89
4
Then,
η
π
=
( )
R
P
Ql
4
32
∆
(4.28)
Equation 4.28 can be rearranged as
η
π
=
R P
L
Q
R
2
32
3
∆
(4.29)
Using force balance in a circular pipe, it can be obtained that shear
stress at the wall is given b,
τ at wall of tube
(
)=
R P
L
2
∆
(4.30)
Hence, the shear rate is
γ π
=
32
3
Q
R
(4.31)
As the fluid considered for this analysis is Newtonian, the shear
rate and viscosity are called the apparent shear rate and apparent
viscosity.
4.1.2.2 Shear Free Flow
It has been discussed previously that the rheological properties of
polymeric matrix in composites have a direct impact on many
manufacturing processes. Shear free flows are also known as elongational flows or stretching. They are a specific characteristic of
polymeric matrices, and polymer melts due to their viscoelastic
behaviour. This is the flow in which it is possible to select elongational stresses for every fluid element and orthogonal set of unit
vectors (shear stress) so that the strain rate also has diagonal components. These flows have no non-zero, off-diagonal components
in shear stress (τ), and the shear rate (γ) in elongational flow is
shown as
τ
τ
τ
γ
γ
γ
11
22
33
11
22
33
0
0
0
0
0
0
0
0
0
0
0
0
.
(4.32)
In elongation flows of incompressible fluids such as polymer matrix
and polymer melts, only ( τ 33 − τ 11 ) and ( τ 22 − τ 11 ) can be measured. The elongational flow is shown schematically in . Fig. 4.5.
The flows occurring during these processes are complicated and
will usually involve some combination of planar Couette flow
(PCF), planar elongational flow (PEF), and biaxial/uniaxial stretching flow (BSF/USF). PCF is the laminar flow in the space between
4.1 · Fundamentals of Rheology
4
Then,
η
π
=
( )
R
P
Ql
4
32
∆
(4.28)
Equation 4.28 can be rearranged as
η
π
=
R P
L
Q
R
2
32
3
∆
(4.29)
Using force balance in a circular pipe, it can be obtained that shear
stress at the wall is given b,
τ at wall of tube
(
)=
R P
L
2
∆
(4.30)
Hence, the shear rate is
γ π
=
32
3
Q
R
(4.31)
As the fluid considered for this analysis is Newtonian, the shear
rate and viscosity are called the apparent shear rate and apparent
viscosity.
4.1.2.2 Shear Free Flow
It has been discussed previously that the rheological properties of
polymeric matrix in composites have a direct impact on many
manufacturing processes. Shear free flows are also known as elongational flows or stretching. They are a specific characteristic of
polymeric matrices, and polymer melts due to their viscoelastic
behaviour. This is the flow in which it is possible to select elongational stresses for every fluid element and orthogonal set of unit
vectors (shear stress) so that the strain rate also has diagonal components. These flows have no non-zero, off-diagonal components
in shear stress (τ), and the shear rate (γ) in elongational flow is
shown as
τ
τ
τ
γ
γ
γ
11
22
33
11
22
33
0
0
0
0
0
0
0
0
0
0
0
0
.
(4.32)
In elongation flows of incompressible fluids such as polymer matrix
and polymer melts, only ( τ 33 − τ 11 ) and ( τ 22 − τ 11 ) can be measured. The elongational flow is shown schematically in . Fig. 4.5.
The flows occurring during these processes are complicated and
will usually involve some combination of planar Couette flow
(PCF), planar elongational flow (PEF), and biaxial/uniaxial stretching flow (BSF/USF). PCF is the laminar flow in the space between
4.1 · Fundamentals of Rheology