Skip to main content
SpringerLink
Log in

Small Mass Ratio Limit of Boltzmann Equations in the Context of the Study of Evolution of Dust Particles in a Rarefied Atmosphere

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We propose a model based on the coupling of two Boltzmann-like equations for the study of the evolution of dust particles in a rarefied atmosphere, such as it can be found in the context of safety studies for the ITER project of nuclear fusion.

When the typical size of a dust speck becomes too large, the numerical simulation of the system under study becomes too expensive, and one needs to introduce an asymptotic model in which the mass ratio between molecules and dust specks tends to 0. This model is constituted of a coupling (by a drag force term) between a Boltzmann equation and a Vlasov equation.

A rigorous proof of the passage to the limit is given in the spatially homogeneous setting. It includes a new variant of Povzner’s inequality in which the vanishing mass ratio is taken into account.

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. Arkeryd, L.: On the Boltzmann equation, I and II. Arch. Ration. Mech. Anal. 45, 1–34 (1972)

    MATH  MathSciNet  Google Scholar 

  2. Bird, G.A.: Monte Carlo simulation of gas flows. Annu. Rev. Fluid Mech. 10, 11–31 (1978)

    Article  ADS  Google Scholar 

  3. Bobylev, A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Stat. Phys. 88, 1183–1214 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, Berlin (1988)

    MATH  Google Scholar 

  5. Cercignani, C.: Measure solutions of the steady Boltzmann equation in a slab. Commun. Math. Phys. 142(2), 285–296 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Charles, F.: Kinetic modelling and numerical simulations using particle methods for the transport of dust in a rarefied gas. In: Rarefied Gas Dynamics: Proceedings of the 26th International Symposium on Rarefied Gas Dynamics, vol. 1084, pp. 409–414 (2008)

  7. Charles, F.: Ph.D. thesis, in Preparation

  8. Degond, P.: Asymptotic continuum models for plasmas and disparate mass gaseous binary mixtures. In: Capriz, G., Mariano, P.-M. (eds.) Material Substructures in Complex Bodies: From Atomic Level to Continuum. Elsevier, Amsterdam (2007)

    Google Scholar 

  9. Degond, P., Lucquin-Desreux, B.: The asymptotics of collision operators for two species of particles of disparate masses. Math. Models Methods Appl. Sci. 6(3), 405–410 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Degond, P., Lucquin-Desreux, B.: Transport coefficients of plasmas and disparate mass binary gases. Transp. Theory Stat. Phys. 25(6), 595–633 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Desvillettes, L.: Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Ration. Mech. Anal. 123, 387–404 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Desvillettes, L., Mouhot, C.: About Lp estimates for the spatially homogeneous Boltzmann equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, 127–142 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Frezzotti, A., Ostmo, S., Ytrhus, T.: Kinetic theory of steady evaporation from a spherical condensed phase containing inert solid particles. Phys. Fluids 9, 211–225 (1997)

    Article  ADS  Google Scholar 

  14. Gamba, I., Panferov, V., Villani, C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246(3), 503–541 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Lu, X.: Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation. J. Stat. Phys. 96, 765–796 (1999)

    Article  MATH  Google Scholar 

  16. Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16(4), 467–501 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Nanbu, K.: Theoretical basis of the direct simulation Monte Carlo method. In: Rarefied Gas Dynamics: Proceedings of the 15th International Symposium on Rarefied Gas Dynamics, pp. 369–383 (1986)

  18. Povzner, A.J.: On the Boltzmann equation in the kinetic theory of gases. Mat. Sb. 58(100), 65–86 (1962)

    MathSciNet  Google Scholar 

  19. Takase, K.: Three-dimensional numerical simulation of dust mobilization and air ingress characteristics in a fusion reactor during a LOVA event. Fusion Eng. Des. 54(3–4), 605–615 (2001)

    Article  Google Scholar 

  20. Wennberg, B.: Entropy dissipation and moment production for the Boltzmann equation. J. Stat. Phys. 86, 1053–1066 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEA Saclay, DEN/DANS/DM2S/SFME/LSET, 91191, Gif-sur-Yvette Cedex, France

    Frédérique Charles

  2. CMLA, ENS Cachan, CNRS, PRES UniverSud, 61 Av. du Pdt. Wilson, 94235, Cachan Cedex, France

    Frédérique Charles

  3. CMLA, ENS Cachan, CNRS & IUF, PRES UniverSud, 61 Av. du Pdt. Wilson, 94235, Cachan Cedex, France

    Laurent Desvillettes

Authors
  1. Frédérique Charles
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laurent Desvillettes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laurent Desvillettes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Charles, F., Desvillettes, L. Small Mass Ratio Limit of Boltzmann Equations in the Context of the Study of Evolution of Dust Particles in a Rarefied Atmosphere. J Stat Phys 137, 539–567 (2009). https://doi.org/10.1007/s10955-009-9858-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-009-9858-2

Search

Navigation