Biomedical Engineering Reference
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this model is summarized below. In this model, the polarity vector is assumed to
represent the velocity of the active particle; the background velocity is assumed
solenoidal, and inertia is neglected. Without external forces, the governing system
of equations consists of:
r
D
0;
r
D
0;
D
a
v
C
r
C
d
;
D
r
v
T
;
d
v
Cr
2
.
ph
r
D
C
hp
/
C
…
I
;
D
Wc.
x
;t/
pp
;
2
I
3
a
k
p
k
C
1
.
p
2
@
p
@t
C
1
2
.
r
v
r
p
v
/
p
r
/
p
C
2
.
r
p
/
p
C
3
rk
p
k
2
r
v
T
D
v
Cr
p
r
c
C
h
;
@c
@t
Cr
.c.
v
C
p
//
D
0;
(5)
where
1;2;3
;;…;W;; are model parameters. The sign of W determines the
nature of the elementary force dipoles. Here,
h
is the molecular field for the polar
vector
p
and is given by
h
D
c
˛
p
2
p
;
2
p
ˇ
k
p
k
C
K
r
(6)
where ˛ and ˇ are model parameters, and K is the analog of the Frank elastic
constant of the Ericksen-Leslie theory for liquid crystals in the one-constant approx-
imation [
23
]. The stress tensors
d
;
r
;
a
are the dissipative, reversible (or reactive)
and active stress, respectively. The reversible stress is due to the response to the
polar order gradient. The terms containing
1;2;3
and are the symmetry-allowed
polar contribution to the nematodynamics of
p
. The corresponding free energy for
the system is identified as
Z
c
2
˛
k
2
d
x
:
2
4
F
D
k
C
ˇ
k
k
C
K
kr
k
p
p
p
(7)
ıF
The molecular field is defined by
h
ı
p
. The moving polar particle velocity
and the background fluid flow velocity are fully coupled. With this model, Muhuri
et al. studied shear-induced isotropic to nematic phase transition of active filament
suspensions as a model of reorientation of endothelial cells. This model neglects the
impact of energy changes to migration of polar filaments.
D
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