Geoscience Reference
In-Depth Information
Regardless of the derivation, however, it should be remembered that Equations (1.8) and
(1.9) describe the flow at a point; therefore, in principle, the integration of (1.8) or (1.9)
should allow the determination of the distribution of the amount and transport of water in
space and in time.
Finite control volume
In a second but equally valid approach, the control volume is assumed to occupy the entire
flow domain by integrating out the spatial dependence of the flux terms. Thus all flux terms
are located on the boundaries of the flow domain and they can be grouped into bulk inflow
rates
Q
i
and outflow rates
Q
e
. As a result, the continuity equation takes on the
lumped
form
of the
storage equation
, as follows
dS
dt
Q
i
−
Q
e
=
(1.10)
where
S
is the amount of water stored in the control volume, and the ordinary derivative
indicates that the time
t
is the only remaining independent variable. When Equation (1.10)
describes the flow of liquid water with an assumed constant density, these variables can
have the dimensions of [
Q
]
=
[L
3
/
T] and [
S
]
=
[L
3
], where L and T represent the basic
dimensions of length and time, respectively; if the
Q
-terms include precipitation and evap-
oration, it is often convenient to take these dimensions as [
Q
]
=
[L
/
T] and [
S
]
=
[L]. In
the lumped formulation of Equation (1.10), all interior variables and parameters represent
spatial averages over the entire control volume.
1.5.3
Conservation of momentum: Euler and Navier-Stokes equations
The flow of a fluid is also subject to the principle of conservation of momentum. Again,
there are several ways of obtaining a mathematical formulation of this principle.
At a point
The simplest method is probably to consider, as before, a small element of an ideal fluid with
a mass (
ρδ
∀
), as illustrated in Figure 1.4, and to apply Newton's second law to it. This states
that the rate of change of momentum is equal to the sum of the impressed forces. The pres-
sure and the velocity at the center (
x
,
y
,
z
) of this element are
p
(
x
,
y
,
z
,
t
) and
v
(
x
,
y
,
z
,
t
),
respectively. Accordingly, the property of the fluid element in this case is its momen-
tum, or
C
=
(
ρδ
∀
v
), and the rate of change is
D
(
ρδ
∀
v
)
/
Dt
, as given by Equation (1.3).
Because the fluid is assumed to be ideal, the only relevant forces are those owing to pres-
sure and to the acceleration of gravity. The latter is a vector,
g
=
i
g
x
+
j
g
y
+
k
g
z
, whose
direction defines the local vertical, and whose absolute value is commonly denoted by
g
;
the coordinates shown in Figure 1.4 are oriented in such a way that
g
x
=−
g
sin
θ
,
g
y
=
0
and
g
z
=−
g
cos
θ
. As illustrated in Figure 1.4, the
x
-component of the net force acting
on the fluid element is the sum of the forces acting on AD and BC plus the force due the
Earth's gravity; this sum equals
−
∂
p
/∂
x
)
+
ρ
g
sin
θ
δ
x
δ
y
δ
z
. Similarly, the sum of the
[(
]
impressed forces in the
z
-direction is
−
[(
∂
p
/∂
z
)
+
ρ
g
cos
θ
]
δ
x
δ
y
δ
z
. Adding to these an
analogous
y
-component, one has in vector notation,
D
Dt
(
ρδ
∀
v
)
=−
(
∇
p
−
ρ
g
)
δ
∀
