Biomedical Engineering Reference
In-Depth Information
Figure 6.37
Mixing of two microdrops confined by electrodes in an EWOD microsystem.
(Courtesy of CEA/LETI.)
reduced. A first step in this approach is to calculate the trajectories of the particles
in a deterministic (i.e., without taking account of the Brownian motion).
6.3.1 Trajectories of Particles in a Microflow
Larger particles experience less diffusion under the action of Brownian motion. In
such a case, these particles follow have trajectories determined by the forces acting
upon them. At a macroscopic scale, the kinematics theory relates the mass accelera-
tion of a body to the resultant of the external forces that act upon it. This is the
well-known Newton's theorem.
�
�
dV
p
=
å
m
F
(6.85)
e
d t
�
F
are the external forces.
We will treat here the case of particles submitted to gravity force and hydrodynamic
drag force. Newton's equation can then be written under the form
�
V
is the velocity, and
where
m
is the mass of the particle,
�
�
�
dV
p
m
=
F
+
F
(6.86)
hyd
grav
d t
The hydrodynamic drag is derived from the velocity field according to the
equation
�
� �
� �
F
= -
C V V
(
-
)
= -
6
πη
r V V
(
-
)
(6.87)
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
is
the hydrodynamic diameter of the particle, and
V
f
is the velocity of the carrier fluid.
It is assumed here that the velocity field of the carrier fluid is not affected by the pres-
ence of the particles, which is the general case, except if the volume fraction of parti-
cles is important, leading to the formation of aggregates. Under this assumption, 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 typical situation in microfluidics is the Hagen-Poiseuille flow
between two plates or in a rounded capillary. The gravity term is given by
