Hydrodynamics, Grad’s Moments Method and the Structure Of Shock Waves
The propagation of a plane shock wave through a monatomic gas is considered. A study based on Grad's thirteen moment approximation to the solution of Boltzmann's equation is performed for a stationary shock. Comparison of the results obtained with two different sets of dynamical variables leads to some inconsistencies. In fact, what we exemplify is that Grad's method does not uniquely determines the underlying dynamical systems of equations for each choice of dynamical variables. The consequences of this incosistency are seen in the shock wave problem.
KeywordsBoltzmann Equation Chapman-Enskog Method Grad's Moments Method Kinetic Theory Shock Waves
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- 1.Giovanni P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Tracts in Natural Philosophy Vol. 38, corrected second printing, (Springer, New York, 1998)Google Scholar
- 5.F. J. Uribe, R. M. Velasco, L. S. García-Colín and E. Díaz-Herrera, Shock wave profiles in the Burnett approximation, Phys. Rev. E 62, 6648 (2000).Google Scholar
- 6.F. J. Uribe and A. L. Garcia, Burnett description for plane Poiseuille flow, Phys. Rev. E 60, 4063 (1999).Google Scholar
- 7.S. Chapman and T. G. Cowling, The Mathematical Theory of Non- Uniform Gases (Cambridge University Press, Cambridge, 1970, 1990).Google Scholar
- 8.C. Truesdell and R. G. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, (Academic Press, London, 1980).Google Scholar
- 9.Charles L. Fefferman, Existence & Smoothness of the Navier-Stokes equations, http://www.claymath.org/prizeproblems/navierstokes.htm.Google Scholar
- 12.B. L. Holian, C. W. Patterson, M. Mareschal and E. Salomons, Modeling shock waves in an ideal gas: Going beyond the Navier-Stokes level, Phys. Rev. E 47, R24 (1993).Google Scholar
- 14.F. J. Uribe, R. M. Velasco and L. S. García-Colín, Two kinetic temperature description for shock waves, Phys. Rev. E 58, 3209 (1998).Google Scholar
- 16.G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, (Clarendon, Oxford, 1994).Google Scholar
- 17.F. J. Uribe and L. S. García-Colín, Nonlinear viscosity and Grad's method, Phys. Rev. E 60, 4052 (1999).Google Scholar