Biomedical Engineering Reference
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Figure  6.70  (a) Particles dispersed in the flow submitted to lift forces and (b) particles at
equilibrium are focused at a fixed distance from the wall.
6.5.4.3 Farhaeus Effect
At this point, we mention that the Farhaeus-Lindqvist effect should not be attrib-
uted to lift forces. The Farhaeus effect denotes the property of blood cells to move
away from the walls [35]. This effect is linked to the non-Newtonian behavior of
blood (as we have discussed in Chapter 2). With the viscosity being smaller at the
walls due to the shear rate, the more liquid plasma circulates preferentially along
the wall, pushing the cells towards the channel center.
6.5.5  Dean Flows in Curved Microchannels
The hydrodynamics of a Dean flow was presented in Chapter 2. In this section, we
show what use can be made of Dean flows for the focusing of particles and cells [36,
37]. We recall here that the Dean effect is a vortex effect in curved microchannels.
This rotational effect appears when fluid inertia is sufficient and curvature is large.
It is characterized by the nondimensional Dean number defined as
De U R
=
ν
R R
= Re
R R
(6.121)
c
c
where R c is the curvature of the microchannel and R is its hydraulic diameter. A
Dean number in the range of 0.1-1 realizes the rotational effect. Let us investigate
how a Dean flow acts on particles and cells. Consider a spiral microchannel as
shown in Figure 6.71 [38]. As we saw in Chapter 2, the effect of the curvature on
the flow is the formation of two vortex tubes in the channel. Let us assume that the
particles or cells are neutrally buoyant. Three forces are exerted by the flow on the
cells (Figure 6.72): (1) hydrodynamic drag that contributes to transport the cells
from the inlet to the outlet, (2) lift forces that tend to bring together the cells in four
equilibrium positions (in a cylindrical tube, we have seen that lift forces maintain
the particles on a tube a some distance of the walls), and (3) the Dean vortex that
reduces the equilibrium positions to only one near the inner wall.
6.5.6  Bifurcation Channels
In this section we investigate the behavior of cells transported in microchannel net-
works, especially in a “branched” geometry such as the one schematized in Figure
6.73.
 
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