Biomedical Engineering Reference
In-Depth Information
6.2.9 Taylor-Aris Approach
6.2.9.1 Taylor-Aris Model
At a time when flow velocimetry methods were developing, it was found that the
measurement of flow velocity inside tubes may be achieved using small particles.
The principle was to introduce a radioactive substance or an electrolyte in the flow
at a certain cross section and to follow its translation inside the pipe. However, it
appeared immediately that the substance introduced into the stream diffused at the
same time as it moved with the fluid. A model was then developed by Taylor [5] and
completed by Aris to take into account in a simple manner both diffusion and ad-
vection. This model, although it was born in the 1950s, has found a recent renewal
in biotechnology where advection and diffusion of particles and macromolecules
carried by a fluid are an everyday concern.
The basic idea behind Taylor's approach is that it is not possible to superpose
advection and diffusion for a liquid flowing in a pipe under the action of pressure
gradient. It is not correct to assume a mere translation where the particles would
move as a whole with the fluid and diffuse with their usual diffusion coefficient.
In a Lagrangian coordinate system moving at the average velocity of the fluid, the
particles diffuse much more than it is predicted by the usual diffusion theory. As
will be seen, this is due to the radial gradient of velocity in the flow. The strength
of Taylor's approach is to show that a superposition may be done by allowing the
particles to move at the average velocity of the fluid and using an effective or appar-
ent diffusion coefficient for the particle—larger than the usual coefficient derived
from the Einstein's formula.
Suppose a capillary tube of radius
R
in which a liquid flows at a mean velocity
U
carrying particles with a concentration
c
(Figure 6.15).
The advection-diffusion equation written under an axisymmetric (
x
,
r
) form is
2
2
é
ù
¶
c
¶
c
¶
c
1
¶
c
¶
c
+
u r
( )
=
D
+
+
(6.47)
ê
ú
2
2
¶
t
¶
x
r
¶
r
¶
x
¶
r
ë
û
The local velocity
u
(
r
) is given by the Hagen-Poiseuille solution
æ
2
2
ö
r
u r
( ) 2
=
U
1
-
(6.48)
ç
÷
R
è
ø
Note that the velocity is zero at the wall and is maximum and equal to 2
U
on the
central axis. The term
¶
¶
2
c
x
in (6.47) may be omitted if it is assumed that the axial
change in concentration is much less than the radial change. After substituting
(6.48) in (6.47), we obtain
2
Figure 6.15
Schematic view of the cylindrical capillary and the diffusion front.
