Biomedical Engineering Reference
In-Depth Information
These forces are
dA
i
dp
i
2
F
ss
p
i
A
i
2
ð
p
i
dp
i
Þð
A
i
dA
i
Þ
1
p
i
A
i
dp
i
5
1
1
1
52
ds
dz
ð
3
:
66
Þ
dA
i
2
dA
i
2
F
bs
g
s
ρ
dV
5
ð
2
g
sin
θÞρ
A
52
ρ
gA
i
5
1
1
because sin
dz
. The differential volume term (
dV
in the
F
bs
equation) is an approxima-
tion of the volume using the midpoint area. The same analysis was used to approximate
the pressure on the surrounding fluid; that is, use the midpoint pressure as the approxi-
mate pressure on the fluid. The flux term in
Equation 3.65
is equal to
ð
θ
ds
5
d
-
-
v
s
ρ
v
i
ð
2
ρ
v
i
A
i
Þ
1
ð
v
i
1
dv
i
Þðρð
v
i
1
dv
i
Þð
A
i
1
dA
i
ÞÞ
U
5
area
making use of the inflow/outflow conditions. From the continuity equation (prior to sim-
plifying the term),
v
i
ð
2
ρ
v
i
A
i
Þ
1
ð
v
i
1
dv
i
Þðρð
v
i
1
dv
i
Þð
A
i
1
dA
i
ÞÞ
5
v
i
ð
2
ρ
v
i
A
i
Þ
1
ð
v
i
1
dv
i
Þðρ
v
i
A
i
Þ
5
ρ
v
i
A
i
dv
i
ð
3
:
67
Þ
Substituting
Equations 3.66/3.67
into the momentum equation (
Equation 3.65
),
dz
dA
i
2
A
i
dp
2
ρ
gA
i
1
A
i
dp
i
2
ρ
gA
i
dz
5
ρ
v
i
A
i
dv
i
2
52
ρ
A
i
, and simplify the velocity derivative,
If we divide this equation by
1
v
i
2
dp
i
ρ
1
dp
i
ρ
1
0
v
i
dv
i
gdz
d
gdz
5
1
5
Integrating this equation and dropping the subscripts, we obtain the Bernoulli equation:
v
2
2
1
p
ρ
1
gz
constant
ð
3
:
68
Þ
5
As we stated before, the Bernoulli equation is a powerful equation which relates the
flow speed, the hydrostatic pressure, and the height to a constant. It can only be applied
to a situation where the flow is steady, invisicid, and incompressible. In developing this
relationship, we used a differential element, where these three criteria were valid. In most
cases, it will not be easy to justify the use of the Bernoulli equation instead of the Navier-
Stokes equations, the Conservation of Mass, and the Conservation of Momentum.
However, as our example will show, we can use the Bernoulli equation as an approxima-
tion for various flow situations. In this simplified form, the Bernoulli equation is a state-
ment of the conservation of energy for an invisicid fluid.
Example
Blood flow from the left ventricle into the aorta can be modeled as a reducing nozzle (see
Figure 3.23
). Model both the left ventricle and the aorta as a tube with diameter of 3.1 cm and
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