Theoretical and Mathematical Physics

, Volume 191, Issue 3, pp 856–869 | Cite as

Some solvability problems for the Boltzmann equation in the framework of the Shakhov model

Article
  • 19 Downloads

Abstract

We consider the nonlinear Boltzmann equation in the framework of the Shakhov model for the classical problem of gas flow in a plane layer. The problem reduces to a system of nonlinear integral equations. The nonlinearity of the studied system can be partially simplified by passing to a new argument depending on the solution of the problem itself. We prove the existence theorem for a unique solution of the linear system and the existence theorem for a positive solution of the nonlinear Urysohn equation. We determine the temperature jumps on the lower and upper walls in the linear and nonlinear cases, and it turns out that the difference between them is rather small.

Keywords

nonlinearity monotonicity model equation iteration temperature jump kinetic thickness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. L. Bhatnagar, E. P. Gross, and M. Krook, “A model for collision processes in gases,” Phys. Rev., 94, 511–525 (1954).ADSCrossRefMATHGoogle Scholar
  2. 2.
    L. H. Holway, “New statistical model for kinetic theory: Methods of construction,” Phys. Fluids, 9, 1658–1673 (1966).ADSCrossRefGoogle Scholar
  3. 3.
    E. M. Shakhov, “On the generalization of the Krook kinetic equation [in Russian],” Izv. AN SSSR. Ser. MZhG, No. 5, 142–145 (1968).Google Scholar
  4. 4.
    V. A. Titarev, “Conservative numerical methods for advanced model kinetic equations,” in: ECCOMAS CFD 2006: Proc. European Conference on Computational Fluid Dynamics (TU Delft, The Netherlands, 2006, E. Onate, P. Wessling, and, J. Periaux, eds.), Delft University of Technology, Netherlands (2006), pp. 1–13.Google Scholar
  5. 5.
    G. Liu, “A method for constructing a model form for the Boltzman equation,” Phys. Fluids, 2, 277–280 (1990).ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Y. Zheng and H. Struchtrup, “Ellipsoidal statistical Bhatnagar–Gross–Krook model with velocity-dependent collision frequency,” Phys. Fluids, 17, 127103 (2005).ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    A. V. Latyshev and A. A. Yushkanov, “Thermal and isothermal slip in a new Liu model transport-equation,” Tech. Phys. Lett., 23, 540–542 (1997).ADSCrossRefGoogle Scholar
  8. 8.
    A. V. Latyshev and A. A. Yushkanov, “The temperature jump and slow evaporation in molecular gases,” JETP, 87, 518–526 (1998).ADSCrossRefGoogle Scholar
  9. 9.
    M. N. Kogan, Rarefied Gas Dynamics [in Russian], Nauka, Moscow (1967); English transl., Plenum, New York (1969).CrossRefGoogle Scholar
  10. 10.
    C. Cercignani, The Boltzmann Equation and Its Applications (Appl. Math. Sci., Vol. 67), Springer, New York (1988).CrossRefMATHGoogle Scholar
  11. 11.
    A. V. Latyshev and A. A. Yushkanov, “Analytic solution of boundary value problems for the Shakhov equation with the collision frequency proportional to the molecule velocity [in Russian],” Izv. RAN. Ser. MZhG, 38, 632–645 (2003).MATHGoogle Scholar
  12. 12.
    N. B. Engibaryan and A. Kh. Khachatryan, “Questions of the nonlinear theory of the dynamics of rarefied gas [in Russian],” Matem. Mod., 16, 67–74 (2004).MATHGoogle Scholar
  13. 13.
    V. A. Ambartsumyan, “On some nonlinear problems of the theory of radiative transfer [in Russian],” in: Theory of Stellar Spectra (V. V. Sobolev, V. G. Gorbatskii, and V. V. Ivanov, eds.), Nauka, Moscow, pp. 91–104.Google Scholar
  14. 14.
    N. B. Engibaryan, “On a problem in nonlinear radiative transfer,” Astrophysics, 2, 12–14 (1966).ADSCrossRefGoogle Scholar
  15. 15.
    A. Kh. Khachatryan, “On the analytical-numerical solution of the Couette problem in the framework of the BGK model of the Boltzmann equation in the nonlinear and linear cases [in Russian],” Vestn. Rossiisko–Armyanskogo (Slavyanskogo) Universiteta, 2, 16–32 (2014).Google Scholar
  16. 16.
    O. A. Kolenchits, Thermal Accomodation of Gas–Solid Systems [in Russian], Izdat. Nauka i Tekhnika, Minsk (1977).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Higher Mathematics and Theoretical MechanicsArmenian National Agrarian UniversityErevanArmenia

Personalised recommendations