### Abstract:

An introduction to the generalized measures of deformation-
rates involving not only velocity gradients but also
acceleration gradients is given and constitutive equations
using these measures have been discussed. These new constitutive
equations are then used to study torsional
flows of viscoelastic fluids between two infinite parallel
planes. In order to assess the advantage of this theory
over the existing theories, a brief review of the existing
nonlinear theories of continuous media has been made. In
this review, it has been pointed out that the existing
theories involve a number of terms in powers and products
of the ordinary measures of strain or strain-rate and
several unknown response functions of invariants of kinematic
matrices. This is owing to the fact that the order
of the measures of strain and strain-rates have not been
fixed and their generalized measures have not been used in
the formulation of constitutive equations.
In the present investigation we have, by fixing a priori
the orders of the measures of the deformation-rates
mentioned above, developed the concept of the generalized
measures and been able to obtain a suitable constitutive
equation for viscoelastic fluids. The orders of the measures
are so chosen that the resulting constitutive equation
describes pseudoplastic fluids, for which the apparent
coefficient of viscosity decreases with the increase in
rate of shear. These new constitutive equations have been
found to contain only four terms in the deformation-rate
tensors and four rheological constants, and no unknown
functions of the invariants.
The constitutive equation obtained thus is applied to
torsional flows of a viscoelastic fluid between two infinite
parallel planes. The velocity components and pressure
have been expanded in power series of a small parameter,
and first- and second-order approximations to the velocity
components have been obtained. The normal stress differences,
velocity profiles, apparent coefficient of viscosity
and their behavior depending on the rheological
constants nave been investigated. The phenomena of reversed
flows have also been discussed. Besides the freedom to choose the order of the measures, the rheological parameter
entering the constitutive equations can also be suitably
varied so as to correlate the theory with experiments.
It is found that the behavior of a viscoelastic fluid depends
on the sign as well as the magnitude of the rheological
constants. Thus, the constitutive equations based on
the concept of combined generalized measures predict adequately
qualitative as well as quantitative information
on the behavior of viscoelastic fluids and eliminate the
need for assuming unknown response coefficients.