fluids similar to De Waele–Ostwald equation is often called theconstitutive equa-
tionof the fluids (De Waele 1923 ; Ostwald 1925 ). Oldroyd proposed the constitu-
tive equation in another way (Oldroyd 1950 ; Bird et al. 1987 ), as given by
sþt 1
@s
@t¼Kðg^0 þt 2
@g^0
@tÞ (7.8)
wheret 1 andt 2 are the relaxation time of the stress and the strain rate, respectively,
Kincludes both solute and solvent viscosity, and∂/∂t denotes the Oldroyd convec-
tive derivatives. In (7.8), either the left-hand-side derivative can be split into the
linear and nonlinear viscoelastic contributions, or the right-hand-side derivative can
be split into the inertia transport and the acceleration contributions. Oldroyd-A
model is used to describe the Newtonian fluids, while Oldroyd-B model is used to
describe the non-Newtonian fluids. For more accurate descriptions of the practical
fluids, the current rheology has developed many empirical and complicate consti-
tutive equations. Various mathematical tools and skills have been developed to find
proper solutions of these equations (Bird et al. 1987 ; Larson 1988 ).
7.2 Characteristics of Polymer Flow
The shear flow behaviors of polymer melts are quite complicated with various
topological architectures of polymers. For linear polymers, their shear flows can be
described by a universal curve (Fig.7.5). After yielding at an initial stress, the bulk
polymer behaves as a pure viscous fluid in the first Newtonian-fluid region. In the
following pseudo-plastic region, the bulk polymer deforms and orients subject to
the stronger shear stress, and meanwhile the viscosity decreases with the increase of
shear rate (the shear-thinning phenomenon). The next is the second Newtonian-
fluid region, in which the polymer reaches their up-limit of entropic elasticity for
deformation, behaving again as a pure viscous fluid. The final stage is the dilatant
region, in which the polymer unsteadily retracts back upon excessive deformation.
The fluid structure becomes unstable, and therefore its viscosity increases with the
further increase of shear rates.
Upon a large shear rate, the polymer flow exhibits nonlinear viscoelasticity. In
this case, the Boltzmann superposition principle becomes invalid, and the fluid
appears as a non-Newtonian fluid. A typical treatment is to consider the nonlinear
response as separate processes at two different time scales: the first one is the rapid
elastic recovery in association with the shear rate, which can relax part of the stress
instantaneously; the second one is the slow relaxation of the rest stress in associa-
tion with time. Thus, the nonlinear relaxation modulus can be expressed as
Eðg;tÞ¼hðgÞEðtÞ (7.9)132 7 Polymer Flow