Civil Engineering Reference
In-Depth Information
As we will see later, the flow in a direction normal to a boundary defined by a vector
n
is also required. For three-dimensional isotropic problems, the flow is computed by
w
U
§
w
U
w
U
w
U
·
T
P
,
Q
k
k
¨
©
n
n
n
¸
¹
(4.11)
x
y
z
w
n
w
x
w
y
w
z
The derivatives of
U
in the global directions are
r
r
r
w
U
w
U
w
U
,
x
,
y
,
z
;
;
(4.12)
2
2
2
w
x
w
y
w
z
4
S
rk
4
S
rk
4
S
rk
where
x
x
y
y
z
z
QP
Q P
QP
r
;
r
;
r
(4.13)
,
x
,
y
,
z
r
r
r
Equation (4.11) can be rewritten as
cos
T
T
P
,
Q
(4.14)
2
4
S
r
where T is defined as the angle between the normal vector
n
and the distance vector
r
,
i.e.
1
^
`
T
^
`
T
(4.15)
cos
T x
nr
with
n
n ,n ,n
and
r
r ,r ,r
xyz
xyz
r
The variation of kernels
U
and
T
is plotted in Figures 4.3 and 4.4. It can be seen that
both solutions decay very rapidly from the value of infinity at the source. Whereas the
fundamental solution for
U
is symmetric with respect to polar coordinates, the solution
for
T
with the vector
n
pointing in x-direction (thus meaning flow in x-direction) is
antisymmetric.
For a two-dimensional problem, the source is assumed to be distributed along a line
of infinite length from z = - f to z = + f and the fundamental solutions are given by
1
S
1
§
·
(4.16)
U
P
,
Q
ln
©
¹
2
k
r
and
w
U
§
w
U
w
U
·
(4.17)
¨
©
¸
¹
T
P
,
Q
k
k
n
n
x
y
w
n
w
x
w
y
where
r
r
w
U
w
U
,
x
,
y
;
(4.18)
w
x
2
S
rk
w
y
2
S
rk
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