Abstract
Viscoelastic theories due to Oldroyd, Pao, Bogue, and Bernstein, Kearsley and Zapas are examined for their ability to quantitatively correlate linear dynamic data with nonlinear viscosity and normal stress data. Data are presented for solutions of 10 wt per cent polyisobutylene in Decalin and 12 wt per cent polystyrene in Aroclor. Some comments on the essential experiments necessary to characterize viscoelastic fluids are made.
Introduction
Constitutive equations for viscoelastic fluids attempt to relate the stress response of these fluids to the strain history. In the tensor equations from continuum. mechanics one has material parameters [similar to viscosity] which bring in viscous, normal stress and elastic effects. These parameters can be rearranged into parameters with dimensions of time and stress in continuum mechanics no effort is made to relate them to molecular properties. Rather, the objective is to interrelate flow behavior from, diverse instruments and geometries. On the other hand, molecular theories deal with the relation of rheological behavior to molecular properties but the deformations studied are simple, linear ones.
Despite these gaps in the theoretical structure, the material parameters in continuum mechanics do in some way reflect underlying molecular mechanisms and thus are not completely independent of each other or of the parameters from molecular theory. One expects materials which show normal stresses in steady shear to also show elasticity in a sinusoidal deformation. Materials that are highly elastic will show strongly non-Newtonian viscosities. The dynamic viscosity as a function of frequency [in a linear sinusoidal deformation] is quantitatively similar to the viscosity as a function of shear rate [in a large shearing deformation]. If these various effects could be brought together in a unifying phenomenological theory, one would be closer to having identified basic material parameters and the basic experiments to measure them.
As discussed in an earlier paper, the present work has been focused on several theories with a continuum mechanics origin. They are related to molecular theories in that they can be reduced to linear viscoelasticity and thus can be made to incorporate the results which molecular researchers display in this mathematical framework. It cannot be said that the theories incorporate a truly mechanistic [molecular] explanation of non-Newtonian viscosities since the arguments which bring in this effect are made on a macroscopic or on a purely mathematical level. However, having established the shear-dependent viscosity, the theories make explicit predictions for the normal stresses.
In the present work four theories are examined for quantitative correlation of experimental data from two polymeric solutions. In one-sentence descriptions these theories may be classified as follows: Oldroyd's three-constant theory: a differential theory involving a generalization of linear viscoelasticity [relations between stress, strain and first time derivatives of them] using a convected time derivative; Pao's theory: an integral theory involving superposition of a nonlinear strain tensor, a Maxwell-type relaxation of stress, and stress-strain behavior tracked in a rotating coordinate system; Bogue's theory: an integral theory of the form of Coleman-Noll second-order theory but with a nonlinear memory function, constructed empirically to give reasonable shear rate-dependent viscosities; and [41 Bernstein, Kearsley and Zapas's [BKZ] theory.
