Shock wave profiles in the Burnett approximation

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 62, Issue 5, 2000, Pages 6648-6666

  Uribe, F J ^a   Velasco, R M ^a   Garcia Colin L S ^a   Diaz Herrera E ^a  

^a DEPARTAMENTO DE QUÍMICA   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; FLUID MECHANICS; GASES; HYDRODYNAMICS; MONTE CARLO METHODS; NAVIER STOKES EQUATIONS; SPHERES;

BURNETT EQUATIONS; MACH NUMBERS;

SHOCK WAVES;

EID: 0034317019     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.62.6648     Document Type: Article

References (114)

1. R. Becker, Z. Phys. 8, 321 (1922).
2. L. H. Thomas, J. Chem. Phys. 12, 449 (1944).
3. D. Gilbarg and D. Paolucci, J. Ration Mech. Anal. 2, 617 (1953).
4. L. Landau and E. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1986).
5. H. W. Liepmann, R. Narashima, and M. T. Chahine, Phys. Fluids 5, 1313 (1962).
6. Y. B. Zeldovich and Y. P. Raizer, Physics of Shock Waves, Vol. I (Academic Press, New York, 1966); Vol. II (1968).
7. H. Alsmeyer, J. Fluid Mech. 74, 497 (1976).
8. D. Jou and D. Pavon, Phys. Rev. A 44, 6496 (1991).
9. J. D. Powell and J. H. Batteh, Phys. Fluids 20, 734 (1977).
10. C. Cercignani, in Nonequilibrium Phenomena I: The Boltzmann Equation, edited by J. L. Lebowitz and E. W. Montroll (North-Holland, Amsterdam, 1983), p. 121.
11. W. C. Griffith, J. Fluid Mech. 106, 81 (1981).
12. S. A. Markovskii and B. V. Somov, Space Sci. Rev. 78, 443 (1996).
13. H. Grad, Commun. Pure Appl. Math. 5, 257 (1952).
14. C. S. Wang-Chang, in Studies in Statistical Mechanics, edited by J. de Boer and G. E. Uhlenbeck (North Holland, Amsterdam, 1970), Vol. V, p. 27.
15. H. M. Mott-Smith, Phys. Rev. 82, 855 (1951).
16. M. N. Kogan, Rarefied Gas Dynamics (Plenum Press, New York, 1969).
17. M. Al-Ghoul and B. C. Eu, Phys. Rev. E 56, 2981 (1997). The introduction to this paper has valuable information concerning kinetic theoretical approaches to the study of shock waves.
18. E. P. Muntz, Annu. Rev. Fluid Mech. 21, 387 (1989).
19. See Ref. 7 and literature cited therein.
20. A. G. Bashkirov and A. V. Orlov, Phys. Rev. E 53, R17 (1996), and references therein.
21. Y. G. Ohr, Phys. Rev. E 57, 1723 (1998).
22. C. Cercignani, A. Frezzotti, and P. Grosfils, Phys. Fluids 11, 2757 (1999). The introduction to this work has valuable information about the Mott-Smith ansatz and its variants.
23. E. Rebhan, Phys. Rev. A 42, 781 (1990).
24. A. N. Gorban and I. V. Karlin, Physica A 190, 393 (1992).
25. L. H. Holway, Phys. Fluids 7, 911 (1964).
26. T. Ruggeri, Phys. Rev. E 47, 4135 (1993).
27. W. Weiss, Phys. Fluids 8, 1659 (1996).
28. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, 1970).
29. J. D. Foch, in The Boltzmann Equation; Theory and Applications, edited by E. G. D. Cohen and W. Thirring (Springer-Verlag, Vienna, 1973), p. 123.
30. J. C. Tannehill and G. R. Eisler, Phys. Fluids 19, 9 (1976).
31. G. C. Pham-Van-Diep, D. A. Erwin, and E. P. Muntz, J. Fluid Mech. 232, 403 (1991), and references therein.
32. K. A. Fiscko and D. R. Chapman, in Rarefied Gas Dynamics, edited by E. P. Muntz, D. P. Weaver, and D. H. Campbell, Progress in Astronautics and Aeronautics Vol. 118 (AAIA, Washington, D.C., 1989), p. 374.
33. F. E. Lumpkin III and R. D. Chapman, J. Thermophys. Heat Transfer 6, 419 (1992).
34. X. Zhong, R. W. MacCormack, and D. R. Chapman, AIAA J. 31, 1036 (1993).
35. M. López de Haro and V. Garzó, Phys. Rev. E 52, 5688 (1995).
36. J. M. Montanero, M. López de Haro, V. Garzó, and A. Santos, Phys. Rev. E 60, 7592 (1999).
37. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon, Oxford, 1994).
38. Wm. G. Hoover, Molecular Dynamics (Springer-Verlag, Berlin, 1986).
39. The Numerical Algorithm Group FORTRAN library documentation has important information and many references regarding the numerical methods used in this work [The NAG Fortran Library Manual, Mark 16 (The Numerical Algorithms Group Ltd., Oxford, UK, 1993)].
40. W. H. Press, B. P. Flannery, S. A. Teukolski, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1992).
41. Modern Numerical Methods for Ordinary Differential Equations, edited by G. Hall and J. M. Watt (Clarendon Press, Oxford, 1976).
42. A. L. Garcia, Numerical Methods for Physics (Prentice-Hall, Englewood Cliffs, NJ, 1999).
43. A. Heck, Introduction to MAPLE (Springer-Verlag, Amsterdam, 1993).
44. B. L. Holian, W. L. Hoover, B. Moran, and G. K. Straub, Phys. Rev. A 22, 2798 (1980).
45. B. L. Holian, Phys. Rev. A 37, 2562 (1988).
46. B. L. Holian, in Microscopic Simulations of Complex Hydrodynamic Phenomena, edited by M. Mareschal and B. L. Holian (Plenum Press, New York, 1992), p. 75.
47. F. J. Uribe, R. M. Velasco, and L. S. García-Colín, Phys. Rev. E 58, 3209 (1998).
48. E. Salomons and M. Mareschal, Phys. Rev. Lett. 69, 269 (1992).
49. B. L. Holian, C. W. Patterson, M. Mareschal, and E. Salomons, Phys. Rev. E 47, R24 (1993).
50. F. J. Uribe, R. M. Velasco, and L. S. García-Colín, Phys. Rev. Lett. 81, 2044 (1998).
51. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York, 1994).
52. Nonlinear Partial Differential Equations, edited by J. A. Smoller (American Mathematical Society, Providence, R.I., 1983). Several papers in this volume deal with noncontinuous solutions for hyperbolic systems.
53. T.-P. Liu, Nonlinear Stability of Shock Waves for Viscous Conservation Laws (American Mathematical Society, Providence RI, 1985).
54. Lord Rayleigh, Proc. R. Soc. London, Ser. A 84, 247 (1910).
55. G. I. Taylor, Proc. R. Soc. London, Ser. A 84, 371 (1910).
56. T.-P. Liu, Commun. Pure Appl. Math. 39, 565 (1986).
57. R. E. Caflisch and B. Nicolaenko, Commun. Math. Phys. 38, 161 (1982).
58. R. E. Caflisch and B. Nicolaenko, in Nonlinear Partial Differential Equations (Ref. 52), p. 35.
59. R. E. Caflisch, in Rarefied Gas Dynamics (Ref. 32), p. 3.
60. A discussion of these matters can be found in G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Ref. 37), Sec. 2.4.
61. D. Burnett, Proc. London Math. Soc. 39, 385 (1935)
62. Proc. London Math. Soc.D. Burnett40, 382 (1936).
63. We used Bird’s public domain code for the shock wave problem 37.
64. D. A. Erwin, G. C. Pham-Van-Diep, and E. P. Muntz, Phys. Fluids 3, 697 (1991).
65. T. Holtz and E. P. Muntz, Phys. Fluids 26, 2425 (1983).
66. C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases (Springer-Verlag, New York, 1994).
67. G. Pham-Van-Diep, D. Erwin, and E. P. Muntz, Science 245, 624 (1989).
68. B. L. Hicks, S. M. Yen, and B. J. Reilly, J. Fluid Mech. 53, 85 (1972).
69. S. M. Yen, Annu. Rev. Fluid Mech. 16, 67 (1984).
70. S. M. Yen, in Rarefied Gas Dynamics (Ref. 32), p. 323.
71. R. Gatinol, Phys. Fluids 18, 153 (1975).
72. D. Goldstein, B. Sturtevant, and J. E. Broadwell, in Rarefied Gas Dynamics (Ref. 32), p. 100. Chapter 2 of this book contains several papers dealing with discrete velocity models.
73. A. E. Beylich, Phys. Fluids 12, 444 (2000).
74. A. L. Garcia, Phys. Rev. A 34, 1454 (1986).
75. A. L. Garcia, M. Malek Mansour, G. C. Lie, M. Mareschal, and E. Clementi, Phys. Rev. A 36, 4348 (1987).
76. A. L. Garcia, in Microscopic Simulations of Complex Flows, edited by M. Mareschal (Plenum Press, New York, 1990), p. 177.
77. W. Garen, R. Synofzik, and G. Wortberg, in Rarefied Gas Dynamics, edited by J. L. Potter (AIAA, New York, 1977), p. 519.
78. F. J. Blatt, Modern Physics (McGraw-Hill, Singapore, 1992).
79. J. Kestin, K. Knierim, E. A. Mason, B. Najafi, S. T. Ro, and M. Waldman, J. Phys. Chem. Ref. Data 13, 229 (1984).
80. P. Glendining, Stability, Instability, and Chaos (Cambridge University Press, New York, 1994).
81. M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, London, 1974).
82. D. K. Arrowsmith and C. P. Place, An Introduction to Dynamical Systems (Cambridge University Press, Cambridge, 1990).
83. I. V. Arnold and Yu. S. Il’yashenko, in Encyclopedia of Mathematical Sciences, edited by D. V. Anosov and I. V. Arnold (Springer-Verlag, Berlin, 1988), Vol. 1, p. 1.
84. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
85. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York, 1990).
86. J. T. Montgomery, Phys. Fluids 18, 148 (1975).
87. C. Conley, Isolated Invariant Sets and the Morse Index (American Mathematical Society, Providence, RI, 1972).
88. C. E. Simon and J. D. Foch, in Rarefied Gas Dynamics (Ref. 76), p. 493.
89. P. Rosenau, Phys. Rev. A 40, 7193 (1989).
90. A. N. Gorban and I. V. Karlin, Phys. Rev. Lett. 77, 282 (1996).
91. I. V. Karlin, G. Dukek, and T. F. Nonnenmacher, Phys. Rev. E 55, 1573 (1997).
92. J. A. McLennan, Phys. Fluids 8, 1580 (1965).
93. I. Prigogine, Physica (Amsterdam) 15, 242 (1949).
94. S. R. deGroot and P. Manzur, Non-Equilibrium Thermodynamics (Dover, Mineola, NY, 1984), Chap. IX.
95. L. C. Woods, An Introduction to the Kinetic Theory of Gases and Magnetoplasmas (Oxford University Press, New York, 1993).
96. J. R. Dorfman and H. van Beijeren, in Statistical Physics, edited by B. J. Berne (Plenum Press, New York, 1977), p. 65.
97. F. J. Uribe and L. S. García-Colín, Phys. Fluids A 5, 1814 (1993).
98. L. S. García-Colín, J. A. Robles-Domíguez, and G. Fuentes, Phys. Lett. 84A, 169 (1984).
99. J. A. Robles-Domínguez, B. Silva, and L. S. García-Colín, Physica A 106, 539 (1981).
100. L. S. García-Colín, Physica A 118, 341 (1983).
101. L. S. García-Colín, and M. López de Haro, J. Non-Equilib. Thermodyn. 7, 95 (1982).
102. F. J. Uribe and A. L. Garcia, Phys. Rev. E 60, 4063 (1999).
103. D. W. Mackowski, D. H. Papadopoulos, and D. E. Rosner, Phys. Fluids 11, 2108 (1999).
104. R. Peralta-Fabi and R. Zwanzig, J. Chem. Phys. 78, 2525 (1983).
105. D. L. Morris, L. Hannon, and A. L. Garcia, Phys. Rev. A 46, 5279 (1992).
106. J. López-Lemus and R. M. Velasco, Physica A 274, 454 (1999).
107. L. Rosenhead, Proc. R. Soc. London, Ser. A 175, 436 (1940).
108. P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1985).
109. F. J. Uribe, Enrique Díaz-Herrera, A. Bravo, and R. Peralta-Fabi, Phys. Fluids 9, 2798 (1997).
110. A. N. Bobylev, Dokl. Akad. Nauk SSSR 262, 71 (1982) [Sov. Phys. Dokl. 27, 29 (1982)].
111. F. J. Uribe, R. M. Velasco, and L. S. García-Colín, Phys. Rev. E 62, 5835 (2000).
112. H. Grad, in Transport Theory, SIAM-AMS Proc. No. 1 (American Mathematical Society, Providence RI, 1969), p. 269.
113. R. E. Caflisch, Commun. Pure Appl. Math. 38, 529 (1985).
114. H. Grad, Phys. Fluids 6, 147 (1963).