Skip to main content
SpringerLink
Log in

Asymptotic analysis for a homogeneous bubbling regime Vlasov–Fokker–Planck/Navier–Stokes system

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

Abstract

The evolution of a cloud of particles in a compressible fluid can be modeled with a Vlasov–Fokker–Planck equation for the distribution function of the particles coupled with Navier–Stokes or Euler equations for the density and velocity of the fluid. Formal calculations have established the convergence of solution to the mesoscopic model to solutions to the macroscopic Navier–Stokes or Euler model coupled with a Smoluchowski equation as the ratio of the settling time for the microscopic velocity fluctuation of the particles to the characteristic macroscopic time scale goes to zero. This paper provides a rigorous asymptotic analysis for a homogeneous mesoscopic fluid–particle interaction model for particles dispersed in a compressible fluid is provided for the bubbling regime. A relative entropy inequality for a mixed hyperbolic/parabolic system of equations is employed.

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. Amsden, A.A.: Kiva-3V Release 2, Improvements to Kiva-3V. Tech. Report, Los Alamos National Laboratory (1999)

  2. Amsden, A.A., O’Rourke, P.J., Butler, T.D.: Kiva-2, A computer program for chemical reactive flows with sprays. Tech. Report, Los Alamos National Laboratory (1989)

  3. Ballew, J.: Vanishing viscosity solutions for the Navier–Stokes–Smoluchowski system for particles in a compressible fluid (2016) Preprint

  4. Ballew, J., Trivisa, K.: Suitable weak solutions and low stratification singular limit for a fluid particle interaction model. Q. Appl. Math. 70(3), 469–494 (2012)

    Article  MathSciNet  Google Scholar 

  5. Ballew, J., Trivisa, K.: Viscous and inviscid models in fluid-particle interaction. Commun. Inf. Syst. 13(1), 45–78 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Ballew, J., Trivisa, K.: Weakly dissipative solutions and weak-strong uniqueness for the Navier–Stokes–Smoluchowski system. Nonlinear Anal. 91, 1–19 (2013)

    Article  MathSciNet  Google Scholar 

  7. Baranger, C., Boudin, L., Jabin, P.-E., Mancini, S.: A modeling of biospray for the upper airways. In: CEMRACS 2004—mathematics and applications to biology and medicine, ESAIM Proc., vol. 14, EDP Sci., Les Ulis, 2005, pp. 41–47 (electronic)

  8. Berres, S., Bürger, R., Karlsen, K.H., Tory, E.M.: Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64(1), 41–80 (2003)

    Article  MathSciNet  Google Scholar 

  9. Bouchut, F.: Construction of bgk models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95(1), 113–170 (1999)

    Article  MathSciNet  Google Scholar 

  10. Carrillo, J.A., Karper, T., Trivisa, K.: On the dynamics of a fluid-particle interaction model: the bubbling regime. Nonlinear Anal. 74(8), 2778–2801 (2011)

    Article  MathSciNet  Google Scholar 

  11. Carrillo, J.A., Goudon, T.: Stability and asymptotic analysis of a fluid-particle interaction model. Commun. Partial Diff. Equ. 31(7–9), 1349–1379 (2006)

    Article  MathSciNet  Google Scholar 

  12. Christoforou, C., Tzavaras, A.E.: Relative entropy for hyperbolic-parabolic systems and application to the constitutive theory of thermoviscoelasticity. Arch. Ration. Mech. Anal. 229(1), 1–52 (2018)

    Article  MathSciNet  Google Scholar 

  13. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer, Berlin (2010)

  14. Feireisl, E., Novotný, A., Sun, Y.: Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60(2), 611–631 (2011)

    Article  MathSciNet  Google Scholar 

  15. Goudon, T., Jabin, P.-E., Vasseur, A.: Hydrodynamic limit for the Vlasov–Navier–Stokes equations. I. Light particles regime. Indiana Univ. Math. J. 53(6), 1495–1515 (2004)

    Article  MathSciNet  Google Scholar 

  16. Kružkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228–255 (1970)

    MathSciNet  Google Scholar 

  17. Mellet, A., Vasseur, A.: Global weak solutions for a Vlasov–Fokker–Planck/Navier–Stokes system of equations. Math. Models Methods Appl. Sci. 17(7), 1039–1063 (2007)

    Article  MathSciNet  Google Scholar 

  18. Mellet, A., Vasseur, A.: Asymptotic analysis for a Vlasov–Fokker–Planck/compressible Navier–Stokes system of equations. Commun. Math. Phys. 281(3), 573–596 (2008)

    Article  MathSciNet  Google Scholar 

  19. Williams, F.A.: Spray combustion and atomization. Phys. Fluids 1(6), 541–545 (1958)

    Article  Google Scholar 

  20. Williams, F.A.: Combustion Theory: The Fundamental Theory of Chemically Reacting Flow Systems, 2nd edn. Benjamin/Cummings Pub, Menlo Park, CA (1985)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical and Statistics, Slippery Rock University of Pennsylvania, Slippery Rock, PA, 16057, USA

    Joshua Ballew

Authors
  1. Joshua Ballew
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joshua Ballew.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

J. B. gratefully acknowledges support from the National Science Foundation under Grant DMS-1401732.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ballew, J. Asymptotic analysis for a homogeneous bubbling regime Vlasov–Fokker–Planck/Navier–Stokes system. Z. Angew. Math. Phys. 71, 131 (2020). https://doi.org/10.1007/s00033-020-01359-9

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-020-01359-9

Keywords

Mathematics Subject Classification

Search

Navigation