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

52 CHAPTER 3. THE POM-POM MODEL IN EXPONENTIAL SHEAR.
S xy
0.25
0.2
0.15
τ b
[sec]
3
1
0.3
0.1
0.03
0.01
0.1
0.05
0
0 2 4 6 8
time [sec]
Figure 3.2: Evolution of S xy for nearly exponential shear with varying τ b values. α =
1sec −1 .
the orientation behaves essentially as for an affine deformation since the exponentially
growing shear rate rapidly becomes very large compared to τ b . In this limit S xy is very
well approximated by the affine orientation solution
S xy (t) =
e α t − 1
4 + e 2 α t − 2 e α t . (3.5)
This expression is obtained by taking the τ b → ∞ limit of equation 3.4 or by solving
the orientation equation with the relaxation terms removed.
In the opposite limit where 1/τ b ≫ α the expression for S xy can be simplified to
S xy (t) =
ατ b e αt
2α 2 τ 2 b e2αt + 3 . (3.6)
In this limit the initial shear rate is too small to produce any appreciable orientation.
The backbone section must “wait” until the instantaneous shear rate is comparable to
the reciprocal of its orientation time. Eventually the shear rate will become sufficiently
large to orientate the backbone section and so S xy grows, peaks and then tends to zero
at long times. In this limit the solutions for different values of τ b and α are related by
S xy
( t
α 1
, τ b1 , α 1
)
= S xy
( t + ∆t
α 2
, τ b2 , α 2
)
where ∆t is a time shift factor given by
∀t. (3.7)
( )
τb1 α 1
∆t = ln . (3.8)
τ b2 α 2