Physics Reports 465 (2008) 149–189 Contents lists available at ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Beyond the Navier–Stokes equations: Burnett hydrodynamics L.S. García-Colín a,b , R.M. Velasco a , F.J. Uribe a,∗ a Department of Physics, Universidad Autónoma Metropolitana - Iztapalapa, México D. F. 09340, Mexico b El Colegio Nacional, Centro Histórico, México, 06020, Mexico article info Article history: Accepted 15 April 2008 Available online 22 May 2008 editor: I. Procaccia PACS: 05.20.Dd 51.10.+y 47.10.+g 47.40.-x Keywords: Boltzmann equation Chapman–Enskog method Hydrodynamics abstract This work is mainly concerned with the extension of hydrodynamics beyond the Navier–Stokes equations, a regime known as Burnett hydrodynamics. The derivation of the Burnett equations is considered from several theoretical approaches. In particular we discuss the Chapman–Enskog, Grad’s method, and Truesdell’s approach for solving the Boltzmann equation. Also, their derivation using the macroscopic approach given by extended thermodynamics is mentioned. The problems and successes of these equations are discussed and some alternatives proposed to improve them are mentioned. Comparisons of the predictions coming from the Burnett equations with experiments and/or simulations are given in order to have the necessary elements to give a critical assessment of their validity and usefulness. © 2008 Elsevier B.V. All rights reserved. Contents 1. Introduction............................................................................................................................................................................................. 150 2. Origin of the Burnett equations.............................................................................................................................................................. 150 3. Grad’s Method ......................................................................................................................................................................................... 155 4. Thermodynamical context ..................................................................................................................................................................... 157 5. Phenomenological calculation ............................................................................................................................................................... 158 6. Other derivations .................................................................................................................................................................................... 161 7. Integration of the Burnett equations ..................................................................................................................................................... 162 8. Plane Poiseuille flow ............................................................................................................................................................................... 164 9. Shock waves ............................................................................................................................................................................................ 168 10. Sound propagation in gases.................................................................................................................................................................... 175 11. Strongly nonisothermal gas and micro Couette flow ........................................................................................................................... 178 12. Final remarks ........................................................................................................................................................................................... 180 Acknowledgements................................................................................................................................................................................. 183 Appendix.................................................................................................................................................................................................. 183 References................................................................................................................................................................................................ 185 ∗ Corresponding author. Tel.: +52 55 58044949; fax: +52 55 58044611. E-mail addresses: lgcs@xanum.uam.mx (L.S. García-Colín), rmvb@xanum.uam.mx (R.M. Velasco), paco@xanum.uam.mx (F.J. Uribe). 0370-1573/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2008.04.010