Biomedical Engineering Reference
In-Depth Information
If the starting velocity of the particle is zero, we obtain a simple relation be-
tween the coordinates of the particle
d x
c
P
1
2
=
u
(6.95)
f
d r
c
P
where the ratio
c
1
/
c
2
is
c
C
1
2
D
=
c
g m
D
Assuming a Hagen-Poiseuille flow in the duct, (6.95) becomes
2
æ
ö
r
d x
V
c
P
0
1
P
=
1
-
(6.96)
ç
÷
2
d r
2
c
R
è
ø
P
2
Integration of this relation gives the relation
3
3
é
ù
r
r
V c
p
p
,0
0
1
x
=
r
-
r
+
-
(6.97)
ê
ú
p
p
,0
2
2
2
c
3
R
3
R
ê
ú
2
ë
û
where (0,
r
p
0
) is the starting location of the particle. A particle starting from the
middle of the duct (
r
p
0
= 0) will contact the wall at an axial distance of
C
V
0
3
L
R
=
(6.98)
g m
D
It is interesting to compare this result (6.98) with (6.68). The two results are
quite different. In the present case, we calculate the distance of travel of a particle
submitted to hydrodynamic drag considering that the particle is supposed suffi-
ciently large (or heavy) to neglect the Brownian motion in front of the gravity force.
In the previous case, we calculated the same distance for a particle submitted to
hydrodynamic drag force but small enough to neglect gravity in front of Brownian
motion. The ratio of the two calculated lengths is
L
3
R m
D
3
R
3
R
diff
grav
=
=
g m
D
=
L
(6.99)
L
2
D C
2
k T
2
D
B
B
where
L
B
is the Bolzman length.
Interestingly, the average buffer fluid velocity has disappeared from (6.99),
which is just a balance between gravity forces and Brownian motion energy (
k
B
T
).
Depending on the relative particle mass D
m
, the travel distance will be either the
gravity model distance defined by (6.98) or the diffusion model distance given by
relation (6.68).
Note that it is very seldom that the trajectory equation can be solved analyti-
cally, such as in this case, but it is always interesting to spend some time investigat-
ing if an analytical solution may exist—even to the price of some simplification
(initial velocities set to zero). Most of the time, a numerical approach is required.
Different methods such as Runge Kutta or predictor-corrector can be used. We give
an example of the predictor-corrector scheme in Chapter 9.
