Biomedical Engineering Reference
In-Depth Information
At the molecular level, viscoelastic fluids can be described by two models: network model and
single-molecule model
[2]
. The network model is based on the formation and rupture of junctions
between polymer molecules. The network model is suitable for solutions with high polymer
concentration. Dilute solutions are better described with a single-molecule model where inter-
actions between the molecules are not frequent. The polymer molecule is represented by
a “dumb-bell” or “bead-string” model where two spheres are connected by a spring. Kinetic
theory with Stokes drag theory and Brownian motion can be used with this model for deriving
macroscopic properties.
Fluids with large molecules display elastic behavior due to the stretching and coiling of the
polymer chain. Here, these fluids and their behaviors are called viscoelastic fluids and viscoelastic
effects respectively. The most apparent viscoelastic effect is the change of velocity profile in a chan-
nel. The dimensionless velocity profile of a viscoelastic fluid in a circular capillary can be approxi-
mated as
[2]
:
r
r
0
1
þ
1
n
u
u
max
z
u
¼
1
(2.122)
where 0
<
n
<
1 is a parameter unique for the fluid and
r
0
is the radius of the capillary. If
n
¼
1, the
fluid becomes Newtonian and the velocity profile is parabolic.
The next viscoelastic effect, which is relevant to mixing in microscale, is the entry flow at
a contraction. The operation point of a viscoelastic flow is represented by theWi-Re diagram, whereWi
is the Weissen
b
erg number and Re is the Reynolds number. With a characteristic length scale
L
c
, the
mean velocity
u
, the density
r
, and the zero-stress viscosity
m
0
, the Reynolds number is defined here as:
ruL
c
m
0
:
Re
¼
(2.123)
The Weissenberg number represents the elastic character of the fluid by using the ratio between the
relaxation time
l
of the fluid and the characteristic residence time
s
flow
:
l
s
flow
:
Wi
¼
(2.124)
_
The characteristic residence time is the inverse value of the characteristic shear rate
g
and is
defined as:
1
g
¼
L
c
s
flow
¼
u
:
(2.125)
Because both Reynolds number and Weissenberg number are proportional to the average velocity
u
, it is useful to define the elasticity number, which is independent of the flow velocity:
Elastic effect
Inertial effect
¼
Wi
Re
¼
lm
0
E1
¼
rL
c
:
(2.126)
Elasticity number represents the importance of elastic effect over the inertial effect. For the small
Reynolds number in microfluidics, the inertial effect is negligible. However, if the elasticity number of
the fluid is large enough, elastic effect may also be large enough to compete with the dominant viscous
effect.
Figure 2.28
shows the typical operation region of a possible micromixer based on viscoelastic
instabilities of a 4-to-1 contraction. Shear-thinning viscoelastic fluids are, for instance, concentrated
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