Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models

Vivette Girault; L. Ridgway Scott

doi:10.5802/crmath.269

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 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

Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1207-1215.

Résumé
Abstract

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 $α \to 0$ (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 $α \to 0$ (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 : 2021-05-25
Accepté le : 2021-09-07
Publié le : 2021-11-25

DOI : 10.5802/crmath.269

Classification : 76A05, 76A10, 76D03, 35B40

Affiliations des auteurs :

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

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     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},
}

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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/

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