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On Some Properties of Generalized Burnett Equations of Hydrodynamics

Abstract

We continue to work on the study of possibilities to describe flows of rarefied gas by higher (compared to the Navier–Stokes level) equations of hydrodynamics. The main question is how to make the next step with respect to Knudsen number and to derive the well-posed equations at the Burnett level. Our method allows to construct well-posed equations at this level. However, they are not unique, since they include some indefinite parameters. In our previous paper on this topic we have studied the group properties of these equations. In the present work we show that invariant solutions (travelling waves and bounded in the half-space stationary solutions) of our equations clearly indicate some disadvantages of the Navier–Stokes equations. These solutions allow to choose the optimal set of parameters in Generalized Burnett Equations (GBEs). This system was already used in our studies on the basis of earlier numerical results, but we present in the paper the full theoretical proof. It is proved that there is a unique set of parameters in GBEs such that (a) main properties (related to dimensions of stable and unstable manifolds for large x) of the kinetic and hydrodynamic description are qualitatively similar and (b) the number of third derivatives in the equations has a minimal possible value \( n = 1 \). All considerations are made for monoatomic gases with arbitrary intermolecular potential (with standard restrictions).

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Acknowledgements

The research was supported by Russian Science Foundation Grant No. 18-11-00238-\(\Pi \).

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Correspondence to A. V. Bobylev.

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Communicated by Eric A. Carle.

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Bobylev, A.V. On Some Properties of Generalized Burnett Equations of Hydrodynamics. J Stat Phys 188, 6 (2022). https://doi.org/10.1007/s10955-022-02927-1

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Keywords

  • Boltzmann kinetic equation
  • Knudsen number
  • Hydrodynamics
  • Generalized Burnett equations