Abstract:
Integral representations for the generalized measures of deformation
and deformation-rates have been obtained and suitable constitutive
equations using these measures have been developed for
viscoelastic materials, These new constitutive equations are then
applied to study some flow problems, In order to assess the comparative
advantage of this approach over the existing nonlinear
theories of continuous media, a brief review of the latter has been
first made. In this review, special attention has been drawn to the
ever-increasing complexity of the constitutive equations which
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 complexity in
these constitutive equations has arisen since generalized measures
have not been used and consequently the order of the measures could
not be fixed. With a view to bring simplicity and at the same time to retain
generality, to ensure the effectiveness of the present theory, generalized
measures of deformation-rates have been suitably extended
before using them in the constitutive equations for isotropic incompressible
fluids. After the orders of the generalized measures have
been fixed, these new constitutive equations have been found to contain
only four terms in the deformation-rate tensors and four
rheological constants, but no unknown functions of the invariants,
This new constitutive theory based on generalized measures
has been applied in the solution of the following three types of problems:
rectilinear flows,
helical flows and
torsional flow.
The normal stress effects including swelling and thinning in
Poiseuille flow, and climbing in Couette flows, velocity profiles,
pressure variations, etc. have been studied in much greater detail
and added precision than has been done in the literature so far. The
phenomena of back flow between two parallel plates and helical flow
in a narrow annular gap have also been studied. The results have
been compared with the classical theory. To enhance the value of
this investigation and to make it more useful for practical purposes,
graphs of velocity profiles, pressure variations, etc. have also been drawn. A special feature of this analysis is to bring out important
non-Newtonian effects in real fluids with an unparalleled precision
and simplicity. All this has been accomplished because of the use
of generalized measures.
Possible scope of future work, where the idea of generalized
measures may be profitably exploited, has also been discussed.
