Skip to main content
Log in

Riemann Problem for van der Waals Fluids in Nozzle with Cross-Sectional Jump

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

The Riemann problem for isentropic van der Waals fluid flows in a nozzle with discontinuous cross-section is investigated. The pressure, expressed as a function of the density, is increasing and admits two inflection points. The model is not strictly hyperbolic, and the characteristic fields are not genuinely nonlinear. Since the model is written in Eulerian coordinates, it is hard to directly examine the Liu entropy condition. There may exist up to four stationary jumps from a given state. After imposing the admissibility criterion, two admissible stationary jumps may also co-exist. This leads to the multiple solutions of the Riemann problem for van der Waals fluids.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Ambroso, A., Chalons, C., Raviart, P.-A.: A Godunov-type method for the seven-equation model of compressible two-phase flow. Comput. Fluids 54, 67–91 (2012)

    Article  MathSciNet  Google Scholar 

  • Andrianov, N., Warnecke, G.: On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64, 878–901 (2004)

    Article  MathSciNet  Google Scholar 

  • Andrianov, N., Warnecke, G.: The Riemann problem for the Baer–Nunziato two-phase flow model. J. Comput. Phys. 195, 434–464 (2004)

    Article  MathSciNet  Google Scholar 

  • Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flows 12, 861–889 (1986)

    Article  Google Scholar 

  • Bedjaoui, N., Chalons, C., Coquel, F., LeFloch, P.G.: Non-monotone traveling waves in van der Waals fluids. Ann. Appl. 3, 419–446 (2005)

    Article  Google Scholar 

  • Bzil, J.B., Menikoff, R., Son, S.F., Kapila, A.K., Steward, D.S.: Two-phase modeling of a deflagration-to-detonation transition in granular materials: A critical examination of modelling issues. Phys. Fluids 11, 378–402 (1999)

    Article  Google Scholar 

  • Cuong, D.H., Thanh, M.D.: A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section. Appl. Math. Comput. 256, 602–629 (2015)

    MathSciNet  MATH  Google Scholar 

  • Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. (9) 74, 483–548 (1995)

    MathSciNet  MATH  Google Scholar 

  • Fan, H.: A vanishing viscosity approach on the dynamics of phase transitions in van der Waals fluids. J. Differ. Equations 103, 179–204 (1993)

    Article  MathSciNet  Google Scholar 

  • 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, 881–902 (2004)

    Article  MathSciNet  Google Scholar 

  • Isaacson, E., Temple, B.: Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52, 1260–1278 (1992)

    Article  MathSciNet  Google Scholar 

  • Isaacson, E., Temple, B.: Convergence of the \(2\times 2\) Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55, 625–640 (1995)

    Article  MathSciNet  Google Scholar 

  • Keyfitz, B.L., Sander, R., Sever, M.: Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete Contin. Dyn. Syst. Ser. B 3, 541–563 (2003)

    MathSciNet  MATH  Google Scholar 

  • LeFloch, P.G.: Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form, Institute for Mathematics and its Application. Minneapolis (1989) (Preprint 593)

  • LeFloch, P.G., Thanh, M.D.: The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci. 1, 763–797 (2003)

    Article  MathSciNet  Google Scholar 

  • LeFloch, P.G., Thanh, M.D.: Nonclassical Riemann solvers and kinetic relations III: A nonconvex hyperbolic model for van der Waals fluids. Electron. J. Differ. Equations 72, 1–19 (2000)

    MathSciNet  MATH  Google Scholar 

  • LeFloch, P.G., Tzavaras, A.E.: Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30, 1309–1342 (1999)

    Article  MathSciNet  Google Scholar 

  • Marchesin, D., Paes-Leme, P.J.: A Riemann problem in gas dynamics with bifurcation. Hyperbolic partial differential equations, III. Comput. Math. Appl. Part A 12, 433–455 (1986)

  • Menikoff, R., Plohr, B.: The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130 (1989)

    Article  MathSciNet  Google Scholar 

  • Ripa, P.: Conservation laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn. 70, 85–111 (1993)

    Article  MathSciNet  Google Scholar 

  • Ripa, P.: On improving a one-layer ocean model with thermodynamics. J. Fluid Mech. 303, 169–201 (1995)

    Article  MathSciNet  Google Scholar 

  • Rosatti, G., Begnudelli, L.: The Riemann problem for the one-dimensional, free-surface shallow water equations with a bed step: theoretical analysis and numerical simulations. J. Comput. Phys. 229, 760–787 (2010)

    Article  MathSciNet  Google Scholar 

  • Schwendeman, D.W., Wahle, C.W., Kapila, A.K.: The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212, 490–526 (2006)

    Article  MathSciNet  Google Scholar 

  • Slemrod, M.: Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Ration. Mech. Anal. 81, 301–315 (1983)

    Article  MathSciNet  Google Scholar 

  • Thanh, M.D.: A phase decomposition approach and the Riemann problem for a model of two-phase flows. J. Math. Anal. Appl. 418, 569–594 (2014)

    Article  MathSciNet  Google Scholar 

  • Thanh, M.D.: The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area. SIAM J. Appl. Math. 69, 1501–1519 (2009)

    Article  MathSciNet  Google Scholar 

  • Thanh, M.D.: The Riemann problem for the shallow water equations with horizontal temperature gradients. Appl. Math. Comput. 325, 159–178 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Reviewers for their very constructive comments and helpful suggestions. This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2018-28-01.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mai Duc Thanh.

Additional information

Publisher's Note

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

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vinh, D.X., Thanh, M.D. Riemann Problem for van der Waals Fluids in Nozzle with Cross-Sectional Jump. Bull Braz Math Soc, New Series 51, 597–639 (2020). https://doi.org/10.1007/s00574-019-00166-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-019-00166-9

Keywords

Mathematics Subject Classification

Navigation