Inconsistency in the Moment’s method for solving the Boltzmann equation

References

1 Uribe, F.J., Velasco, R.M., García-Colín, L.S., Hydrodynamics, Grad’s moments method and the structure of shock waves, in: Developments in Mathematical and Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Eds. A. Macías, F.J. Uribe and E. Díaz, pp. 53. Kluwer Academic, New York, 2003.10.1007/978-1-4615-0199-2_4Suche in Google Scholar

2 Velasco, R.M., Uribe, F.J., García-Colín, L.S., Inconsistencies in moment methods, Phys. Rev. E, 66 (2002), 032103.10.1103/PhysRevE.66.032103Suche in Google Scholar PubMed

3 Grad, H., On the Kinetic Theory of Rarefied Gases, Comm. Pure Applied Math., 2 (1949), 331.10.1002/cpa.3160020403Suche in Google Scholar

4 Grad, H., Principles of the Kinetic Theory of Gases, in: Handbuch der Physik, Vol. XII, Ed. S. Flügge, pp. 205. Springer-Verlag, Berlin, 1958.10.1007/978-3-642-45892-7_3Suche in Google Scholar

5 Bhatnagar, P.L., Gross, E.P., Krook, M., A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems, Phys. Rev., 94 (1954), 511.10.1103/PhysRev.94.511Suche in Google Scholar

6 Mott-Smith, H.M., The Solution of the Boltzmann Equation for a Shock Wave, Phys. Rev., 82 (1951), 885.10.1103/PhysRev.82.885Suche in Google Scholar

7 Chapman, S., Cowling, T.G., The Mathematical Theory of Non-uniform Gases, Cambridge University Press, Cambridge, 3rd ed., 1970.Suche in Google Scholar

8 Cercignani, C., The Boltzmann equation and its applications, Springer-Verlag, New York, 1988.10.1007/978-1-4612-1039-9Suche in Google Scholar

9 Harris, S., An introduction to the theory of the Boltzmann equation, Holt Rinehart and Winston Inc., New York, 1971.Suche in Google Scholar

10 We will use alternatively the notation of 1, 2, 3 or x, y, z, to denote the components. The components of the vectors or tensors will be denoted in various ways, for example C₁ ≡ C₁ ≡ C_x.Suche in Google Scholar

11 Uhlenbeck, G.E., Ford, G.W., in: Lectures in Statistical Mechanics, Chap. VI. American Mathematical Society, Providence R.I., 1963.Suche in Google Scholar

12 Pham-Van-Diep, G.C., Erwin, D.A., and Muntz, E.P., Nonequilibrium Molecular Motion in a Hypersonic Shock Wave, Science, 243 (1989), 624.10.1126/science.245.4918.624Suche in Google Scholar PubMed

13 Erwin, D.A., Pham-Van-Diep, G.C., Muntz, E.P., Nonequilibrium gas flows I: A detailed validation of Monte Carlo Direct Simulation for monatomic gases, Phys. Fluids A, 3 (1991), 697.10.1063/1.858075Suche in Google Scholar

14 Mazouffre, S., Vankan, P., Engeln, R., Schram, D.C., Behavior of the H atom velocity distribution function within the shock wave of a hydrogen plasma jet, Phys. Rev. E, 64, (2001), 066405.10.1103/PhysRevE.64.066405Suche in Google Scholar PubMed

15 Cercignani, C., Frezzotti, A., The structure of an infinitely shock wave, Phys. Fluids, 11 (1999), 2757.10.1063/1.870134Suche in Google Scholar

16 Bashkirov, A.G., Orlov, A.V., Shock-Wave structure for some nonanalytical in-velocity closures, Phys. Rev. E, 53 (1996), R17.10.1103/PhysRevE.53.R17Suche in Google Scholar

17 Ohr, Y.G., Iterative method to improve the Mott-Smith shock-wave structure theory, Phys. Rev E, 57 (1998), 1723.10.1103/PhysRevE.57.1723Suche in Google Scholar

18 Alsmeyer, H., Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, J. Fluid Mech., 74 (1976), 497.10.1017/S0022112076001912Suche in Google Scholar

19 Pham-Van-Diep, G.A., Erwin, E.A., Muntz, E.P., Testing continuum descriptions of low Mach number shock structures, J. Fluid Mech., 232 (1991), 403.10.1017/S0022112091003749Suche in Google Scholar

20 Uribe, F.J., Velasco, R.M., García-Colín, L.S., Díaz Herrera, E., Shock wave profiles in the Burnett approximation, Phys. Rev. E, 62 (2000), 6648.10.1103/PhysRevE.62.6648Suche in Google Scholar

21 Garen, W., Synofzik, R., Wortberg, G., in: Rarefied Gas Dynamics, Ed. J.L. Potter, p. 519. AIAA, New York, 1977.Suche in Google Scholar

22 Grad, H., The Profile of a Steady Plane Shock Wave, Comm. Pure Applied Math., 5 (1952), 257.10.1002/cpa.3160050304Suche in Google Scholar

23 Karlin, I.V., Dukek, G., Nonnenmacher, T.F., Reply to “Comment on ‘Invariance principle for extension of hydrodynamics’ ”, Phys. Rev. E, 57 (1998), 3674.10.1103/PhysRevE.57.3674Suche in Google Scholar

24 Uribe, F.J., Piña, E., Comment on “Invariance principle for extension of hydrodynamics”, Phys. Rev. E, 57 (1998), 3672.10.1103/PhysRevE.57.3672Suche in Google Scholar

25 Holway, L.H., Existence of Kinetic Theory of Solutions to the Shock Structure Problem, Phys. Fluids, 7 (1964), 911.10.1063/1.1711307Suche in Google Scholar

26 Weiss, W., Comments on “Existence of kinetic theory solutions to the shock structure problem”, Phys. Fluids, 8 (1996), 1689.10.1063/1.868947Suche in Google Scholar

27 Weiss, W., Continuous Shock structure in extended thermodynamics, Phys. Rev. E, 52 (1995), R5760.10.1103/PhysRevE.52.R5760Suche in Google Scholar

28 Müller, I., and Ruggeri, T., Extended Thermodynamics, Springer-Verlag, New York, 1993.10.1007/978-1-4684-0447-0Suche in Google Scholar

29 Ruggeri, T., Breakdown of shock-wave-structure solutions, Phys. Rev. E, 47 (1993), 4135.10.1103/PhysRevE.47.4135Suche in Google Scholar

30 Al-Gohul, M., Eu, B.C., Phys. Rev. E, 56 (1997), 2981.10.1103/PhysRevE.56.2981Suche in Google Scholar

31 Ohr, Y.G., Improvement of the Grad 13 moment method for strong shock waves, Phys. Fluids, 13 (2001), 2105.10.1063/1.1373681Suche in Google Scholar

32 Jou, D., Pavón, D., Nonlocal and nonlinear effects in shock waves, Phys. Rev. A, 44 (1991), 6496.10.1103/PhysRevA.44.6496Suche in Google Scholar

33 Waldmann, L., Transporterskheimer in Gases von mittlerem Druck, in: The Handbuch der Physik Vol. 12, Ed. S. Füdge, pp. 295. Springer-Verlag, Berlin, 1958.10.1007/978-3-642-45892-7_4Suche in Google Scholar

34 Losse, W., and S. Hess, Nonequilibrium velocity distribution function of Gases: Kinetic theory and molecular dynamics, Phys. Rev. A, 37 (1988), 2099.10.1103/PhysRevA.37.2099Suche in Google Scholar

35 Hess, S., and M. Malek Mansour, Temperature profile of a dilute gas undergoing a plane Poiseuille flow, Physica A, 272 (1999), 481.10.1016/S0378-4371(99)00254-XSuche in Google Scholar

36 In some of the integrals we have taken advantage of the condition that the trace of the viscous pressure tensor is zero.Suche in Google Scholar

37 Uribe, F.J., García-Colín, L.S., Nonlinear viscosity and Grad’s method, Phys. Rev. E, 60 (1999), 4052.10.1103/PhysRevE.60.4052Suche in Google Scholar