Recasting a Brinkman-based acoustic model as the damped Burgers equation

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September 2016, 5(3): 463-474. doi: 10.3934/eect.2016014

Recasting a Brinkman-based acoustic model as the damped Burgers equation

David Rossmanith ^1, and Ashok Puri ^1,

1.	University of New Orleans, Department of Physics, New Orleans, LA 70148, United States, United States

Received April 2016 Revised May 2016 Published August 2016

Abstract
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In order to gain a better understanding of the behavior of finite-amplitude acoustic waves under a Brinkman-based poroacoustic model, we make use of approximations and transformations to recast our model equation into the damped Burgers equation. We examine two special case solutions of the damped Burgers equation: the approximate solution to the damped Burgers equation and the boundary value problem given an initial sinusoidal signal. We study the effects of varying the Darcy coefficient, Reynolds number, and coefficient of nonlinearity on these solutions.

Keywords: Brinkman's equation, Burgers equation, traveling wave solution, initial-boundary value problem., nonlinear PDE.

Mathematics Subject Classification: Primary: 76S99, 35Q35; Secondary: 35Q3.

Citation: David Rossmanith, Ashok Puri. Recasting a Brinkman-based acoustic model as the damped Burgers equation. Evolution Equations and Control Theory, 2016, 5 (3) : 463-474. doi: 10.3934/eect.2016014

References:

[1]	J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics, Quarterly of Applied Mathematics, 9 (1951), 225-236.
[2]	D. G. Crighton, Model equations of nonlinear acoustics, Annual Review of Fluid Mechanics, 11 (1979), 11-33. doi: 10.1146/annurev.fl.11.010179.000303.
[3]	P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009, Mechanics Research Communications, 73 (2016), 127-139. doi: 10.1016/j.mechrescom.2016.02.014.
[4]	P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena, Mathematics and Computers in Simulation (MATCOM), 80 (2009), 202-211. doi: 10.1016/j.matcom.2009.06.004.
[5]	S. V. Korsunskii, Propagation of nonlinear magnetoacoustic waves in electrically conductlnq dissipative media with drag, Sov. Phys. Acoust., 37 (1991), 373-375.
[6]	W. Malfiet, Approximate solution of the damped Burgers equation, Journal of Physics A: Mathematical and General, 26 (1993), L723-L728.
[7]	D. A. Nield and A. Bejan, Convection in Porous Media, $2^{nd}$ edition, Springer, New York, 1999. doi: 10.1007/978-1-4757-3033-3.
[8]	L. E. Payne, J. Rodrigues and B. Straughan, Effect of anisotropic permeability on Darcy's law, Mathematical Methods in the Applied Sciences, 24 (2001), 427-438. doi: 10.1002/mma.228.
[9]	D. A. Rossmanith and A. Puri, The role of Brinkman viscosity in poroacoustic propagation, International Journal of Non-Linear Mechanics, 67 (2014), 1-6. doi: 10.1016/j.ijnonlinmec.2014.07.002.
[10]	D. A. Rossmanith and A. Puri, Non-linear evolution of a sinusoidal pulse under a Brinkman-based poroacoustic model, International Journal of Non-Linear Mechanics, 78 (2016), 53-58. doi: 10.1016/j.ijnonlinmec.2015.09.014.
[11]	S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium, Sov. Phys. Acoust., 8 (1962), 170-175.
[12]	P. A. Thompson, Compressible-fluid Dynamics, McGraw-Hill, New York, 1972.

show all references

References:

[1]	J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics, Quarterly of Applied Mathematics, 9 (1951), 225-236.
[2]	D. G. Crighton, Model equations of nonlinear acoustics, Annual Review of Fluid Mechanics, 11 (1979), 11-33. doi: 10.1146/annurev.fl.11.010179.000303.
[3]	P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009, Mechanics Research Communications, 73 (2016), 127-139. doi: 10.1016/j.mechrescom.2016.02.014.
[4]	P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena, Mathematics and Computers in Simulation (MATCOM), 80 (2009), 202-211. doi: 10.1016/j.matcom.2009.06.004.
[5]	S. V. Korsunskii, Propagation of nonlinear magnetoacoustic waves in electrically conductlnq dissipative media with drag, Sov. Phys. Acoust., 37 (1991), 373-375.
[6]	W. Malfiet, Approximate solution of the damped Burgers equation, Journal of Physics A: Mathematical and General, 26 (1993), L723-L728.
[7]	D. A. Nield and A. Bejan, Convection in Porous Media, $2^{nd}$ edition, Springer, New York, 1999. doi: 10.1007/978-1-4757-3033-3.
[8]	L. E. Payne, J. Rodrigues and B. Straughan, Effect of anisotropic permeability on Darcy's law, Mathematical Methods in the Applied Sciences, 24 (2001), 427-438. doi: 10.1002/mma.228.
[9]	D. A. Rossmanith and A. Puri, The role of Brinkman viscosity in poroacoustic propagation, International Journal of Non-Linear Mechanics, 67 (2014), 1-6. doi: 10.1016/j.ijnonlinmec.2014.07.002.
[10]	D. A. Rossmanith and A. Puri, Non-linear evolution of a sinusoidal pulse under a Brinkman-based poroacoustic model, International Journal of Non-Linear Mechanics, 78 (2016), 53-58. doi: 10.1016/j.ijnonlinmec.2015.09.014.
[11]	S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium, Sov. Phys. Acoust., 8 (1962), 170-175.
[12]	P. A. Thompson, Compressible-fluid Dynamics, McGraw-Hill, New York, 1972.

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