Biomedical Engineering Reference
In-Depth Information
when the polymer chain starts stretching. The relaxation time usually depends on
the concentration according to a power law
[ ]
)
p
(
τ
»
c
η
(2.30)
where the exponent
p
depends on the liquid.
2.2.2.6
Non-NewtonianMicrolows
Physical Behavior
The flow field of viscoelastic liquids is sometimes surprising and very different from
that of Newtonian liquids. Some striking examples, at the macroscopic scale, are
shown in [16].
At the microscopic scale, there are many examples of shear thinning behavior.
Friction on the walls of a channel is very important due to the small dimensions
of the channel cross section. Even if the average velocity is small, shear rate can be
large due to the vicinity of the walls. A typical example is that of blood flowing in
the human body. It has been shown that streamlines at the entrance of a constriction
can be very different for non-Newtonian fluids if the viscoelastic Mach number is
larger than 1. Fluidic diodes can be made using visco-elastic liquids [18]. Finally,
encapsulation of biologic object in gelling polymers is conditioned by the visco-
elastic behavior of the encapsulating liquid [19]. Hence, it is important to be able to
predict the behavior of visco-elastic fluids in microsystems. In this section, we give
some insights for the computation of visco-elasticity liquid shear flows.
Modeling Non-Newtonian Microflows
Liquid flows are characterized by their deformation tensor
D
(or rate-of-deformation
or rate-of-strain tensor)
1
2
(
)
T
D
=
Ñ + Ñ
V
V
(2.31)
where
V
is the velocity vector. One defines the shear rate associated to a fluid de-
formation by
γ
=
(2.32)
Let us recall that the 1D shear rate corresponding to a flow along a planar solid
surface is
�
2
D D
=
∂
V
y
//
�
γ
(2.33)
where
V
//
is the fluid velocity close to the wall (and parallel to it), and
y
is the normal
distance. For a 2D Cartesian coordinate system, the expression of the shear rate is
2
2
2
æ
u
ö
æ
u
v
ö
æ
v
ö
∂
∂ ∂
∂
�
γ
=
2
+
+
+
2
(2.34)
ç
÷
ç
÷
ç
÷
∂
x
∂ ∂
y
x
∂
y
è
ø
è
ø
è
ø
