Comptes Rendus
Équations aux dérivées partielles
Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models
[La dualité de Tanner entre les Modèles de Fluides de Oldroyd–Maxwell et de Grade deux]
Vivette Girault1 ; L. Ridgway Scott2
1 Sorbonne Universités, UPMC University Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, Place Jussieu 75005 Paris, France
2 Departments of Computer Science and Mathematics, University of Chicago, Chicago, Illinois, USA
Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1207-1215.

On démontre une relation asymptotique entre un modèle de fluides de grade deux et une classe de modèles de fluides non Newtoniens proposés par Oldroyd, comprenant les modèles de Maxwell de convection supérieure et convection inférieure. Ceci confirme une observation faite à l’origine par Tanner. On donne une interprétation nouvelle de l’instabilité en temps du modèle de fluides de grade deux lorsque ses coefficients sont négatifs. Notre approche inclut une démonstration simple de la convergence de la solution du modèle stationnaire de fluides de grade deux vers celle du modèle de Navier–Stokes quand α→0 (sous des hypothèses convenables) en dimension trois. Elle donne aussi une preuve de la convergence de la solution des modèles stationnaires de Oldroyd, quand ses paramètres tendent vers zéro, vers celle du modèle de Navier–Stokes.

We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwell models. This confirms an earlier observation of Tanner. We provide a new interpretation of the temporal instability of the grade-two fluid model for negative coefficients. Our techniques allow a simple proof of the convergence of the steady grade-two model to the Navier–Stokes model as α→0 (under suitable conditions) in three dimensions. They also provide a proof of the convergence of the steady Oldroyd models to the Navier–Stokes model as their parameters tend to zero.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.269
Classification : 76A05, 76A10, 76D03, 35B40
Vivette Girault 1 ; L. Ridgway Scott 2

1 Sorbonne Universités, UPMC University Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, Place Jussieu 75005 Paris, France
2 Departments of Computer Science and Mathematics, University of Chicago, Chicago, Illinois, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2021__359_9_1207_0,
     author = {Vivette Girault and L. Ridgway Scott},
     title = {Tanner {Duality} {Between} the {Oldroyd{\textendash}Maxwell} and {Grade-two} {Fluid} {Models}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1207--1215},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {9},
     year = {2021},
     doi = {10.5802/crmath.269},
     language = {en},
}
TY  - JOUR
AU  - Vivette Girault
AU  - L. Ridgway Scott
TI  - Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 1207
EP  - 1215
VL  - 359
IS  - 9
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.269
LA  - en
ID  - CRMATH_2021__359_9_1207_0
ER  - 
%0 Journal Article
%A Vivette Girault
%A L. Ridgway Scott
%T Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models
%J Comptes Rendus. Mathématique
%D 2021
%P 1207-1215
%V 359
%N 9
%I Académie des sciences, Paris
%R 10.5802/crmath.269
%G en
%F CRMATH_2021__359_9_1207_0
Vivette Girault; L. Ridgway Scott. Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1207-1215. doi : 10.5802/crmath.269. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.269/

[1] Jean-Marie Bernard Problem of Second Grade Fluids in Convex Polyhedrons, SIAM J. Math. Anal., Volume 44 (2012) no. 3, pp. 2018-2038 | DOI | MR | Zbl

[2] Adriana Valentina Busuioc From second grade fluids to the Navier–Stokes equations, J. Differ. Equations, Volume 265 (2018) no. 3, pp. 979-999 | DOI | MR

[3] Lamberto Cattabriga Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Semin. Mat. Univ. Padova, Volume 31 (1961), pp. 308-340 | Numdam | MR | Zbl

[4] Doina Cioranescu; Vivette Girault; Kumbakonam R. Rajagopal Mechanics and Mathematics of Fluids of the Differential Type, Advances in Mechanics and Mathematics, 35, Springer, 2016 | DOI

[5] Monique Dauge Stationary Stokes and Navier–Stokes systems on two-or three-dimensional domains with corners. Part I. Linearized equations, SIAM J. Math. Anal., Volume 20 (1989) no. 1, pp. 74-97 | DOI | Zbl

[6] Jerald L. Ericksen; Ronald S. Rivlin Stress-deformation relations for isotropic materials, J. Ration. Mech. Anal., Volume 4 (1955), pp. 323-425 | MR | Zbl

[7] Enrique Fernández-Cara; Francisco Guillén; Rubens R. Ortega Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, Numerical Methods for Fluids (Part 2) (Handbook of Numerical Analysis), Volume 8, Elsevier, 2002, pp. 543-660 | DOI

[8] Vivette Girault; L. Ridgway Scott Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition, J. Math. Pures Appl., Volume 78 (1999), pp. 981-1011 | DOI | MR

[9] Vivette Girault; L. Ridgway Scott Wellposedness of some Oldroyd models that lack explicit dissipation (2017) no. TR-2017-04 (Research Report UC/CS)

[10] Vivette Girault; L. Ridgway Scott Oldroyd models without explicit dissipation, Rev. Roum. Math. Pures Appl., Volume 63 (2018) no. 4, pp. 401-446 | MR | Zbl

[11] Vivette Girault; L. Ridgway Scott An asymptotic duality between the Oldroyd-Maxwell and grade-two fluid models (2021) no. TR-2021-08 (Research Report UC/CS)

[12] Vivette Girault; Luc Tartar Régularité dans L p et W 1,p de la solution d’une équation de transport stationnaire, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 15-16, pp. 885-890 | Zbl

[13] James G. Oldroyd Non-Newtonian effects in steady motion of some idealized elastico-viscous fluids, Proc. R. Soc. Lond., Ser. A, Volume 245 (1958), pp. 278-297

[14] Michael Renardy Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech., Volume 65 (1985), pp. 449-451 | DOI | MR | Zbl

[15] Michael Renardy Existence of slow steady flows of viscoelastic fluids of integral type, Z. Angew. Math. Mech., Volume 68 (1988) no. 4, p. T40-T44 | MR | Zbl

[16] Roger I. Tanner The stability of some numerical schemes for model viscoelastic fluids, J. Non-Newton. Fluid Mech., Volume 10 (1982), pp. 169-174 | DOI

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Circumventing the lack of dissipation in certain Oldroyd models

Vivette Girault; L. Ridgway Scott

C. R. Math (2017)

Numerical study of Dean vortices in developing Newtonian and viscoelastic flows through a curved duct of square cross-section

Mohammed Boutabaa; Lionel Helin; Gilmar Mompean; ...

C. R. Méca (2009)