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International Conference on Finite Volumes for Complex Applications

FVCA 2017: Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems pp 163–170Cite as

A Finite-Volume Discretization of Viscoelastic Saint-Venant Equations for FENE-P Fluids

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 200)

Abstract

Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows. A Finite-Volume discretization for the 1D Saint-Venant system generalized to Upper-Convected Maxwell (UCM) fluids was proposed in Bouchut and Boyaval (M3AS 23(08): 1479–1526, 2013, [6]), which preserved a physically-natural stability property (i.e. free-energy dissipation) of the full system. It invoked a relaxation scheme of Suliciu type for the numerical computation of approximate solutions to Riemann problems. Here, the approach is extended to the 1D Saint-Venant system generalized to the finitely-extensible nonlinear elastic fluids of Peterlin (FENE-P). We are currently not able to ensure all stability conditions a priori, but the scheme is fully computable. And, using numerical simulations, it may help understand the famous High-Weissenberg number problem (HWNP) well-known in computational rheology.

Keywords

  • Saint-venant equations
  • Fene-p viscoelastic fluids
  • Finite-volume
  • Simple Riemann solver
  • Suliciu relaxation scheme

MSC (2010):

  • 65M08
  • 65N08
  • 35Q30

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References

  1. Barrett, J.W., Boyaval, S.: Existence and approximation of a (regularized) Oldroyd-B model. M3AS 21(9), 1783–1837 (2011)

    Google Scholar 

  2. Berthon, C., Coquel, F., LeFloch, P.G.: Why many theories of shock waves are necessary: kinetic relations for non-conservative systems. Proc. R. Soc. Edinb. Sect. A Math. 142, 1–37 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Bird, R., Dotson, P., Johnson, N.: Polymer solution rheology based on a finitely extensible beadspring chain model. J. Non Newton. Fluid Mech. 7, 213–235 (1980)

    CrossRef  MATH  Google Scholar 

  4. Bouchut, F.: Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94, 623–672 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Bouchut, F.: Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004)

    Google Scholar 

  6. Bouchut, F., Boyaval, S.: A new model for shallow viscoelastic fluids. M3AS 23(08), 1479–1526 (2013)

    Google Scholar 

  7. Bouchut, F., Boyaval, S.: Unified derivation of thin-layer reduced models for shallow free-surface gravity flows of viscous fluids. Eur. J. Mech. B Fluids Part 1 55, 116–131 (2016)

    Google Scholar 

  8. Bouchut, F., Klingenberg, C., Waagan, K.: A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves. Numer. Math. 115(4), 647–679 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Boyaval, S.: Johnson-Segalman – Saint-Venant equations for viscoelastic shallow flows in the pure elastic limit. Proceedings in Mathematics and Statistics (PROMS) of the International Conference on Hyperbolic Problems: Theory, Numeric and Applications in Aachen 2016, (2017), ArXiv e-prints (2016)

    Google Scholar 

  10. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, vol. GM 325. Springer, Berlin (2000)

    Google Scholar 

  11. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. LeFloch, P.G.: Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2002)

    Google Scholar 

  13. Owens, R.G., Philips, T.N.: Computational Rheology Imperial. College Press/World Scientific, London (2002)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire D’hydraulique Saint-Venant (Ecole des Ponts ParisTech – EDF R&D – CEREMA), Université Paris-Est, EDF’lab 6 Quai Watier, 78401, Chatou Cedex, France

    Sébastien Boyaval

  2. INRIA MATHERIALS, Paris, France

    Sébastien Boyaval

Authors
  1. Sébastien Boyaval
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Correspondence to Sébastien Boyaval .

Editor information

Editors and Affiliations

  1. Equipe RAPSODI, Inria Lille - Nord Europe, Villeneuve-d’Ascq, France

    Prof. Clément Cancès

  2. Commissariat à l'énergie atomique et aux énergies alternatives, Centre de Saclay, Gif-sur-Yvette, France

    Prof. Pascal Omnes

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Boyaval, S. (2017). A Finite-Volume Discretization of Viscoelastic Saint-Venant Equations for FENE-P Fluids. In: Cancès, C., Omnes, P. (eds) Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems. FVCA 2017. Springer Proceedings in Mathematics & Statistics, vol 200. Springer, Cham. https://doi.org/10.1007/978-3-319-57394-6_18

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