Biomedical Engineering Reference
In-Depth Information
Figure 6.71
Sketch of a spiral microchannel.
We shall distinguish two cases: low velocities, with negligible inertia forces, and
larger velocities where inertial effects and lift forces intervene.
6.5.6.1 Bifurcation Channels at Low Flow Rates
At low low rates, the assumption holds that a cell/spherical particle follows the
streamline passing by its centroid [39, 40]. Let us consider spherical particles fo-
cused near a wall and approaching a bifurcation (Figure 6.74).
For simplicity, let us assume a 2D situation and neglect the effect of the channel
depth. The 3D calculation is similar, using the velocity expression given in Chapter
2. The velocity field can be approximated by the Poiseuille-Hagen quadratic profile
y w y
(
-
)
u y
( ) 6
=
U
(6.122)
2
w
where
U
is the mean axial velocity, related to the flow rate by
U
+
Q
/(
dw
). At the
bifurcation, the flow rate conservation equation requires that
w
1
*
ò
(6.123)
d
u y dy Q
( )
=
1
0
Figure 6.72
The cells, initially dispersed in the channel, progressively concentrate under the effect
of the Dean flow. (a) Initially the particles are dispersed in the channel, (b) lift forces impose four
equilibrium positions, and (c) the Dean vortex leaves only one equilibrium position.
