A Modified Functional Expansion for Viscoelastic Fluids
Authors: Goddard J.D. 
Published: December 2007

Abstract
A rheological equation for viscoelastic fluids is derived as an expansion of Noll's functional for sample fluids. The present expansion is developed in terms of the deformation rate referred to a co-rotational material frame. The resulting series of approximations for stress differs significantly from those proposed previously by Green and Rivlin and by Coleman and Noll.
Applications to both simple-extensional and viscometric flow are considered. For viscometric flows the present model exhibits non-Newtonian effects similar to those observed in real fluids, with relatively few terms in the functional expansion. The first term in the expansion yields a "first-order" fluid similar to that proposed by Pao. Expressions are also given for the viscometric material functions of the present "third-order" fluid. As with higher-order fluids, these functions are Fourier integrals of certain hereditary or memory functions for the fluid.
Introduction
In a previous paper it was shown how one could construct an inverse for the co-rotational or Jaumann derivative. This is operation, termed "Jaumann integration" was then used in a somewhat intuitive manner to derive a rheological model for viscoelastic fluids, by means of a Boltzmann superposition integral. As discussed in the above paper, the resulting fluid model is essentially a generalization of the "Quasilinear" model proposed earlier by Oldroyd. Moreover, the rheological equation bears some resemblance to certain others which have been derived from molecular considerations. However, while the model was shown to exhibit some of the features of real-fluid behavior, including shear-dependent viscosity and normal-stress functions, both related to the linear-viscoelastic "memory" function, it was concluded that the model was in general not sufficient to describe nonlinear viscoelastic effects in their entirety.
The present work is first of all an attempt to provide the above model with a somewhat more rigorous continuum-mechanical basis, by deriving it from Noll's general theory of materials with a "memory". More importantly, the intent of the present paper is to arrive systematically at certain improvements on the "quasi-linear" model previously proposed and, then, to relate the resulting fluid models to the "higher-order" fluids proposed by Green and Rivlin and by Coleman and Noll. Finally, it is hoped that the present work might suggest other types of functional expansions and perhaps stimulate investigation of the convergence of such expansions, as it relates to the physical properties of fluids.
FORMULATION OF THE RHEOLOGICAL EQUATION
The Material Matrizant
For the present work it will be convenient to recall the notion of a material matrizant, which was introduced in an earlier paper. Thus, for a given second-order tensor field(x,t) and a material velocity field the material matrizant of A relative to v is a tensor field Mtt [A] which is to satisfy
........................................(2.1)
where I is the unit [metric] tensor or idem factor and D/Dt is the material derivative:
........................................(2.2)