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

DOI: 10.1007/s002050050137

- Cite this article as:
- Slemrod, M. Arch Rational Mech Anal (1999) 146: 73. doi:10.1007/s002050050137

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## Abstract

. 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 \(\mathbf{A}_1, \mathbf{A}_2,\ldots\). If this function is ordered in terms of the number of space plus time derivatives and appropriately scaled, one obtains \(\mathbf{f}(\mathbf{A}_1, \mathbf{A}_2,\ldots)=\mu\mathbf{A}_1+\mu^2 (\alpha_1\mathbf{A}_2 +\alpha_2 \mathbf{A}^2_1)+\cdots\). 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 \(\alpha_1<0\) in polymeric fluids. However, the requirement \(\alpha_1<0\) causes the rest state of the second‐order fluid to be unstable. This paper shows how the approximation of \(\mathbf{f}\) via generalized rational functions eliminates the instability paradox.