Partial differential equations
Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models
Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1207-1215.

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 α0 (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.

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 α0 (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.

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DOI: 10.5802/crmath.269
Classification: 76A05, 76A10, 76D03, 35B40
Girault, Vivette 1; Scott, L. Ridgway 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
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Girault, Vivette; Scott, L. Ridgway. 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. http://www.numdam.org/articles/10.5802/crmath.269/

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