Biomedical Engineering Reference
In-Depth Information
Let us develop how to quantify stress on a surface. Any flowing surface is in contact
with either another flowing surface (termed a “lamina of fluid”) or the bounding surface
of the fluid volume. The movement of this fluid surface will generate a force on the neigh-
boring surface (either fluid or boundary). The force that is generated on the two laminae
of fluid must be in balance with each other so that conservation of momentum (Newton's
laws) is satisfied (
Figure 2.13
). This figure illustrates that the areas in contact with each
other will have the same forces acting upon them (i.e.
-
2
), whereas the surfaces not in
contact do not necessarily have the same forces acting upon them (
δ
-
3
). Also, the
actual force on the opposite lamina surface does not need to be balanced because each
moving surface is in contact with some other surface that has some quantifiable area asso-
ciated with it. These areas may be different, which will generate different stresses on the
surface that balance over the entire differential element (i.e., the same force on a larger
area induces a lower stress). However, as the height of these laminae approaches the dif-
ferential element size (
-
1
and
δ
δ
0), the forces will nearly be balanced, as described previously,
so that momentum is conserved over the differential element. Let us define the area of the
original lamina (lamina 2) in contact with a fluid lamina above it (lamina 1) as A
2
. Lamina
1 is in contact with lamina 2 and the blood vessel wall in this example. It has the same
area in contact with lamina 2 as lamina 2 has with lamina 1 (i.e. A
2
). Because of this con-
tact, there is some force distributed over the area of each element. Consider a small ele-
ment of area,
δ
h
-
-
2
, which has a force component acting on it (
-
2
). This force can be
δ
δ
-
2
and one that is tangential to
-
2
.
resolved into two elements; one that is normal to
δ
δ
F
-
and
F
-
, respectively. The normal stress,
These forces are denoted as
δ
δ
σ
n
(
Equation 2.7
),
and the shear stress,
τ
n
(
Equation 2.8
), can then be defined as
F
-
0
δ
σ
n
5
lim
ð
2
:
7
Þ
A
-
A
-
δ
δ
-
F
-
0
δ
τ
5
lim
ð
2
:
8
Þ
n
A
-
A
-
δ
δ
-
where n denotes the unit normal vector with regard to the differential area element. As
the area approaches zero, the normal stress equation (
Equation 2.9
) and the shear stress
equation (
Equation 2.10
) can be represented in differential form.
dF
-
dA
-
σ
5
ð
2
:
9
Þ
n
FIGURE 2.13
δ
F
1
acting on
δ
A
2
Internal laminae of fluid exert
forces on neighboring laminae.
These forces must
be balanced within the fluid. Areas of the individual
laminae are the same and are represented as
Lamina
1
δ
h
δ
A
2
as
δ
F
2
acting on
δ
A
2
in the text.
δ
F
2
acting on
δ
A
2
Lamina
2
δ
F
3
acting on
δ
A
2
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