Biomedical Engineering Reference
In-Depth Information
Figure 2.54
RecirculationconditionsasfunctionsofthechannelReynoldsnumberandthegating
ratio.
been shown that such flows in curved tubes have spiral streamlines in the curved
regions caused by a centrifugal effect. We shall see in Chapter 6 that these spiral-
ing streamlines are useful to guide and concentrate particles or cells [57], and to
enhance mixing [58].
Let us consider the example of Figure 2.55. The main component of the veloc-
ity is directed along the direction of the tube, but the transverse component of the
velocity (here
v
y
), which is zero at low Reynolds number, is positive in two quad-
rangles and negative in the two others. Hence, there is a recirculating component of
the velocity that induces a spiral flow.
If we define the Dean number by
De U R
=
ν
R R
=
Re
R R
(2.97)
c
c
where
R
c
is the curvature radius of the channel. The inertia-induced spiral motion
is noticeable when the Dean number is larger than 1.
Another way of pinpointing this recirculation motion is by plotting the
z
-
vorticity contour, as shown in Figure 2.56.
For a cylindrical, curved channel such as that of Figure 2.55, the rotational
velocity components can be analytically computed [58]. Dean showed that, in a
toroidal coordinate system (Figure 2.57), secondary flow velocities are given by the
following equations