Biomedical Engineering Reference
In-Depth Information
FIGURE 3.22
A differential volume of fluid fol-
lowing expanding streamlines (streamlines are the
curved arrows in the figure). The expansion causes
an increase in area at the outlet as compared to the
inlet.
Outlet
g
ds
Inlet
Z
θ
X
Y
Applying the conservation of mass to this condition, we find that
ð
V
ρ
ð
area
ρ
5
@
@
d
-
-
0
dV
1
U
t
ð
area
ρ
d
-
-
0
52
ρ
v
i
A
i
1
ρð
v
i
dv
i
Þð
A
i
dA
i
Þ
5
U
1
1
because we assumed that the flow is steady and incompressible. Simplifying the previous
equation, we can obtain
ρ
v
i
A
i
5
ρð
v
i
A
i
v
i
dA
i
A
i
dv
i
dv
i
dA
i
Þ
5
ρ
v
i
A
i
1
ρ
v
i
dA
i
1
ρ
A
i
dv
i
1
ρ
dv
i
dA
i
ð
3
:
63
Þ
1
1
1
Remember that the product of two differentials (
dv
i
dA
i
) is going to be negligible compared
to the remaining terms, which allows us to simplify
Equation 3.63
to
0
v
i
dA
i
1
A
i
dv
i
ð
3
:
64
Þ
5
Now we will simplify the conservation of momentum equation in the streamline direc-
tion (s). The conservation of momentum states that
ð
ð
F
ss
5
@
@
d
-
-
F
bs
1
v
s
ρ
dV
v
s
ρ
1
U
t
V
area
for
the streamline direction. Because we are assuming steady flow,
this formula
simplifies to
ð
d
-
-
F
bs
1
F
ss
5
v
s
ρ
ð
3
:
65
Þ
U
area
Because the fluid is invisicid, the only forces that arise are from the pressure acting
on the two surfaces and the surrounding fluid and the body force due to gravity.
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