×

On group symmetries of the hydrodynamic equations for rarefied gas. (English) Zbl 1476.35194

Summary: The invariant group transformations of three-dimensional hydrodynamic equations derived from the Boltzmann equation are studied. Three levels (with respect to the Knudsen number) of hydrodynamic description are considered and compared: (a) Euler equations, (b) Navier-Stokes equations, (c) Generalized Burnett equations (GBEs), which replace the original (ill-posed) Burnett equations. The main attention is paid to group analysis of GBEs in their most general formulation because this and related questions have not been studied before in the literature. The results of group analysis of GBEs and, for comparison, of similar results for Euler and Navier-Stokes equations are presented in two theorems and discussed in detail. It is remarkable that the use of computer code greatly simplifies the proof of the results for GBEs, which are very cumbersome equations with many undetermined parameters.

MSC:

35Q35 PDEs in connection with fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] V. I. Arnol’d, Geometrical Methods in the Theory of Ordinary Differential Equations, Fundamental Principles of Mathematical Science, 250, Springer-Verlag, New York-Berlin, 1983.
[2] M. Bisi; M. P. Cassinari; M. Groppi, Qualitative analysis of the generalized Burnett equations and applications to half-space problems, Kinet. Relat. Models, 1, 295-312 (2008) · Zbl 1140.76036
[3] A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys., 132, 569-580 (2008) · Zbl 1144.82057
[4] A. V. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys., 124, 371-399 (2006) · Zbl 1134.82031
[5] A. V. Bobylev, Kinetic Equations. Boltzmann Equation, Maxwell Models, and Hydrodynamics Beyond Navier-Stokes, De Gruyter Series in Applied and Numerical Mathematics, 1, De Gruyter, Berlin/Boston, 2020.
[6] A. V. Bobylev, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262, 71-75 (1982)
[7] A. V. Bobylev and N. K. Ibragimov, Relationships between the symmetry properties of the equations of gas kinetics and hydrodynamics, Math. Modeling Comput. Experiment, 1 (1993), 291-300. · Zbl 1039.82507
[8] A. V. Bobylev; S. V. Meleshko, Group analysis of the generalized Burnett equations, J. Nonlinear Math. Phys., 27, 494-508 (2020) · Zbl 1436.35019
[9] A. V. Bobylev; Å. Windfäll, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models, 5, 237-260 (2012) · Zbl 1247.35195
[10] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, Cambridge University Press, New York 1960. · JFM 65.1541.01
[11] A. A. Cherevko, Optimal system of subalgebras of the Lie algebra of generators admitted by the system of the gas dynamics equations of a polytropic gas with the state equation \(p = f(s)\rho^{5/3}\), preprint of Lavrent’ev Institute of Hydrodynamics, Novosibirsk, 1996, 4-96.
[12] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, CRC Press, Boca Raton, FL, 2011.
[13] Y. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev and S. V. Meleshko, Symmetries of Integro-Differential Equations, Lecture Notes in Physics, 806, Springer, Dordrecht, 2010.
[14] A. C. Hearn, REDUCE Users Manual, Ver. 3.3, The Rand Corporation CP 78, Santa Monica, 1987.
[15] N. H. Ibragimov (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, CRC Press, Boca Raton, 1996.
[16] M. N. Kogan, Rarefied Gas Dynamics, Springer, Boston, MA, 1969.
[17] S. V. Meleshko, Methods for Constructing Exact Solutions of Partial Differential Equations, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. · Zbl 1081.35001
[18] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986.
[19] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, Inc., Harcourt Brace Jovanovich, Publishers, New York-London, 1982. · Zbl 0485.58002
[20] L. V. Ovsiannikov, The program ’Submodels’. Gas dynamics, J. Appl. Math. Mech., 58, 601-627 (1994) · Zbl 0890.76070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.