, 2019, 8 (1) : 101-116. doi: 10.3934/eect.2019006
References:
[1]

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1966.

[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, 2009. doi: 10.1090/surv/156.

[3]

S. BargmannP. 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), 4418-4424. doi: 10.1016/j.physleta.2008.04.010.

[4]

R. T. Beyer, The parameter B/A, in: Nonlinear Acoustics (eds. M. F. Hamilton and D. T. Blackstock), Academic Press, 1998, 25–39.

[5]

J. Bissell and B. Straughan, Discontinuity waves as tipping points: Applications to biological & sociological systems, Discrete Cont. Dyn. Sys., Ser. B (DCDS-B), 19 (2014), 1911-1934. doi: 10.3934/dcdsb.2014.19.1911.

[6]

D. R. Bland, Wave Theory and Applications, Oxford University Press, 1988.

[7]

B. A. Boley and R. B. Hetnarski, Propagation of discontinuities in coupled thermoelastic problems, J. Appl. Mech. (ASME), 35 (1968), 489-494.

[8]

J. P. Boyd, A proof, based on the Euler sum acceleration, of the recovery of an exponential (geometric) rate of convergence for the Fourier series of a function with Gibbs phenomenon, Spectral and High Order Methods for Partial Differential Equations, 131-139, Lect. Notes Comput. Sci. Eng., 76, Springer, Heidelberg, 2011, (https://arXiv.org/abs/1003.5263v1). doi: 10.1007/978-3-642-15337-2_10.

[9]

J. P. Boyd, Dynamics of the Equatorial Ocean, Springer-Verlag, 2018, $\S\S$ A.13, A.14. doi: 10.1007/978-3-662-55476-0.

[10]

J. P. Boyd, Private communication, 24 February 2018.

[11]

H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Dover, 1963.

[12]

I. Christov and P. M. Jordan, Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation, New J. Phys., 10 (2008), 043027.

[13]

M. Ciarletta and B. Straughan, Poroacoustic acceleration waves, Proc. R. Soc. A, 462 (2006), 3493-3499. doi: 10.1098/rspa.2006.1730.

[14]

D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.

[15]

D. G. Crighton, Nonlinear waves in aerosols and dusty gases, in: Nonlinear Waves in Real Fluids (ed. A. Kluwick), Springer-Verlag, 1991, 69-82.

[16]

D. G. Crighton, Propagation of finite-amplitude waves in fluids, in: Handbook of Acoustics (ed. M. J. Crocker), Wiley, 1998, Chap. 17.

[17]

D. G. Crighton and J. T. Scott, Asymptotic solutions of model equations in nonlinear acoustics, Phil. Trans. R. Soc. London A, 292 (1979), 101-134. doi: 10.1098/rsta.1979.0046.

[18]

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J., 16 (2008), 247-270, (http://arXiv.org/abs/math.NT/0506319v3). doi: 10.1007/s11139-007-9102-0.

[19]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Cont. Dyn. Sys., Ser. B (DCDS-B), 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189.

[20]

P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009, Mech. Res. Commun., 73 (2016), 127-139.

[21]

P. M. JordanR. 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.

[22]

B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Eq. Control Theory (EECT), 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447.

[23]

B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control and Cybernetics (C & C), 40 (2011), 971-988.

[24]

R. S. KeifferP. M. Jordan and I. C. Christov, Acoustic shock and acceleration waves in selected inhomogeneous fluids, Mech. Res. Commun., 93 (2018), 80-88.

[25]

H. Lamb, The Dynamical Theory of Sound, 2nd ed. Dover Publications, Inc., New York, 1960.

[26]

J. D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, 1994.

[27]

F. E. Marble, Dynamics of dusty gases, Ann. Rev. Fluid Mech., 2 (1970), 397-446.

[28]

J. P. Moran and S. F. Shen, On the formation of weak plane shock waves by impulsive motion of a piston, J. Fluid Mech., 25 (1966), 705-718.

[29]

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. doi: 10.2514/8.11882.

[30]

A. Morro, Jump relations and discontinuity waves in conductors with memory, Math. Comput. Modelling, 43 (2006), 138-149. doi: 10.1016/j.mcm.2005.04.016.

[31]

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

[32]

G. G. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Phil. Mag. (Ser. 4), 1 (1851), 305-317.

[33]

B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, vol. 165, Springer, 2008, Chap. 8.

[34]

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

[35]

D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, 1997, $\S$ 2.5.1. doi: 10.1002/9781118818275.

[36]

J. von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21 (1950), 232-237. doi: 10.1063/1.1699639.

[37]

E. W. Weisstein, Lerch Transcendent, From MathWorld-A Wolfram Web Resource (http://mathworld.wolfram.com/LerchTranscendent.html).

[38]

G. B. Whitham, Non-linear dispersive waves, Proc. R. Soc. London A, 283 (1965), 238-261. doi: 10.1098/rspa.1965.0019.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1966.

[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, 2009. doi: 10.1090/surv/156.

[3]

S. BargmannP. 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), 4418-4424. doi: 10.1016/j.physleta.2008.04.010.

[4]

R. T. Beyer, The parameter B/A, in: Nonlinear Acoustics (eds. M. F. Hamilton and D. T. Blackstock), Academic Press, 1998, 25–39.

[5]

J. Bissell and B. Straughan, Discontinuity waves as tipping points: Applications to biological & sociological systems, Discrete Cont. Dyn. Sys., Ser. B (DCDS-B), 19 (2014), 1911-1934. doi: 10.3934/dcdsb.2014.19.1911.

[6]

D. R. Bland, Wave Theory and Applications, Oxford University Press, 1988.

[7]

B. A. Boley and R. B. Hetnarski, Propagation of discontinuities in coupled thermoelastic problems, J. Appl. Mech. (ASME), 35 (1968), 489-494.

[8]

J. P. Boyd, A proof, based on the Euler sum acceleration, of the recovery of an exponential (geometric) rate of convergence for the Fourier series of a function with Gibbs phenomenon, Spectral and High Order Methods for Partial Differential Equations, 131-139, Lect. Notes Comput. Sci. Eng., 76, Springer, Heidelberg, 2011, (https://arXiv.org/abs/1003.5263v1). doi: 10.1007/978-3-642-15337-2_10.

[9]

J. P. Boyd, Dynamics of the Equatorial Ocean, Springer-Verlag, 2018, $\S\S$ A.13, A.14. doi: 10.1007/978-3-662-55476-0.

[10]

J. P. Boyd, Private communication, 24 February 2018.

[11]

H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Dover, 1963.

[12]

I. Christov and P. M. Jordan, Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation, New J. Phys., 10 (2008), 043027.

[13]

M. Ciarletta and B. Straughan, Poroacoustic acceleration waves, Proc. R. Soc. A, 462 (2006), 3493-3499. doi: 10.1098/rspa.2006.1730.

[14]

D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.

[15]

D. G. Crighton, Nonlinear waves in aerosols and dusty gases, in: Nonlinear Waves in Real Fluids (ed. A. Kluwick), Springer-Verlag, 1991, 69-82.

[16]

D. G. Crighton, Propagation of finite-amplitude waves in fluids, in: Handbook of Acoustics (ed. M. J. Crocker), Wiley, 1998, Chap. 17.

[17]

D. G. Crighton and J. T. Scott, Asymptotic solutions of model equations in nonlinear acoustics, Phil. Trans. R. Soc. London A, 292 (1979), 101-134. doi: 10.1098/rsta.1979.0046.

[18]

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J., 16 (2008), 247-270, (http://arXiv.org/abs/math.NT/0506319v3). doi: 10.1007/s11139-007-9102-0.

[19]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Cont. Dyn. Sys., Ser. B (DCDS-B), 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189.

[20]

P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009, Mech. Res. Commun., 73 (2016), 127-139.

[21]

P. M. JordanR. 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.

[22]

B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Eq. Control Theory (EECT), 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447.

[23]

B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control and Cybernetics (C & C), 40 (2011), 971-988.

[24]

R. S. KeifferP. M. Jordan and I. C. Christov, Acoustic shock and acceleration waves in selected inhomogeneous fluids, Mech. Res. Commun., 93 (2018), 80-88.

[25]

H. Lamb, The Dynamical Theory of Sound, 2nd ed. Dover Publications, Inc., New York, 1960.

[26]

J. D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, 1994.

[27]

F. E. Marble, Dynamics of dusty gases, Ann. Rev. Fluid Mech., 2 (1970), 397-446.

[28]

J. P. Moran and S. F. Shen, On the formation of weak plane shock waves by impulsive motion of a piston, J. Fluid Mech., 25 (1966), 705-718.

[29]

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. doi: 10.2514/8.11882.

[30]

A. Morro, Jump relations and discontinuity waves in conductors with memory, Math. Comput. Modelling, 43 (2006), 138-149. doi: 10.1016/j.mcm.2005.04.016.

[31]

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

[32]

G. G. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Phil. Mag. (Ser. 4), 1 (1851), 305-317.

[33]

B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, vol. 165, Springer, 2008, Chap. 8.

[34]

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

[35]

D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, 1997, $\S$ 2.5.1. doi: 10.1002/9781118818275.

[36]

J. von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21 (1950), 232-237. doi: 10.1063/1.1699639.

[37]

E. W. Weisstein, Lerch Transcendent, From MathWorld-A Wolfram Web Resource (http://mathworld.wolfram.com/LerchTranscendent.html).

[38]

G. B. Whitham, Non-linear dispersive waves, Proc. R. Soc. London A, 283 (1965), 238-261. doi: 10.1098/rspa.1965.0019.

Figure 1.  Blue: $ u $ vs. $ x $ (based on Eq. (21) and using $ M = 5000 $), Orange: $ \upsilon $ vs. $ x $ (based on Eq. (22) and using $ M = 500 $), Green-solid lines: $ [\![ u ]\!] $ vs. $ x $, Green-dashing lines: $ (x-t) [\![ \upsilon_{x} ]\!] $ vs. $ x $. Here, $ \hat{\tau} = 0.1 $ and $ \kappa = 0.15 $ were assumed.
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