### Abstract:

By various experiments, it has been found that the response
of real materials to external forces is, in general,
nonlinear in character. In classical continuum mechanics,
the use of ordinary measures of strain have forced the
constitutive equations to take complex forms and since the
orders of these measures are not fixed, many unknown response
coefficients have to be introduced into the constitutive
equations. In general, there is no basis of choosing
these coefficients. Seth attempted to resolve this
difficulty by introducing generalized measures in continuum
mechanics and Narasimhan and Sra extended these measures in
such a way as to adequately explain some rheological behavior
of materials. The constitutive equation of Narasimhan
and Sra essentially contains two terms and four rheological
constants and, unlike some previous theories, it does not contain any unknown functions of the invariants of kinematic
matrices while at the same time explains many viscoelastic
phenomena.
In the present investigation, a theorem has been
proved establishing certain criteria for fixing the orders
of generalized measures suitably so as to predict different
types of viscoelastic phenomena, such as dilatancy. We
have found during the course of this investigation that the
constitutive equation of Narasimhan and Sra does not adequately
explain such physical phenomena as pseudoplasticity.
However, in order to construct a constitutive equation so
that it does explain such phenomena, we have found it necessary
to construct combinations of sets of generalized
measures. The resulting constitutive equation is found to
be quite general and is able to explain a vast range of
physical behavior of fluids.
To illustrate the use of this constitutive equation
based upon combined generalized measures, we have investigated
the important problem of secondary flows for fluids
in the presence of moving boundaries. This problem is very
important, since the investigation of secondary flows allows
us to obtain a clearer picture of the actual motion of
the fluid. For the problem of flow of a fluid in the annulus
of two rotating spheres, we have obtained the solution for the velocity and pressure fields. In order to investigate
the secondary flow pattern more thoroughly, we
have obtained the streamline function of the flow. The
streamlines in meridian planes containing the axis of rotation
are found to be closed loops and the nature of the
closed loops is found to be strongly dependent upon the
viscoelastic parameter S. For S less than critical
value, the flow is found to be very much like that of a
Newtonian fluid, with the fluid advancing toward the inner
sphere along the pole and outward along the equation. At
the critical value the flow region is found to split into
two subregions, each containing closed loops of streamlines.
As S is increased further, another critical value
is reached whereby the streamlines again become one set of
closed loops, with the sense of rotation reversed from that
of Newtonian fluid.