Biomedical Engineering Reference
In-Depth Information
The transport equation for the orientation tensor
Q
is proposed as following
d
Q
dt
C
a.1
C
/
D
3
W
Q
Q
W
aŒ
D
Q
C
Q
D
DC
2a
D
W
.
Q
C
.1
C
/
I
=6/.
Q
C
.1
C
/
I
=6/
C
H
C
2
Q
;
(72)
where
2
is an active parameter,
W
is the vorticity tensor,
D
is the rate of strain
tensor,
H
D
Œ
ı
ı
Q
tr.
ı
ı
Q
/.1
C
/
I
=6 is the so-called molecular field and f
n
is the
free energy density associated with the orientational dynamics given by
D
A
0
1
C
2
r
1
2
.1
N=3/
Q
Q
/
2
3
tr
Q
3
N
N
4
.
Q
f
n
W
Q
C
W
r
Q
.1
C
/
r
K
2
r
C
1
Q
:
:
:
r
C
C
f
anch
;
(73)
where r
D
1 is a positive integer, N is the dimensionless concentration, K is a
elastic constant, and f
anch
is the anchoring potential [
11
].
Elastic stress
The elastic stress is calculated by the virtual work principle [
27
]. Consider a virtual
deformation given by
E
Dr
v
ıt. The corresponding change in the free energy is
given by
ıf
D
@
@
Q
@t
ıt:
@t
ıt
H
W
(74)
The variation of ;
Q
are given, respectively, by
@
@t
Dr
.
v
/ıt;
ı
D
ı
Q
D
r
.
vQ
/
C
W
Q
Q
W
C
aŒ
D
Q
C
Q
D
C
.1
C
/
I
=6/
ıt: (75)
a.1
C
/
3
C
D
2a
D
W
.
Q
C
.1
C
/
I
=6/.
Q
So, the elastic stress is calculated as
F
e
D
r
./
Cr
.
H
ij
/
Q
ij
;
p
D
.
H
Q
Q
H
/
a.
H
.
Q
C
.1
C
/
I
=6/
C
.
Q
C
.1
C
/
I
=6/
H
/
C
2a.
Q
C
.1
C
/
I
=6/
W
H
.
Q
C
.1
C
/
I
=6/
Q
;
(76)
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