Biomedical Engineering Reference
In-Depth Information
theta direction and
1
R
in the radial direction. Using these conditions to solve for the integration
constants,
dv
z
ð
0
Þ
0
2
@
p
c
1
0
5
5
z
1
0
dr
μ
@
Due to the discontinuity at
r
0, the only way for this equation to be valid is that
c
1
5
5
0.
Therefore, the discontinuity is removed. Using the second boundary condition,
R
2
4
@
p
v
z
ð
R
Þ
5
z
1
c
2
5
0
μ
@
R
2
4
@
p
c
2
52
μ
@
z
Substituting the values for the integration constants into the velocity equation,
R
2
r
2
4
@
p
R
2
4
@
p
R
2
4
@
p
r
v
z
ð
r
Þ
5
1
z
2
z
5
2
μ
@
μ
@
μ
@
z
For this particular flow scenario, using the same values as the previous example,
R
2
R
2
4
r
2
@
p
r
v
z
ð
r
Þ
5
1
11
:
9cm
=
s
1
2
52
m
2
2
μ
@
z
62
;
500
μ
The shear stress distribution is
@
v
r
@
z
1
@
v
z
@
5
μ
@
v
z
@
τ
zr
5
μ
r
r
5
r
2
@
p
2
@
r
p
cm
2
τ
zr
5
μ
0
:
1332 dyne
=
μ
m
r
z
52
μ
@
z
@
3.8 BERNOULLI EQUATION
The Bernoulli equation is a useful formula that relates the hydrostatic pressure, the fluid
height, and the speed of a fluid element. However, there are a few important assumptions
that are made to derive this formula, which makes this powerful equation not necessarily
useful in many biofluid mechanics applications. Although as a back-of-the-envelope calcu-
lation, the Bernoulli equation can approximate the real flow situation reasonably well. To
derive this equation, the conservation of mass and conservation of momentum equations
are simplified by making the assumptions that the flow is steady, incompressible, and
invisicid (has no viscosity).
To derive the Bernoulli equation, let us follow a differential volume of fluid in an
expanding streamline (
Figure 3.22
). The fluid properties at the inlet will be denoted as
p
i
,
v
i
,
A
i
, and
ρ
. The fluid properties at the outlet will be denoted as
p
i
1
dp
i
,
v
i
1
dv
i
,
A
i
1
dA
i
,
ρ
and
. This same analysis can be conducted for a reducing streamline, where the solution
would include negative differential changes, as necessary.
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