Biomedical Engineering Reference
In-Depth Information
Figure 2.43
Evolutionofthevelocityproileattheentranceofatube.Outsidetheboundarylayer,
thelowieldpresentsnoshearing.
length where the boundary layer grows till it reaches the value
R
=
D
/2, we can use
Blasius' boundary layer correlation [47]
1
2
δ
-
@
5Re
x
(2.92)
x
where
δ
is the boundary layer thickness and
x
is the axial distance. With the en-
trance length
h
being the distance at which the two boundary layers merge,
h
is
obtained from (2.92) by setting
δ
=
R
=
D
/2
h
D
@
0.01 Re
D
(2.93)
Actually, this approximation does not take into account the acceleration of the
fluid in the core region. A more accurate expression has been obtained by Sparrow
and Schlichting [47]
h
D
@
0.04 Re
D
(2.94)
There is an important difference between macroscopic and microscopic flows.
At a macroscopic scale, the establishment length may be long, whereas it is very
short in microfluidics. An approximate entrance length for a 100
m
m radius tube
is of the order of 8
m
m for a Reynolds number of 1, which is quite small. It is of-
ten experimentally observed that during biochemical reactions in capillary tubes,
there is an anomalous region at the entrance of the tube, and the reason is observed
wrongly attributed to the flow pattern in the entrance region. It is much more likely
that there is an anomalous concentration of immobilized chemical species—or
tracers—in the entrance region which results in anomalous reaction or detection
(see Chapter 7).
2.2.11.2
Modeling
When modeling a flow in a microchannel, one wants usually wants to avoid the
problem of the establishment length, for example, if one wants to determine pre-
cisely the linear pressure drop in the channel. In such a case, it is sufficient to spec-
ify for inlet conditions the established (or nearly established) velocity field (Figure
2.44). For a cylindrical tube, the velocity field is given by (2.42).
