Biomedical Engineering Reference
In-Depth Information
6.3.2 Ballistic Random Walk (BRW)
6.3.2.1 Model
In this section, we combine particles entrainment by the flow with Brownian mo-
tion. This approach is sometimes called the
ballistic random walk
(BRW) method.
We have seen in Chapter 5 that discrete models such as the Monte Carlo model
are interesting because they bring new insight to the understanding of the effect
of Brownian motion. With this in mind, a similar approach may be done for mic-
roparticles transport. The behavior of the buffer (or carrier) fluid is still obtained
by solving the Navier-Stokes equations. If the entrainment of the microparticles is
strong enough, one can assume that the transported microparticles are following
trajectories slightly modified by the effect of Brownian motion (Figure 6.38)
The real force balance on a particle is given by Langevin's equation [16]
�
� � �
dV
c
m
=
C V V
-
+
F t
(6.100)
(
)
( )
e
D
f
c
dt
where the function
F
(
t
) represents the Brownian forces. Although this is not strictly
correct, we approximate (6.100) by the superposition of a deterministic trajectory,
modified by the effect of the Brownian motion modeled by a Monte Carlo method.
This is approximately correct if the particle trajectory is not too much affected by
Brownian motion (i.e., if the entrainment is strong in front of the Brownian motion).
In this example, for simplicity we do not take into account gravity force. In
such a case,
æ
C
ö
-
t
(6.101)
V
=
V
1
-
e
m
ç
÷
p
f
ç
÷
è
ø
According to (6.101), the velocity of an extremely small particle is that of the car-
rier fluid. Now we account for the Brownian motion by introducing the relations
Figure 6.38
Sketch of the superposition of advection by the buffer fluid and Brownian motion.
