Biomedical Engineering Reference
In-Depth Information
taking alginates as an example, the Carreau-Yasuda relation fit well the experimen-
tal measurements as shown in Figure 2.13.
Another relation sometimes used for visco-elastic liquids is the Rabinowitch-
Ellis relation [14]
αη
0
η
=
(2.29)
(
)
β
�
1
+
τγ
where
α
and
β
are again some constants to be determined.
Relaxation Time
Due to the chains of polymers, droplets of alginate solutions show elasticity. The
physicist Weissenberg used to present the image of some line tension—like an elastic
filament—that brings elasticity to the flow streamlines. The elasticity of an alginate
solution is characterized by a relaxation time
τ
[15]. Different characteristic times
can be defined for polymers depending on the complexity of the polymeric structure
and the forcing flow (shear or elongational flow) [16, 17].
In the case of a shear flow, the relaxation time appears in the Carreau-Yasuda
and Rabinowitch-Ellis models. As a matter of fact, the term
.
τγ
is the Weissenberg
number. A fit of the experimental data by the Carreau-Yasuda law produces
τ
. From
a physical point of view, the relaxation time is approximately the inverse of the
critical shear rate
τ γ
»
1
cr
, and the critical shear rate corresponds to the situation
Figure 2.13
ViscosityofKeltoneHValginatesolutionsversusshearrate:thedotscorrespondtothe
experimentalresults,andthecontinuouslinecorrespondstotheCarreau-Yasudamodel.Thefour
curvescorrespondtofouralginateconcentrations:1,1.25,1.5,and1.75wt%.Alginateviscosity
increaseswiththeconcentrationanddecreaseswiththeshearrate.Therelaxationtimesarededuced
fromaitof(2.28)onthedifferentexperimentalcurves.
