Biomedical Engineering Reference
In-Depth Information
9.4 Deterministic Trajectory
At a macroscopic scale, kinematics theory relates the mass acceleration of a body
to the resultant of the external forces that act upon it. This is the well-known
Newton's theorem.
�
�
dV
=
å
(9.11)
m
F
e
d t
�
�
the velocity, and
F
where
m
is the mass of the particle,
V
the external forces.
At a macroscopic scale, Brownian motion (random hit by other molecules) is
completely negligible. At a microscopic scale, the effects of the Brownian agita-
tion are more visible and Newton's formula should be replaced by Langevin's law
[10]
�
�
dV
å
(9.12)
m
=
F R t
+
( )
e
d t
where
R(t)
is a white noise due to the Brownian effect. However, it is often the case
that an “average” trajectory can be calculated simply by using Newton's equation,
especially if the size of the beads is larger than 1
μ
m or if the forces that act on the
particle dominate the Brownian motion. In such a case, because Brownian motion
can be considered as a white noise, the real positions of the beads are close to the
“average” trajectory, with a nearly symmetrical dispersion. This “average” trajec-
tory is often sufficient to predict the behavior of the microsystem and to design the
relevant component.
Usually, for microfluidics systems using magnetic beads, three types of forces
are present: gravity, hydrodynamic drag, and magnetic forces. In this case, Newton's
equation can be written under the form
�
�
� �
dV
p
(9.13)
m
=
F
+
F
+
F
mag
hyd
grav
d t
�
where
V
is the velocity of the particle. The hydrodynamic drag is derived from the
velocity field according to the equation
�
� �
� �
F
= -
C V V
(
-
)
= -
6
πη
r V V
(
-
)
(9.14)
hyd
D p
f
h
p
f
where
C
D
is the drag coefficient,
h
is the dynamic viscosity of the carrier fluid,
r
h
the hydrodynamic diameter of the particle,
and
V
f
the velocity of the carrier fluid.
It is assumed here that the velocity field of the carrier fluid is not affected by the
presence of the beads, which is the general case, except if the volume concentration
of the beads is important leading to aggregation of the beads. Under this assump-
tion, the velocity field of the carrier fluid must be calculated before attempting the
calculation of the particles trajectories, using classical hydrodynamics equations
(i.e., Navier-Stokes equations). A classical situation in microfluidics is the Poiseuille
flow between two plates or in a rounded capillary. In such a case, the velocity field
is obtained under a closed form.
