Constitutive Relations for Rivlin‐Ericksen Fluids Based on Generalized Rational Approximation

. A well‐known constitutive expression for the stress in an incompressible non‐Newtonian fluid is provided by the representation of the extra stress as a function of the Rivlin‐Ericksen tensors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf{A}_1, \mathbf{A}_2,\ldots$\end{document}. If this function is ordered in terms of the number of space plus time derivatives and appropriately scaled, one obtains \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf{f}(\mathbf{A}_1, \mathbf{A}_2,\ldots)=\mu\mathbf{A}_1+\mu^2 (\alpha_1\mathbf{A}_2 +\alpha_2 \mathbf{A}^2_1)+\cdots$\end{document}. Truncation at first order yields the usual Newtonian viscous stress while truncation at second order provides the second‐order Rivlin‐Ericksen fluid. Many rheologists believe that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha_1<0$\end{document} in polymeric fluids. However, the requirement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha_1<0$\end{document} causes the rest state of the second‐order fluid to be unstable. This paper shows how the approximation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf{f}$\end{document} via generalized rational functions eliminates the instability paradox.