Constitutive relations for Rivlin-Ericksen fluids based on generalized rational approximation

A well-known constitutive expression for the stress in an incompressible no n-Newtonian fluid is provided by the representation of the extra stress as a function of the Rivlin-Ericksen tensors A(1), A(2),.... If this function is Ordered in terms of the number of space plus time derivatives and approp riately scaled, one obtains f(A(1), A(2),...) = mu A(1) + mu(2)(alpha(1)A(2 ) + alpha(2)A(1)(2)) +.... Truncation at first order yields the usual Newto nian viscous stress while truncation at second order provides the second-or der Rivlin-Ericksen fluid. Many rheologists believe that alpha(1) < 0 in po lymeric fluids. However, the requirement alpha(1) < 0 causes the rest state of the second-order fluid to be unstable. This paper shows how the approxi mation of f via generalized rational functions eliminates the instability p aradox.