Geology Reference
In-Depth Information
Fig. A1.3
Contour lines and gradient for the scalar field of Fig.
A1.1
Fig. A1.4
Fluid flux
through a small volume
operators: the divergence and the curl. The
diver-
gence
of a vector field is defined as follows:
through the right face per unit time is given by
v
x
(
x
C
x
,
y
,
z
)
y
z
. Therefore, the net volume
of fluid
per unit volume
and per unit time through
V
in the
x
direction is given by:
@A
y
@y
C
@A
x
@x
C
@A
z
@
z
r
A
D
(A1.4)
v
x
.x
C
x;y;
z
/y
z
v
x
.x;y;
z
/y
z
xy
z
To understand the physical interpretation of
this scalar field, let us consider a steady velocity
field in a fluid,
v
D
v
(
r
), and a small volume
V
D
x
y
z
at location
r
(Fig.
A1.4
).
The volume of fluid which enters the volume
through the left face in the
x
direction per unit
time is given by
v
x
(
x
,
y
,
z
)
y
z
. Similarly, the
volume of fluid that leaves
V
in the
x
direction
h
@
v
x
@x
x
i
y
z
xy
z
D
@
v
x
@x
D
(A1.5)
Consequently, the net volume of fluid per unit
volume and per unit time through
V
is given
by
r
v
. The second differential operator that
