Skip to main content
SpringerLink
Log in

Asymptotic Behavior of Solutions to a Model for the Flow of a Reacting Fluid

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We give a complete analysis of solutions of a model for the flow of a multispecies reacting fluid occupying a thin cylinder whose walls may be semipermeable with respect to some or all of the chemical species. We prove the global existence of solutions and establish a number of time-independent a priori bounds sufficient to determine the corresponding time-asymptotic steady-state. We then derive necessary conditions and sufficient conditions ensuring that this steady-state reflects complete combustion, that is, that at least one of the reactant species is depleted.

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. Boltzmann L.: Lectures on Gas Theory (English translation by Stephen G. Brush). University of California Press, Berkeley (1964)

    Google Scholar 

  2. Callen H.B.: Thermodynamics. Wiley, New York (1960)

    MATH  Google Scholar 

  3. Chen G.-Q.: Global solutions to the compressible Navier-Stokes equations for a reacting mixture. SIAM J. Math. Anal. 23, 609–634 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Chen G.-Q., Hoff D., Trivisa K.: Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data. Arch. Rational Mech. Anal. 166, 321–358 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Feireisl E., Petzeltova H., Trivisa K.: Multicomponent reactive flows: global existence for large data. Commun. Pure Appl. Anal. 7(5), 1017–1047 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Giovangigli V.: Multicomponent flow modeling. Birkhauser, Boston (1999)

    MATH  Google Scholar 

  7. Hoff D.: Global solutions of the equations of one-dimensional compressible flow with large data and differing end states. Zeit. Ang. Math. Phys. 49, 774–785 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hoff D.: Discontinuous solutions of the Navier-Stokes equations for multidimensional heat-conducting fluids. Arch. Rational Mech. Anal. 139, 303–354 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Hoff D., Tsyganov E.: Time analyticity and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional flow. J. Differ. Equ. 245, 3068–3094 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kazhikhov A.V., Shelukhin V.V.: Unique global solution with respect to time of initial-boundary-value problems for one–dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, 273–282 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  11. Straskaba I.: Asymptotic development of vacuums for one-dimensional Navier-Stokes equations of compressible flows. Nonlinear World 3, 519–535 (1996)

    MathSciNet  Google Scholar 

  12. Williams F.A.: Combustion Theory. Addison-Wesley, Reading (1965)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405-5701, USA

    David Hoff

Authors
  1. David Hoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Hoff.

Additional information

Communicated by S. Müller

Research supported in part by the NSF under grant No. DMS-0758043.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoff, D. Asymptotic Behavior of Solutions to a Model for the Flow of a Reacting Fluid. Arch Rational Mech Anal 196, 951–979 (2010). https://doi.org/10.1007/s00205-009-0260-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-009-0260-7

Keywords

Search

Navigation