Environmental Engineering Reference
In-Depth Information
Since for the
u
(
r
,
θ
,
t
)
in Eq. (6.110),
t
t
W
f
τ
τ
=
t
=
∂
W
f
τ
∂
∂
W
f
τ
∂
u
t
=
d
τ
+
d
τ
,
t
t
0
0
τ
=
t
=
t
t
2
W
f
τ
∂
2
W
f
τ
∂
∂
τ
+
∂
W
f
τ
∂
∂
u
tt
=
d
d
τ
+
f
(
r
,
θ
,
t
)
,
t
2
t
t
2
0
0
t
0
Δ
Δ
u
=
W
f
τ
d
τ
,
t
0
Δ
t
t
W
f
τ
τ
=
t
=
∂
∂
=
∂
∂
∂
∂
∂
∂
t
Δ
u
W
f
τ
d
τ
=
t
Δ
W
f
τ
d
τ
+
Δ
t
Δ
W
f
τ
d
τ
,
t
0
0
a substitution of Eq. (6.110) into the equation of PDS (6.109) yields
u
t
τ
B
2
∂
∂
A
2
0
+
u
tt
−
Δ
u
(
r
,
θ
,
t
)
−
t
Δ
u
(
r
,
θ
,
t
)
1
τ
d
t
2
W
f
τ
∂
0
∂
W
f
τ
∂
t
+
∂
B
2
∂
∂
A
2
=
−
Δ
W
f
τ
−
t
Δ
W
f
τ
τ
t
2
0
+
f
(
r
,
θ
,
t
)=
f
(
r
,
θ
,
t
)
.
Therefore the
u
in Eq. (6.110) is indeed the solution of PDS (6.109).
By the principle of superposition, the solution of
⎧
⎨
(
r
,
θ
,
t
)
u
t
τ
0
+
B
2
∂
∂
A
2
=
(
,
θ
,
)+
(
,
θ
,
)+
(
,
θ
,
)
,
u
tt
Δ
u
r
t
t
Δ
u
r
t
f
r
t
<
<
,
<
θ
<
π
,
<
,
0
r
a
0
2
0
t
(6.112)
⎩
L
(
u
,
u
r
)
|
r
=
a
=
0
,
u
(
r
,
θ
,
0
)=
ϕ
(
r
,
θ
)
,
u
t
(
r
,
θ
,
0
)=
ψ
(
r
,
θ
)
is
1
W
ϕ
(
τ
0
+
∂
B
2
W
k
mn
ϕ
(
u
=
r
,
θ
,
t
)+
r
,
θ
,
t
)
∂
t
t
+
W
ψ
(
r
,
θ
,
t
)+
W
f
τ
(
r
,
θ
,
t
−
τ
)
d
τ
,
(6.113)
0
where
f
τ
=
(
,
θ
,
τ
)
.The
W
ψ
(
,
θ
,
)
is the solution of PDS (6.94) and is available in
Eq. (6.99). Note that Eqs. (6.99) and (6.113) are valid for all three kinds of boundary
conditions. However, the
f
r
r
t
(
n
m
are boundary-conditiondependent and are determined
μ
by Eq. (6.97).
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