Biomedical Engineering Reference
In-Depth Information
“viscous” stress
b
is added analogous to the viscous stress
d
. The free energy
contains additional coupling terms between the polar velocity and the concentration
gradient.
This model is used to study sheared active polar fluids. An extremely rich variety
of phenomena are identified including an effective reduction or increase in the
apparent viscosity, depending on the nature of the active stresses and flow alignment
property of the particles, nonmonotone stress-strain-rate relationship, and yield
stress for large active forcing [
44
]. In the limit of strongly polarized states where
the magnitude of
p
is locked, this formulation can be recast in terms of a unit vector
u
p
k
p
k
D
: The details can be found in [
44
].
2.2
Active Apolar Filament Models
When the polarity on the moving objects is weak, instead of the polarity vector, a
second order nematic tensor can be employed to describe both the nematic order as
well as the active stress. For apolar filament fluids, a coarse-grained model can be
derived with only the nematic order tensor [
15
,
46
]. We summarize the version used
by Cates et al. [
15
] in this section. Let
Q
be a traceless second order tensor denoting
the nematic order in the active filament fluid. The governing system of equations
consist of the following equations.
r
v
D
0;
@
v
@t
C
v
r
Dr
./;
3
Tr
ıF
I
;
ıF
ı
Q
C
1
H
D
ı
Q
C
2
Q
Q
H
Q
Q
3
I
H
I
3
I
3
1
D
P
0
I
C
2
D
C
W
H
C
C
ıF
ı
r
r
Q
W
Q
C
Q
H
H
Q
Q
;
@
@t
C
Q
v
r
Q
C
Q
Œ
D
Q
C
Q
D
D
H
;
(10)
where c is the concentration of the apolar active rod assumed constant in this model,
is the friction coefficient, P
0
is the hydrostatic pressure, is the activity parameter
with >0corresponding to extensile and <0contractile motion. The free energy
density of the material system is given by a simplified Landau-deGennes functional
"
1
Q
2
#
Q
Q
:
:
:
r
N
3
Q
2
W
N
3
N
4
.
Q
K
2
Q
3
Q
/
2
F
D
k
B
Tc
C
W
C
r
; (11)
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