Geoscience Reference
In-Depth Information
and
∂
x
2
δ
x
2
w
+
∂w
∂
x
δ
x
2
+
∂
2
w
1
2
+···
z
=
z
0
+
δ
t
2
Therefore, the length of the segment HF, as projected on the
x
-axis, is given by [
δ
x
+
(
t
]; in a similar way, one obtains for the projection of the length of the segment
EG on the
z
-axis the value [
∂
u
/∂
x
)
δ
x
δ
t
], and for the length of the segment in the
y
-direction (not shown in Figure 1.3) [
δ
y
+
(
∂v/∂
y
)
δ
y
δ
t
]. If
δ
x
,δ
y
,δ
z
and
δ
t
are all small
enough, higher-order terms can be neglected, and the volume occupied by the mass at time
t
+
δ
t
can be taken as the product of these three segments. Thus the change in volume
during
δ
t
becomes
δ
z
+
(
∂w/∂
z
)
δ
z
δ
1
+
∂
u
∂
x
δ
t
δ
x
1
+
∂v
∂
y
δ
t
δ
y
1
+
∂w
∂
z
δ
t
D
(
δ
x
δ
y
δ
z
)
Dt
δ
t
=
δ
z
−
δ
x
δ
y
δ
z
so that
∂
u
∂
D
(
δ
∀
)
Dt
x
+
∂v
y
+
∂w
=
δ
∀
(1.5)
∂
∂
z
In more concise vector notation this can also be written as
D
(
δ
∀
)
Dt
=∇·
v
δ
∀
(1.6)
where
∇
is the operator
∇=
(
∂/∂
x
)
i
+
(
∂/∂
y
)
j
+
(
∂/∂
z
)
k
. Equations (1.5) and (1.6)
show that the divergence
∇·
v
is indeed, as its name suggests, the fractional rate of change
of the fluid element volume. With this result, Equation (1.4) can be written as
∂
u
D
Dt
+
ρ
∂
x
+
∂v
∂
y
+
∂w
=
0
(1.7)
∂
z
or again, in vector notation, as
∂ρ
∂
t
+∇·
(
ρ
v
)
=
0
(1.8)
Equations (1.7) and (1.8) are forms of the classical continuity equation. The form of (1.8) is
applicable to describe the conservation of mass of any substance at a given point (
x
,
y
,
z
),
provided (
ρ
v
) is made to represent the specific mass flux
F
, that is the transport of mass
of that substance per unit cross sectional area and per unit time. Whenever the density of
the substance in question can be considered constant, the continuity equation assumes its
well-known form
∇·
v
=
0
(1.9)
Note that the continuity equation is not usually derived this way; the present derivation is
used to maintain unity and consistency of treatment with the conservation of momentum
in Section 1.5.3; moreover, it is relevant to the study of deforming porous media later on
in Chapter 8. A more common way to derive the continuity equation consists of setting up
a mass balance for a certain volume fixed in space, also called a
control volume
. The mass
balance states that the sum of all the inflow rates into the control volume minus the sum of
all the outflow rates is equal to the time rate of change of mass stored in the control vol-
ume. For an infinitesimally small control volume this procedure also yields Equation (1.8).
