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

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.