Biomedical Engineering Reference
In-Depth Information
Figure 2.23
Modeling a luidic “diode”: stream lines in dissymmetrical convergents (COMSOL
software).Thelowislaminarbutwithrecirculatingregions.
or 31 Pa depending on the direction of the flow. Thus, such a design functions as a
diode for flow velocities larger than approximately 1 mm/s.
Besides linearity and reversibility, the Stokes equation has a unique solution,
meaning that there cannot be bifurcations of the solution linked to the development
of instabilities.
It is worth noticing that care must be taken when deciding to apply Stokes' sim-
plification. The condition
Re
<< 1 must be verified everywhere in the fluid domain.
If there is only one location, however small, where this condition is not realized,
then the simplification may not be valid and will bias the whole solution.
2.2.5 Hagen-Poiseuille Flow
In practice, it is common to deal with cylindrical tubes or rectangular microchan-
nels of different aspect ratios. In these cases, when the flow is laminar, there exists
an analytical exact solution—for the cylindrical duct and the parallel plates—and
an approximated solution—for the rectangular duct [2]. This solution is of inter-
est because it simplifies considerably the PDE system governing the convection of
particles (Chapter 6).
2.2.5.1
CylindricalTube
Navier-Stokes equations can be solved analytically in the particular case of a cylin-
drical duct. This solution is classical [2, 25] and we indicate here only the result.
The velocity in the axial
z
-direction is given by
é
2
ù
r
æ
ö
( )
u r
=
2
U
ê
1
-
ú
(2.42)
ç
÷
R
è
ø
ê
ú
ë
û
where
U
is the average velocity. Equation (2.42) shows that the flow profile is
parabolic and the same in any cross section. The mean velocity
U
is the averaged
velocity in a cross section
R
1
ò
U
=
u r
2
π
r dr
( )
2
R
π
0
