Chemistry Reference
In-Depth Information
Yasuda LC model is defi ned by Er
1 /
a
, which is the value of the Ericksen
number at which fl ow signifi cantly affects the orientation. The second transi-
tion between the power law shear thinning regime and the fl ow alignment
regime is
=
1
(
)
1
/
a
Er
ST-FA
=
c
an
/
(
−
1
)
−
1
(2.24)
τ
where
c
is of the order of 10
− 3
.
Backfl ow
This process is the opposite to orientation-driven fl ow (Rey, 2010).
Except for pure homogeneous twist reorientation, changes in the director
orientation
n
create fl ow. The reorientation viscosities associated with splay,
twist, and bend deformations (shown in Fig. 2.10) are defi ned by Rey (2007,
2009, 2010) and de Andrade Lima and Rey (2003c):
α
η
2
α
η
2
η
=
γ
η
=
γ
−
η
=
γ
−
(2.25)
twist
1
splay
1
bend
1
1
2
These transient reorientation viscosities are given by the rotational viscosity
(
γ
1
) decreased by a factor introduced by the backfl ow effect. The general
expression for the reorientation viscosities can be rewritten in a more reveal-
ing general form (de Andrade Lima and Rey, 2003c; Rey, 2000a-c, 2007, 2009,
2010 ):
=−
(
)
2
TC
i
(2.26)
ηγ
α
1
η
i
where
η
i
denotes the corresponding Miesowicz viscosity and TC
i
the corre-
sponding torque coeffi cient. Since twist is the only mode that creates no
backfl ow (Rey, 2010), then
γ
1
. For a bend distortion the backfl ow is
normal to
n
, and hence the torque coeffi cient is
η
twist
=
α
2
, and the Miesowicz viscosity
is
η
2
. On the other hand for a splay distortion, the backfl ow is parallel to
n
and
hence the torque coeffi cient is
α
3
, and the Miesowicz viscosity is
η
1
. The order-
ing in the reorientation viscosities is
η
>
η
η
>
η
twist
splay
twist
bend
Secondary Flows
Whenever the director deviates from the shear plane, a
transverse fl ow will be generated since the viscosity tensor
C
ijkl
in the extra
stress tensor [see Eq. (2.27)] is a function of the director (Rey, 2007, 2009, 2010;
Stewart, 2004 ):
TCADN
ij
extra
=
+
(2.27)
ijkl
lk
ijk
k
Search WWH ::
Custom Search