Geology Reference
In-Depth Information
Fig. A1.5
Rotors in a pressure-driven fluid flow through a channel
describes the structure of a vector field
A
D
A
(
r
)
is the
curl
, which is a vector field defined as
follows:
divergence have the following important proper-
ties, which can be easily verified by the reader:
2
rr
A
Dr
.
r
A/
r
A
(A1.7)
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
@A
z
@y
i
i jk
@=@x @=@y @=@
z
A
x
A
y
A
z
@A
y
@
z
r
A
D
D
r
.
r
A/
D
0
(A1.8)
@A
x
@
z
@A
y
@x
rr
¥
D
0
(A1.9)
@A
z
@x
@A
x
@y
C
j
C
k
for each vector field
A
and for any scalar field ¥.
In (A1.7), the operator
r
(A1.6)
2
is the
Laplacian
,which
is defined by:
The physical interpretation of the curl can
be understood considering again the case of the
velocity field within a fluid. Let us consider a
steady pressure-driven fluid flow through a chan-
nel, as in Fig.
A1.5
. Two rotors, located close
to the opposite walls would rotate in opposite
directions.
The curl of the velocity field in Fig.
A1.5
is
given by:
r
v
D
(@
v
x
/@
y
)
k
. It is in the negative
z
direction close to the lower wall and in the
positive
z
direction close to the upper wall. In
the first case, a rotor placed in the fluid would
rotate clockwise (negative angular velocity) or
counterclockwise (positive rotation) according to
the sign of
r
v
and with angular velocity pro-
portional to the magnitude of the curl. Curl and
@
2
¥
@x
2
C
@
2
¥
@y
2
C
@
2
¥
@
z
2
2
¥
rr
¥
D
r
(A1.10)
for an arbitrary scalar field
¥
. In the case of a
vector field, it is intended that the operator is
applied to each component independently.
A1.3 Integrals Theorems
3
is a line such that each point
P
2
C
has a position vector
r
defined by the parametric
equation:
A
path C
in
<
r
D
r.t/
D
x.t/i
C
y.t/j
C
z
.t/k
I
t
2
Œa;b
(A1.11)
