A new model for shallow viscoelastic fluids

François Bouchut; Sébastien Boyaval

hal-00628651, version 2

A new model for shallow viscoelastic fluids

François Bouchut (, ) ¹, Sébastien Boyaval () ^2, 3

Mathematical Models and Methods in Applied Sciences 23, 8 (2013) 1479-1526

Résumé : We propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. Additionally to the classical layer depth and velocity in shallow models, our system describes also the evolution of two scalar stresses. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver.

1 : Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout
2 : Laboratoire d'Hydraulique Saint-Venant / Saint-Venant Laboratory for Hydraulics (Saint-Venant)
Université Paris-Est Créteil Val-de-Marne (UPEC) – Ecole des Ponts ParisTech – EDF – CETMEF
3 : MICMAC (INRIA Paris - Rocquencourt)
Ecole des Ponts ParisTech – INRIA

hal-00628651, version 2
http://hal-enpc.archives-ouvertes.fr/hal-00628651
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Contributeur : Sébastien Boyaval <>
Soumis le : Mercredi 8 Août 2012, 23:16:57
Dernière modification le : Mercredi 12 Juin 2013, 17:38:21

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arXiv : 1110.0799

DOI : 10.1142/S0218202513500140

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%% hal-00628651, version 2
%% http://hal-enpc.archives-ouvertes.fr/hal-00628651
@article{bouchut:hal-00628651,
    hal_id = {hal-00628651},
    url = {http://hal-enpc.archives-ouvertes.fr/hal-00628651},
    title = {{A new model for shallow viscoelastic fluids}},
    author = {Bouchut, Fran{\c c}ois and Boyaval, S{\'e}bastien},
    abstract = {{We propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. Additionally to the classical layer depth and velocity in shallow models, our system describes also the evolution of two scalar stresses. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver.}},
    keywords = {Viscoelastic fluids ; Maxwell model ; Oldroyd model ; Saint Venant equations ; shallow-water model ; pseudo-conservative hyperbolic system of equations ; well-balanced finite-volume scheme},
    language = {Anglais},
    affiliation = {Laboratoire d'Analyse et de Math{\'e}matiques Appliqu{\'e}es - LAMA , Laboratoire d'Hydraulique Saint-Venant / Saint-Venant Laboratory for Hydraulics - Saint-Venant , MICMAC - INRIA Paris - Rocquencourt},
    pages = {1479-1526},
    journal = {Mathematical Models and Methods in Applied Sciences},
    volume = {23},
    number = {8 },
    note = {Part of this work was completed while Sebastien Boyaval was an academic host at MATHICSE- ASN chair (EPFL). SB would like to thank Prof. Marco Picasso and Prof. Jacques Rappaz for this invitation. },
    audience = {internationale },
    doi = {10.1142/S0218202513500140 },
    year = {2013},
    month = Jan,
    pdf = {http://hal-enpc.archives-ouvertes.fr/hal-00628651/PDF/ms2.pdf},
}

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