Skip to main content
SpringerLink
Log in

Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Sharipov and V. Seleznev, “Data on internal rarefied gas flows,” J. Phys. Chem. Ref. Data 27 (3), 657–706 (1998).

    Article  Google Scholar 

  2. F. M. Sharipov and V. N. Seleznev, Rarefied Gas Flows through Channels and Microchannels (Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 2008) [in Russian].

    Google Scholar 

  3. V. A. Titarev and E. M. Shakhov, “Efficient method for computing rarefied gas flow in a long finite plane channel,” Comput. Math. Math. Phys. 52 (2), 269–284 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. A. Titarev, “Rarefied gas flow in a planar channel by arbitrary pressure and temperature drops,” Int. J. Heat Mass Transfer 55, 5916–5930 (2012).

    Article  Google Scholar 

  5. F. Sharipov, “Transient flow of rarefied gas through a short tube,” Vacuum 90, 25–30 (2013).

    Article  Google Scholar 

  6. F. Sharipov and I. A. Graur, “General approach to transient flows of rarefied gas through long capillaries,” Vacuum 100, 22–25 (2013).

    Article  Google Scholar 

  7. Yu. Yu. Kloss, F. G. Tcheremissine, and P. V. Shuvalov, “Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels,” Comput. Math. Math. Phys. 50 (6), 1093–1103 (2010).

    Article  MathSciNet  Google Scholar 

  8. D. E. Zeitoun, I. A. Graur, Y. Burtschell, M. S. Ivanov, A. N. Kudrayvtsev, and Ye. A. Bondar, “Continuum and kinetic simulation of shock wave propagation in long microchannel,” 26th International Symposium on Rarefied Gas Dynamics, Melville, New York, 2009; Ed. by T. Abe; AIP Conf. Proc. 1084, 964–969 (2009).

    Google Scholar 

  9. Nobuya Mioshi, Shuhei Nagata, Ikuya Kinefuchi, Kazuya Shimizu, Shu Takagi, and Yoichiro Matsumoto, “Development of ultra small shock tube for high energy molecular beam source,” 26th International Symposium on Rarefied Gas Dynamics, Melville, New York, 2009; AIP Conf. Proc. 1084, 557–562 (2009).

    Google Scholar 

  10. I. N. Larina, V. A. Rykov, and E. M. Shakhov, “Evolution of the rarefied gas flow between parallel plates driven by an initial pressure difference,” Dokl. Akad. Nauk 343 (4), 482–485 (1995).

    MATH  Google Scholar 

  11. I. N. Larina, V. A. Rykov, and E. M. Shakhov, “Unsteady rarefied gas flows between parallel planes,” Fluid Dyn. 32 (2), 289–295 (1997).

    MATH  Google Scholar 

  12. N. A. Konopel’ko, V. A. Titarev, and E. M. Shakhov, “Rarefied gas deceleration in a channel in the case of expansion into a vacuum,” Fluid Dyn. 50 (2), 294–305 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  13. N. A. Konopel’ko, V. A. Titarev, and E. M. Shakhov, “Unsteady rarefied gas flow in a microchannel driven by a pressure difference,” Comput. Math. Math. Phys. 56 (3), 470–482 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  14. E. M. Shakhov and V. A. Titarev, “Nonstationary rarefied gas flow into vacuum from a circular pipe closed at one end,” Vacuum 100, 284–293 (2014).

    Article  Google Scholar 

  15. E. M. Shakhov and V. A. Titarev, “Time-dependent rarefied gas flow into vacuum from a long circular pipe closed at one end,” Proceedings of the 29th International Symposium on Rarefied Gas Dynamics; AIP Conf. Proc. 1628, 1071–1078 (2014). doi doi 10.1063/1.490271210.1063/1.4902712

    Google Scholar 

  16. E. M. Shakhov, “Generalization of the Krook kinetic relaxation equation,” Fluid Dyn. 3 (5), 95–96 (1968).

    Article  MathSciNet  Google Scholar 

  17. V. P. Kolgan, “The principle of minimal derivative values as applied to the construction of finite-difference schemes for the computation of discontinuous gas flows,” Uch. Zap. Tsentr. Aerogidrodin. Inst. 3 (6), 68–77 (1972).

    Google Scholar 

  18. I. N. Larina and V. A. Rykov, “Computation of rarefied diatomic gas flows through a plane microchannel,” Comput. Math. Math. Phys. 52 (4), 637–648 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  19. I. N. Larina and V. A. Rykov, “Numerical investigation of rarefied diatomic gas flows through a plane channel into a vacuum,” Fluid Dyn. 48 (3), 389–401 (2013).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, 141700, Russia

    N. A. Konopel’ko

  2. Bauman Moscow State Technical University, Moscow, 107005, Russia

    E. M. Shakhov

  3. Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control,”, Russian Academy of Sciences, Moscow, 119333, Russia

    E. M. Shakhov

Authors
  1. N. A. Konopel’ko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. M. Shakhov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. A. Konopel’ko.

Additional information

Original Russian Text © N.A. Konopel’ko, E.M. Shakhov, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 10, pp. 1722–1733.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Konopel’ko, N.A., Shakhov, E.M. Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel. Comput. Math. and Math. Phys. 57, 1695–1705 (2017). https://doi.org/10.1134/S0965542517100098

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542517100098

Keywords

Search

Navigation