Molecular modelling of entangled polymer fluids under flow The ...

1.4. DEFORMATION KINEMATICS 5
flow to be in the x direction, the velocity gradient to act in the y direction, and the z
direction to be parallel to the vorticity. Experimental data typically measure a range
of relevant components of the stress tensor. The easiest component to measure is the
force which directly opposes the shear, namely the shear stress, σ xy . Polymer fluids
also typically exert a force normal to the shear plane. This stress is measured relative
to atmospheric pressure and is expressed as differences between diagonal components of
the stress tensor. Two independent stress differences can be defined: the first normal
stress difference, N 1 = σ xx − σ yy and second normal stress difference, N 2 = σ yy −
σ zz . Considerable experimental effort is required to produce reliable normal stress
measurements in shear [Meissner (1972)]. Since shear contains both extensional and
rotational characteristics, the principle stretching direction rotates as the flow proceeds.
As a result, despite being a comparatively simple experiment, it can pose theoretical
difficulties.
Extensional flows are rotation free. They may be achieved by increasing the length
of a sample exponentially in time, producing a linearly increasing velocity profile. This
constant velocity gradient is the extension rate, ˙ɛ, and from this the Hencky extensional
strain, ɛ can be defined as ɛ = ∫ ˙ɛ(t ′ )dt ′ . The actual extensional strain is exp( ∫ ˙ɛdt).
Conventionally, extension is taken to occur in the x direction. To maintain a fixed volume
the two remaining directions can be allowed to contract equally, which is known
as uniaxial extension. Alternatively, one direction can be held fixed, forcing the final
direction to contract sufficiently to maintain the volume, which is called planar extension.
For both flows the first normal stress difference can be measured and in planar
extension there is also a second normal stress difference. These extensional flows, particularly
planar extension, are difficult experiments and the necessary equipment is only
available in a limited number of laboratories. Other extensional flows are feasible, such
as bi-axial flows, however data for such deformations are rare. The velocity gradient
and deformation gradient tensors for these simple flows are shown in table 1.1.
1.4.3 Flow in complex geometries
The aim of many constitutive equations is to produce reliable quantitative predictions
for as many of the above simple flows as possible, using the same parameters for each
flow. This is, of course, conditional on the availability of suitable data for comparison.
However, almost all flows which are of industrial relevance are complex flows. Many
complex flows will be a combination of shear and extensional deformations and so a
model which captures these simple flows might be expected to perform well for flows
under a complex geometry if a suitable numerical implementation can be found. However,
this conjecture is by no means a guarantee. For example, many complex flows