Biomedical Engineering Reference
In-Depth Information
D
Tr./
3
.
We denote
Q
DQ
a
C
d
;
Q
D
2k
B
T
a
c
1
Q
pp
I
C
C
1
c
4
3
I
c
c
I
N
c
c
I
N
k
p
k
2
k
p
k
2
Q
a
Q
C
pp
2
2
C
C
2
c
4
r
p
r
p
T
;
r
p
2
1
r
p
I
D
C
3
1
2
D
2
Tr.
D
/
D
/
; (20)
I
1
3
.
Q
Tr.
D
/
.
D
1
3
.
D
Q
d
C
W
Q
/
I
/
C
Q
C
Q
where C
1
;C
2
;C
3
are model parameters [
72
].
This together with the continuity equation for the average velocity
v
and the
momentum balance equation in the form of Stokes equation constitute the governing
system of equations for the kinetic theory.
r
v
D
0;
r
.
C
2
D
P
0
I
/
D
0;
(21)
where is the viscosity of the solvent and P
0
is the hydrostatic pressure.
Shelley and Santillian studied dilute active rod particle fluids using a kinetic
theory in which only convective transport is accounted for [
91
];
v
0
u
c;
J
R
J
D
v
c
C
D
c
!
:
(22)
In their model, the active stress tensor is given by
Q
a
D
Q
:
(23)
Next, we present one of the latest versions of the kinetic theory in which the
active flux due to rod-rod binary collisions is carefully considered. Baskaran and
Marchetti derived a Smoluchowski equation for self-propelled hard rods in 2-D [
9
].
@c
@t
Cr
J
R
J
C
R
D
0;
D
jj
m
v
0
2k
B
T
1
k
B
T
D
c
r
V
ex
v
0
u
c
D
SP
I
SP
;
J
D
c
v
C
r
c
D
c!
D
r
k
B
T
R
V
ex
D
r
m
v
0
2k
B
T
c
J
R
I
S
r
;
R
c
C
(24)
where
J
is the translational flux and
J
R
is the rotational flux,
D
SP
D
D
?
I
C
.D
jj
C
D
S
D
?
/
uu
(25)
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