Abstract
The Burnett equations for shock structure in a Maxwell gas (inverse fifth power repulsion) are not amenable to numerical integration by the usual methods (fourth order Runge-Kutta) above a Mach number of 1.9. The super-Burnett equations are not amenable to numerical integration by the usual methods for any Mach number. It is not yet clear whether these negative results indicate a fundamental difficulty in the Chapman-Enskog solution of the Boltzmann equation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
This lecture is not a complete review of the subject. The literature is quite extensive and several interesting contributions have been omitted.
B. Schmidt, J. Fluid Mech. 39, 361 (1969).
E.P. Muntz and L.N. Harnett, Phys. Fluids 12, 2027 (1969).
D. Gilbarg and D. Paolucci, J. Rat. Mech. Analysis 2, 617 (1953).
Two trajectories hit the downstream singular point (from opposite directions), but only one is compatible with the expected monotonic change within a shock wave.
F.S. Sherman and L. Talbot, in Proceedings of the First International Symposium on Rarefied Gas Dynamics, edited by F.M. Devienne (Pergamon, New York, 1960), p. 161. The equations used by Sherman and Talbot contain an error, but the error has negligible effect on their results. In the expression for pxx on p. 164 the coefficient of \(\frac{{{{\mu }^{2}}}}{p}\left( {\frac{{du}}{{dx}}} \right)2\) is 40/27; it should be 8/9.
J.D. Foch, Jr. and C.E. Simon, to be published.
A theory of weak shock structure based on the linearized Boltzmann equation would probably not lead to such exacting comparisons with experiment as the theory of sound propagation. Its putative value would be in suggesting now to formulate a theory for arbitrary Mach number.
H.M. Mott-Smith, Phys. Rev. 82, 885 (1951).
Although the predictions of the linearized Burnett equations for sound propagation are in quantitative agreement with experiment, the same cannot quite be said for the linearized super-Burnett equations. We hope to settle this uncertainty soon with absorption measurements at the University of Colorado.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1973 Springer-Verlag
About this paper
Cite this paper
Foch, J.D. (1973). On Higher Order Hydrodynamic Theories of Shock Structure. In: Cohen, E.G.D., Thirring, W. (eds) The Boltzmann Equation. Acta Physica Austriaca, vol 10/1973. Springer, Vienna. https://doi.org/10.1007/978-3-7091-8336-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-7091-8336-6_7
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-8338-0
Online ISBN: 978-3-7091-8336-6
eBook Packages: Springer Book Archive