# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [3] S. Bargmann, P. 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.  Google Scholar [4] R. T. Beyer, The parameter B/A, in: Nonlinear Acoustics (eds. M. F. Hamilton and D. T. Blackstock), Academic Press, 1998, 25–39. Google Scholar [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.  Google Scholar [6] D. R. Bland, Wave Theory and Applications, Oxford University Press, 1988. Google Scholar [7] B. A. Boley and R. B. Hetnarski, Propagation of discontinuities in coupled thermoelastic problems, J. Appl. Mech. (ASME), 35 (1968), 489-494. Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. P. Boyd, Private communication, 24 February 2018. Google Scholar [11] H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Dover, 1963.  Google Scholar [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. Google Scholar [13] M. Ciarletta and B. Straughan, Poroacoustic acceleration waves, Proc. R. Soc. A, 462 (2006), 3493-3499.  doi: 10.1098/rspa.2006.1730.  Google Scholar [14] D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.   Google Scholar [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. Google Scholar [16] D. G. Crighton, Propagation of finite-amplitude waves in fluids, in: Handbook of Acoustics (ed. M. J. Crocker), Wiley, 1998, Chap. 17. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009, Mech. Res. Commun., 73 (2016), 127-139.   Google Scholar [21] 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.  Google Scholar [22] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Eq. Control Theory (EECT), 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447.  Google Scholar [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.  Google Scholar [24] R. S. Keiffer, P. M. Jordan and I. C. Christov, Acoustic shock and acceleration waves in selected inhomogeneous fluids, Mech. Res. Commun., 93 (2018), 80-88.   Google Scholar [25] H. Lamb, The Dynamical Theory of Sound, 2nd ed. Dover Publications, Inc., New York, 1960.  Google Scholar [26] J. D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, 1994.  Google Scholar [27] F. E. Marble, Dynamics of dusty gases, Ann. Rev. Fluid Mech., 2 (1970), 397-446.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [31] A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, 1989. Google Scholar [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. Google Scholar [33] B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, vol. 165, Springer, 2008, Chap. 8.  Google Scholar [34] P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, 1972. Google Scholar [35] D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, 1997, $\S$ 2.5.1. doi: 10.1002/9781118818275.  Google Scholar [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.  Google Scholar [37] E. W. Weisstein, Lerch Transcendent, From MathWorld-A Wolfram Web Resource (http://mathworld.wolfram.com/LerchTranscendent.html). Google Scholar [38] G. B. Whitham, Non-linear dispersive waves, Proc. R. Soc. London A, 283 (1965), 238-261. doi: 10.1098/rspa.1965.0019.  Google Scholar

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##### References:
 [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1966.  Google Scholar [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.  Google Scholar [3] S. Bargmann, P. 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.  Google Scholar [4] R. T. Beyer, The parameter B/A, in: Nonlinear Acoustics (eds. M. F. Hamilton and D. T. Blackstock), Academic Press, 1998, 25–39. Google Scholar [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.  Google Scholar [6] D. R. Bland, Wave Theory and Applications, Oxford University Press, 1988. Google Scholar [7] B. A. Boley and R. B. Hetnarski, Propagation of discontinuities in coupled thermoelastic problems, J. Appl. Mech. (ASME), 35 (1968), 489-494. Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. P. Boyd, Private communication, 24 February 2018. Google Scholar [11] H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics, Dover, 1963.  Google Scholar [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. Google Scholar [13] M. Ciarletta and B. Straughan, Poroacoustic acceleration waves, Proc. R. Soc. A, 462 (2006), 3493-3499.  doi: 10.1098/rspa.2006.1730.  Google Scholar [14] D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), 11-33.   Google Scholar [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. Google Scholar [16] D. G. Crighton, Propagation of finite-amplitude waves in fluids, in: Handbook of Acoustics (ed. M. J. Crocker), Wiley, 1998, Chap. 17. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009, Mech. Res. Commun., 73 (2016), 127-139.   Google Scholar [21] 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.  Google Scholar [22] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Eq. Control Theory (EECT), 4 (2015), 447-491. doi: 10.3934/eect.2015.4.447.  Google Scholar [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.  Google Scholar [24] R. S. Keiffer, P. M. Jordan and I. C. Christov, Acoustic shock and acceleration waves in selected inhomogeneous fluids, Mech. Res. Commun., 93 (2018), 80-88.   Google Scholar [25] H. Lamb, The Dynamical Theory of Sound, 2nd ed. Dover Publications, Inc., New York, 1960.  Google Scholar [26] J. D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, 1994.  Google Scholar [27] F. E. Marble, Dynamics of dusty gases, Ann. Rev. Fluid Mech., 2 (1970), 397-446.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [31] A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, 1989. Google Scholar [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. Google Scholar [33] B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, vol. 165, Springer, 2008, Chap. 8.  Google Scholar [34] P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, 1972. Google Scholar [35] D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, 1997, $\S$ 2.5.1. doi: 10.1002/9781118818275.  Google Scholar [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.  Google Scholar [37] E. W. Weisstein, Lerch Transcendent, From MathWorld-A Wolfram Web Resource (http://mathworld.wolfram.com/LerchTranscendent.html). Google Scholar [38] G. B. Whitham, Non-linear dispersive waves, Proc. R. Soc. London A, 283 (1965), 238-261. doi: 10.1098/rspa.1965.0019.  Google Scholar
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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