Biomedical Engineering Reference
In-Depth Information
where k is the consistency index and n is the flow behavior index. To maintain the sign
convention for shear stress, this expression may also be written as
n
2
1
k
@
v
x
@
y
1
@
v
y
@
@
v
x
@
y
1
@
v
y
@
@
v
x
@
y
1
@
v
y
@
τ
5
5η
ð
2
:
35
Þ
xy
x
x
x
where
is the apparent viscosity of the fluid. For Newtonian fluids, the apparent viscosity
is the same as the dynamic viscosity because n is equal to 1. Pseudoplastic fluids have an
n
η
1. One more classification of fluids is a Bingham plastic
fluid. These types of fluids do not flow when small stresses are applied on them. Some
threshold shear stress level must be applied to the fluid to get the fluid to begin to flow.
An example of this type of fluid is toothpaste. When in the tube, small forces will not
cause the toothpaste to flow out of the tube (e.g., gravity when the tube is held upside
down). A certain yield stress (
,
1 and dilatant fluids have an n
.
τ
y
) must be applied to the tube for the toothpaste to come
out. In the case of toothpaste, the yield stress must exceed the atmospheric pressure.
Bingham plastic fluids are modeled as
@
v
x
@
y
1
@
v
y
@
τ
xy
5τ
y
1μ
ð
2
:
36
Þ
x
where
τ
y
is the yield shear stress.
Blood is a Non-Newtonian fluid that is a combination of a pseudoplastic fluid and a
Bingham plastic fluid. This means that blood must surpass an initial yield stress to begin
to flow and that, as the shear rate increases, once blood is flowing, the viscosity decreases.
Through numerical and experimental studies, it has been shown that using the assump-
tion that blood behaves as a Newtonian fluid overestimates actual shear stress values. This
is significant because, as we will see in later chapters, shear stress is a significant mediator
of cardiovascular diseases. Therefore, an accurate prediction of shear stress must be made
by using the appropriate fluid relationships for blood's properties.
Another classification can be made for fluids that exhibit a time-dependent change in
viscosity. If the apparent viscosity decreases under a constant shear stress, this fluid is
thixotropic. Similar, rheopectic fluids have a time-dependent increase in apparent viscos-
ity, under a constant shear stress. The relationship between shear stress and shear rate for
these fluids cannot be plotted on
Figure 2.20
because the viscosity is also dependent on
time, which is not a variable on this figure.
Inviscid fluids have a zero (or negligible viscosity). While there is no such fluid that
exhibits these properties, some fluid mechanics analysis can be done using this assump-
tion and the obtained results are still meaningful (e.g., gas flows do not typically involve
viscosity). In most of the following examples in this textbook, we will use the no-slip
boundary assumption. This means that the fluid at a wall must have the same velocity as
the wall. (Refer to the previous parallel plate example in this textbook.) In most cases,
when the wall is stationary, the first lamina of fluid along the wall will have a zero veloc-
ity. In some cases, the wall can be moving and that first lamina will match this velocity.
However, this only holds for viscous fluids. Inviscid fluids do not resist flow internally
and, therefore, all of the fluid layers must have the same velocity, regardless of the
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