# Instabilities in the Chapman-Enskog Expansion and Hyperbolic Burnett Equations

- 248 Downloads
- 42 Citations

## Abstract

It is well-known that the classical Chapman-Enskog procedure does not work at the level of Burnett equations (the next step after the Navier-Stokes equations). Roughly speaking, the reason is that the solutions of higher equations of hydrodynamics (Burnett's, etc.) become unstable with respect to short-wave perturbations. This problem was recently attacked by several authors who proposed different ways to deal with it. We present in this paper one of possible alternatives. First we deduce a criterion for hyperbolicity of Burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases. Then we discuss in more detail the problem of truncation of the Chapman-Enskog expansion and show that the way of truncation is not unique. The general idea of changes of coordinates (based on analogy with the theory of dynamical systems) leads finally to nonlinear Hyperbolic Burnett Equations (HBEs) without using any information beyond the classical Burnett equations. It is proved that HBEs satisfy the linearized *H*-theorem. The linear version of the problem is studied in more detail, the complete Chapman-Enskog expansion is given for the linear case. A simplified proof of the Slemrod identity for Burnett coefficients is also given.

## Key Words

Boltzmann equation Chapman-Enskog method Burnett equations hyperbolicity Perturbation theory Hydrodynamics## Preview

Unable to display preview. Download preview PDF.

## References

- V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, (Springer-Verlag, 1988).Google Scholar
- A. A. Arsen'ev, The Cauchy problem for the linearized Boltzmann equation,
*J. Comp. Math. and Math. Phys.***5**, 664–882 (1965) (in Russian).Google Scholar - C. Bardos, F. Golse and C. D. Levermore, On asymptotic limits of kinetic theory leading to incompressible fluid dynamics,
*C. R. Acad. Sci. Paris Sr. I Math.***309**, 727–732 1989.Google Scholar - P. Biscari, C. Cercignani and M. Slemrod, Time derivatives and frame indifference beyond Newtonian fluids,
*C. R. Acad. Sci. Paris Sr. II b***328**, 417–422 (2000).Google Scholar - A. V. Bobylev, The Chapman-Enskog and Grad methods for solving the Boltzmann equation,
*Sov. Phys. Dokl.***27**, 29–31 (1982).Google Scholar - C. Cercignani, Mathematical methods in kinetic theory, (Plenum Press, 1969).Google Scholar
- C. Cercignani, The Boltzmann equation and its applications, (Springer-Verlag, 1988).Google Scholar
- S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases, (Cambrige University Press, 1990).Google Scholar
- A. De Masi, R. Esposito and J. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation,
*Comm. Pure Appl. Math.***42**, 1189–1214 (1989).Google Scholar - J. H. Ferziger and H. G. Kaper, Mathematical theory of transport processes in gases, (North-Holland, 1972).Google Scholar
- A. N. Gorban and I. Karlin, Method of invariant manifolds and regularization of acoustic spectra,
*Transp. Th. Stat. Phys.***23**, 559–632 (1994).Google Scholar - H. Grad, Principles of the kinetic theory of gases, Handbuch der Physik, Vol. 12, (1958) Springer, pp. 205–251.Google Scholar
- Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation,
*Comm. Pure Appl. Math.*(2005) (In press).Google Scholar - S. Jin and M. Slemrod, Regularization of the Burnett equations via relaxation,
*J. Stat. Phys.***103**, 1009–1033 (2001).Google Scholar - T. Kato, Perturbation theory for linear operators, (Springer-Verlag, 1995).Google Scholar
- M. N. Kogan, Rarefied Gas dynamics, (Plenum Press, 1969).Google Scholar
- C. D. Levermore, Gas dynamics beyond Navier-Stokes, Oberwolfach Report No. 51/2003, (2003) pp. 10.Google Scholar
- R. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion,
*Phys. Rev. A,***40**, 7193–7196 (1989).Google Scholar - M. Slemrod, A normalization method for the Chapman-Enskog expansion,
*Physica D***109**, 257–273 (1997).Google Scholar - M. Slemrod, Constitutive relations for monoatomic gases based on a generalized rational approximation to the sum of the Chapman-Enskog expansion,
*Arch. Rat. Mech. Anal.***150**, 1–22 (1999).Google Scholar - M. Slemrod, In the Chapman-Enskog expansion the Burnett coefficients satisfy the universal relation ω
_{3}+ ω_{4}+ θ_{3}= 0,*Arch. Rat. Mech. Anal.***161**, 339–344 (2002).Google Scholar - H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis,
*Phys. Fluids***15**, 2668–2680 (2003).Google Scholar - M. Torrilhon and H. Struchtrup, Regularized 13 moment equations: shock structure calculations and comparison to Burnett models,
*J. Fluid Mech.***513**, 171–198 (2004).Google Scholar