Archive for Rational Mechanics and Analysis

, Volume 146, Issue 1, pp 73–93

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

  • M. Slemrod

DOI: 10.1007/s002050050137

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


. 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • M. Slemrod
    • 1
  1. 1.Center for the Mathematical Sciences, University of Wisconsin‐Madison, 1308 W. Dayton St., Madison, Wisconsin 53715‐1149, USAUS