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

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

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

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.
Citation: David Rossmanith, Ashok Puri. Recasting a Brinkman-based acoustic model as the damped Burgers equation. Evolution Equations & 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.   Google Scholar

[2]

D. G. Crighton, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11.  doi: 10.1146/annurev.fl.11.010179.000303.  Google Scholar

[3]

P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009,, Mechanics Research Communications, 73 (2016), 127.  doi: 10.1016/j.mechrescom.2016.02.014.  Google Scholar

[4]

P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena,, Mathematics and Computers in Simulation (MATCOM), 80 (2009), 202.  doi: 10.1016/j.matcom.2009.06.004.  Google Scholar

[5]

S. V. Korsunskii, Propagation of nonlinear magnetoacoustic waves in electrically conductlnq dissipative media with drag,, Sov. Phys. Acoust., 37 (1991), 373.   Google Scholar

[6]

W. Malfiet, Approximate solution of the damped Burgers equation,, Journal of Physics A: Mathematical and General, 26 (1993).   Google Scholar

[7]

D. A. Nield and A. Bejan, Convection in Porous Media,, $2^{nd}$ edition, (1999).  doi: 10.1007/978-1-4757-3033-3.  Google Scholar

[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.  doi: 10.1002/mma.228.  Google Scholar

[9]

D. A. Rossmanith and A. Puri, The role of Brinkman viscosity in poroacoustic propagation,, International Journal of Non-Linear Mechanics, 67 (2014), 1.  doi: 10.1016/j.ijnonlinmec.2014.07.002.  Google Scholar

[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.  doi: 10.1016/j.ijnonlinmec.2015.09.014.  Google Scholar

[11]

S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium,, Sov. Phys. Acoust., 8 (1962), 170.   Google Scholar

[12]

P. A. Thompson, Compressible-fluid Dynamics,, McGraw-Hill, (1972).   Google Scholar

show all references

References:
[1]

J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics,, Quarterly of Applied Mathematics, 9 (1951), 225.   Google Scholar

[2]

D. G. Crighton, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11.  doi: 10.1146/annurev.fl.11.010179.000303.  Google Scholar

[3]

P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009,, Mechanics Research Communications, 73 (2016), 127.  doi: 10.1016/j.mechrescom.2016.02.014.  Google Scholar

[4]

P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena,, Mathematics and Computers in Simulation (MATCOM), 80 (2009), 202.  doi: 10.1016/j.matcom.2009.06.004.  Google Scholar

[5]

S. V. Korsunskii, Propagation of nonlinear magnetoacoustic waves in electrically conductlnq dissipative media with drag,, Sov. Phys. Acoust., 37 (1991), 373.   Google Scholar

[6]

W. Malfiet, Approximate solution of the damped Burgers equation,, Journal of Physics A: Mathematical and General, 26 (1993).   Google Scholar

[7]

D. A. Nield and A. Bejan, Convection in Porous Media,, $2^{nd}$ edition, (1999).  doi: 10.1007/978-1-4757-3033-3.  Google Scholar

[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.  doi: 10.1002/mma.228.  Google Scholar

[9]

D. A. Rossmanith and A. Puri, The role of Brinkman viscosity in poroacoustic propagation,, International Journal of Non-Linear Mechanics, 67 (2014), 1.  doi: 10.1016/j.ijnonlinmec.2014.07.002.  Google Scholar

[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.  doi: 10.1016/j.ijnonlinmec.2015.09.014.  Google Scholar

[11]

S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium,, Sov. Phys. Acoust., 8 (1962), 170.   Google Scholar

[12]

P. A. Thompson, Compressible-fluid Dynamics,, McGraw-Hill, (1972).   Google Scholar

[1]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[2]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[3]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[4]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[5]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[6]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[7]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[8]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[9]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[10]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[11]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[12]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[13]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[14]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[15]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[16]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[17]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[18]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[19]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[20]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]