Biomedical Engineering Reference
In-Depth Information
At any time
t
, the system is defined by the volume of interest, which keeps the same
shape at all times. At time
t
t
, the system occupies area 2 and 3, instead of area 1 and 2
(for time
t
). Again, regardless of the area encompassed by the volume of interest, the vol-
ume (and its dimensions) remains constant at all time. Therefore, the following definitions
apply for the system properties:
1
Δ
W
t
5
W
volume of interest
5
W
VI
W
t
1
Δt
5
W
2
1
W
3
5
W
VI
2
W
1
1
W
3
Using these definitions in the derivative formulation
dW
dt
5
0
ð
W
VI
W
1
W
3
Þj
t
1
Δt
2
ð
W
VI
Þj
t
2
1
ð
:
Þ
lim
Δ
3
15
Δ
t
t
-
which is equal to
dW
dt
5
0
ð
W
VI
Þj
t
1
Δt
2
ð
W
VI
Þj
t
0
ð
W
3
Þj
t
1
Δt
Δ
0
ð
W
1
Þj
t
1
Δt
Δ
lim
Δ
lim
Δ
lim
Δ
ð
3
:
16
Þ
1
2
Δ
t
t
t
t
-
t
-
t
-
The first term in
Equation 3.16
is equal to
ð
0
ð
W
VI
Þj
t
1
Δt
2
ð
W
VI
Þj
t
5
@
W
VI
@
5
@
@
lim
Δ
w
ρ
dV
ð
3
:
17
Þ
Δ
t
t
t
t
-
V
For the remaining two terms, a similar analysis can be conducted, to obtain
dW
1
t
1
Δt
5
j
dW
3
t
1
Δt
5
w
ρ
dV
t
1
Δt
5
w
ρΔ
xdA
t
1
Δt
w
ρ
dV
t
1
Δt
5
w
ρΔ
x
ð
dA
Þ
t
1
Δt
j
ð
3
:
18
Þ
2
Remember that
dV
can be described as the change in length (i.e., from area 2 to area 3)
multiplied by the differential area (in general, the cube can move in three-dimensional
space). Also recall that a negative sign is included in the second term of
Equation 3.18
to
take care of the direction that the normal area vector is facing. The change in length can
also be considered as the fluid path for any deformation that a fluid element can experi-
ence. The mass is moving to the right (
Figure 3.8
), but the area vector for area 1 is oriented
toward the left. If we integrate the two equations in
3.18
, we get
ð
ð
dW
3
j
t
1
Δt
5
W
3
j
t
1
Δt
5
w
ρΔ
xdA
j
t
1
Δt
area 3
area 3
ð
ð
ð
3
:
19
Þ
dW
1
j
t
1
Δt
5
W
1
j
t
1
Δt
5
w
ρΔ
x
ð
2
dA
Þj
t
1
Δt
area 1
area 1
Substituting these values into
Equation 3.16
,
ð
w
ρΔ
xdA
ð
0
ð
W
3
Þj
t
1
Δt
Δ
d
-
-
area 3
lim
Δ
w
ρ
Δ
5
5
t
Δ
t
t
-
area 3
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