Isentropic fluid dynamics in a curved pipe

Article

Abstract

In this paper we study isentropic flow in a curved pipe. We focus on the consequences of the geometry of the pipe on the dynamics of the flow. More precisely, we present the solution of the general Cauchy problem for isentropic fluid flow in an arbitrarily curved, piecewise smooth pipe. We consider initial data in the subsonic regime, with small total variation about a stationary solution. The proof relies on the front-tracking method and is based on [1].

Keywords

Isentropic fluid dynamics Curved pipe 

Mathematics Subject Classification

Primary: 35L65 Secondary: 45L67 76N15 

References

  1. 1.
    Amadori D., Gosse L., Guerra G.: Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws. Arch. Ration. Mech. Anal. 162(4), 327–366 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Borsche R., Colombo R.M., Garavello M.: On the coupling of systems of hyperbolic conservation laws with ordinary differential equations. Nonlinearity 23(11), 2749–2770 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bressan A.: Hyperbolic Systems of Conservation Laws, Volume 20 of Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2000)Google Scholar
  4. 4.
    Chen, G.-Q.: Euler equations and related hyperbolic conservation laws. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations. Vol. II. Evolutionary Equations, pp. 1–104. Elsevier/North-Holland, Amsterdam (2005)Google Scholar
  5. 5.
    Colombo R.M., Herty M., Sachers V.: On \({2\times2}\) conservation laws at a junction. SIAM J. Math. Anal. 40(2), 605–622 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Colombo R.M., Marcellini F.: Smooth and discontinuous junctions in the \({p}\)-system. J. Math. Anal. Appl. 361(2), 440–456 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cunge J.A., Holly F.M., Verwey A.: Practical Aspects of Computational River Hydraulics. Pitmann Publishing, Bath (1980)Google Scholar
  8. 8.
    Dal Maso G., Lefloch P.G., Murat F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74(6), 483–548 (1995)MathSciNetMATHGoogle Scholar
  9. 9.
    Goatin P., LeFloch P.G.: The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(6), 881–902 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Guerra G., Marcellini F., Schleper V.: Balance laws with integrable unbounded source. SIAM J. Math. Anal. 41(3), 1164–1189 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Holden H., Risebro N.H.: Riemann problems with a kink. SIAM J. Math. Anal. 30(3), 497–515 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Holden H., Risebro N.H.: Front Tracking for Hyperbolic Conservation Laws, 2nd edn. Springer, New York (2016)MATHGoogle Scholar
  13. 13.
    Kröner D., LeFloch P.G., Thanh M.-D.: The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section. M2N Math. Model. Numer. Anal. 42(3), 425–442 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Leon, A.S.: Improved modeling of unsteady free surface, pressurized and mixed flows in storm-sewer systems. Ph.D. thesis, University of Illinois at Urbana-Champaign (2007)Google Scholar
  15. 15.
    Liu T.P.: Transonic gas flow in a duct of varying area. Arch. Ration. Mech. Anal. 80(1), 1–18 (1982)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pagliara, S., Yen, B.C.: Sewer network hydraulic model: NISN. Technical report, Department of Civil Engineering, Urbana, IL (1997)Google Scholar
  17. 17.
    de Saint-Venant, M.: Théorie du mouvement non permanent des eaux, avec application aux crues des riveres et a l’introduction des marees dans leur lit. Acad. Sci. Comptes Rondus, pp. 147–154, 237–240 (1871)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.INDAM UnitUniversity of BresciaBresciaItaly
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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