Skip to main content
Log in

An analytical approach to the strong evaporation problem in rarefied gas dynamics

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this work, we consider a semi-infinite expanse of a rarefied gas bounded by its plane condensed phase on which evaporation takes place. The analysis is based on the BGK model derived from the Boltzmann equation. In particular, the strong evaporation problem is considered, where nonlinear aspects have to be taken into account. We present the complete development of a closed form solution for evaluating density, velocity and temperature perturbations. Numerical results are presented and discussed.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Aoki K., Masukawa N.: Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation. Phys. Fluids 6, 1379–1395 (1994)

    Article  MATH  Google Scholar 

  2. Sone Y., Takata S., Golse F.: Notes on the boundary conditions for fluid-dynamic equations on the interface of a gas and its condensed phase. Phys. Fluids 13, 324–334 (2001)

    Article  Google Scholar 

  3. Frezzotti A., Ytrehus T.: Kinetic theory study of steady condensation of a polyatomic gas. Phys. Fluids 18, 027101(1)–027101(12) (2006)

    Article  MathSciNet  Google Scholar 

  4. Yano T.: Half-space problem for gas flows with evaporation or condensation on a planar interface with a general boundary condition. Fluid Dyn. Res. 40, 474–484 (2008)

    Article  MATH  Google Scholar 

  5. Scherer C.S., Barichello L.B.: An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problem. Z. Angew. Math. Phys. 61, 95–117 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ytrehus, T.: Theory and experiments on gas kinetics in evaporation. In: 10th International Symposium on Rarefied Gas Dynamics, Aspen, USA, pp. 1197–1212 (1976)

  7. Pao Y.: Temperature and density jumps in the kinetic theory of gases and vapors. Phys. Fluids 14, 1340–1346 (1971)

    Article  MathSciNet  Google Scholar 

  8. Yasuda S., Takata S., Aoki K.: Evaporation and condensation of a binary mixture of vapors on a plane condensed phase: numerical analysis of the linearized Boltzmann equation. Phys. Fluids 17, 047105(1)–047105(19) (2005)

    Article  Google Scholar 

  9. Williams M.M.R.: Mathematical Methods in Particle Transport Theory. Butterworth, London (1971)

    Google Scholar 

  10. Cercignani C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)

    Book  MATH  Google Scholar 

  11. Pao Y.: Application of kinetic theory to the problem of evaporation and condensation. Phys. Fluids 14, 306–312 (1971)

    Article  MathSciNet  Google Scholar 

  12. Siewert C.E., Thomas J.R. Jr: Half-space problems in the kinetic theory of gases. Phys. Fluids 16, 1557–1559 (1973)

    Article  MATH  Google Scholar 

  13. Thomas J.R. Jr, Chang T.S., Siewert C.E.: Reverse temperature gradient in the kinetic theory of evaporation. Phys. Rev. Lett. 33, 680–682 (1974)

    Article  Google Scholar 

  14. Thomas J.R. Jr: The F N method in kinetic theory. I. Half-space problems. Transp. Theory Stat. Phys. 14, 485–496 (1985)

    Article  MATH  Google Scholar 

  15. Sone Y., Ohwada T., Aoki K.: Evaporation and condensation on a plane condensed phase: numerical analysis of the linearized Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1, 1398–1405 (1989)

    Article  MATH  Google Scholar 

  16. Loyalka S.K.: Kinetic theory of planar condensation and evaporation. Transp. Theory Stat. Phys. 20, 237–249 (1991)

    Article  MATH  Google Scholar 

  17. Siewert C.E.: Heat transfer and evaporation/condensation problems based on the linearized Boltzmann equation. Euro. J. Mech. B Fluids 22, 391–408 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Arthur M.D., Cercignani C.: Non-existence of a steady rarefied supersonic flow in a half-space. Z. Angew. Math. Phys. 31, 634–645 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)

    Article  MATH  Google Scholar 

  20. Siewert C.E., Thomas J.R. Jr: Strong evaporation into a half space. Z. Angew. Math. Phys. 32, 421–433 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Loyalka S.K., Siewert C.E., Thomas J.R. Jr: An approximate solution concerning strong evaporation into a half space. Z. Angew. Math. Phys. 32, 745–747 (1981)

    Article  Google Scholar 

  22. Siewert C.E., Thomas J.R. Jr.: Strong evaporation into a half space. II. The three-dimensional BGK model. Z. Angew. Math. Phys. 33, 202–218 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Barichello L.B., Siewert C.E.: A discrete-ordinates solution for a non-grey model with complete frequency redistribution. JQSRT 62, 665–675 (1999)

    Article  Google Scholar 

  24. Scherer C.S., Prolo Filho J.F., Barichello L.B.: An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. I. Flow problems. Z. Angew. Math. Phys. 60, 70–115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Scherer C.S., Prolo Filho J.F., Barichello L.B.: An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems. Z. Angew. Math. Phys. 60, 651–687 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Smith B.T., Boyle J.M., Dongarra J.J., Garbow B.S., Ikebe Y., Klema V.C., Moler C.B.: Matrix Eigensystem Routines—EISPACK Guide. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  27. Dongarra J.J., Bunch J.R., Moler C.B., Stewart G.W.: LINPACK User’s Guide. SIAM, Philadelphia (1979)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to C. S. Scherer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Scherer, C.S. An analytical approach to the strong evaporation problem in rarefied gas dynamics. Z. Angew. Math. Phys. 66, 1821–1833 (2015). https://doi.org/10.1007/s00033-014-0462-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-014-0462-1

Mathematics Subject Classification

Keywords

Navigation