Biomedical Engineering Reference
In-Depth Information
Figure 2.14
ComparisonbetweenNewtonianandnon-Newtonianlows(withthesamezero-shear
viscosity):thesamelowrateisimposedinthetwochannels.Thepressuredropismuchsmallerin
thenon-Newtonianluid.Notethedifferencesinthevelocityproile(COMSOLcalculation).
where
u
and
v
are the
x
and
y
components of the velocity. This value of the shear
rate can be introduced in the Carreau-Yasuda law to produce the shear viscosity,
which replaces the Newtonian viscosity in the Navier-Stokes equations.
Examples
Non-Newtonian Microflow in a Microchannel
Let us compare the flow in a
straight microchannel of two liquids, the first one Newtonian and the other non-
Newtonian. Let us consider that the Newtonian fluid has a constant viscosity
η
0
=
2.63 Pa.s and the non-Newtonian fluid has a shear thinning viscosity defined by a
Carreau-Yasuda law
.
1.2
-
0.41
= +
. If two identical channels have the
same imposed inlet velocity, the pressure contour plots are very different in the two
cases, as shown in Figure 2.14. The pressure drop is much smaller for the visco-
elastic fluid because the viscosity at the wall is much smaller at the wall due to the
shear. A viscoelastic fluid can be gel-like at rest (zero velocity), and suddenly flow if
a large driving pressure is applied; once the fluid is in motion, the driving pressure
is substantially reduced.
η η
0
[(1 (0.047 )
γ
]
Non-Newtonian Microflow in a Microchannel with a Constriction
Consider a
microchannel with a constriction (Figure 2.15), like a microscopic Venturi [20].
Due to mass conservation, the velocity is larger in the narrow cross section. Because
of the small transverse dimension of the constriction, the shear rate is important
and the viscosity is small at the wall. These two effects add, and the velocity of the
non-Newtonian fluid is larger than that of the Newtonian fluid in the constriction.
Figure 2.15
(a)Viscositycontourplotinaconstrictionfor1.25%KeltoneHValginatelowingatan
averagevelocityof1mm/s.(b)Ratiooftheelongationratetotheshearrate.Polymersstretchinthe
convergingregionandintheconstrictionandrelaxinthedivergenceregion.(COMSOLcalculation).
