Geoscience Reference
In-Depth Information
C
′
G
D
′
F
H
δ
t
w
C
D
B
′
δ
z
E
A
′
δ
x
A
B
u
δ
t
z
x
Fig. 1.3 At time
t
the mass
ρδ ∀
occupies the volume ABCD, and at time
t
+
δ
t
this same fluid
mass has moved to A
B
C
D
. The center of the volume has moved from (
x
0
,
y
0
,
z
0
)to
(
x
0
+
u
δ
t
,
y
0
+
v
δ
t
,
z
0
+
w
δ
t
). The figure is shown in two dimensions for clarity; the third
coordinate
y
can be imagined as pointing into the plane of the drawing.
does not change and must remain the same. Therefore, if the fluid property is taken as the
mass of the fluid, or
C
=
ρδ
∀
(
), one has with Equation (1.3) that
D
(
ρδ
∀
)
=
0
Dt
(1.4)
or
D
(
δ
∀
)
Dt
D
Dt
=
ρ
+
δ
∀
0
The rate of change of the fluid element volume
D
(
δ
∀
)
/
Dt
can be derived by tracking the
fluid element, shown in Figure 1.3, as it moves from ABCD to A
B
C
D
during the small
time interval
δ
t
. The point H is then located at
u
−
∂
u
∂
x
2
δ
x
2
2
u
x
=
x
0
−
δ
x
∂
x
δ
x
2
+
∂
1
2
−···
2
+
δ
t
2
and at
x
2
δ
x
2
2
w
−
∂w
∂
δ
x
2
+
∂
w
1
2
−···
z
=
z
0
+
δ
t
x
∂
2
The point F is at
u
+
∂
u
∂
x
2
δ
x
2
x
=
x
0
+
δ
x
∂
x
δ
x
2
+
∂
2
u
1
2
+···
2
+
δ
t
2
