Environmental Engineering Reference
In-Depth Information
Substituting Eqs. (6.127), (6.128) and (6.132) into Eq. (6.126) thus yields the solu-
tion of PDS (6.125)
1
W ϕ (
τ 0 +
B 2 W
u
(
r
, θ ,
z
,
t
)=
r
, θ ,
z
,
t
)+
) ϕ (
r
, θ ,
z
,
t
) .
(6.133)
(
k mn + λ
k
t
6.6.3 Solution from f
(
r
, θ ,
z
,
t
)
Theorem 2. Let u
(
r
, θ ,
z
,
t
)=
W ψ (
r
, θ ,
z
,
t
)
be the solution of
u t
τ
B 2
A 2
0 +
u tt =
Δ
u
(
r
, θ ,
z
,
t
)+
t Δ
u
(
r
, θ ,
z
,
t
)
Ω × (
, + ) ,
0
(
,
,
) | ∂Ω =
,
L
u
u r
u z
0
u
(
r
, θ ,
z
,
0
)=
0
,
u t
(
r
, θ ,
z
,
0
)= ψ (
r
, θ ,
z
) .
The solution of
u t
τ 0 +
B 2
A 2
u tt =
Δ
u
(
r
, θ ,
z
,
t
)+
t Δ
u
(
r
, θ ,
z
,
t
)
+
f
(
r
, θ ,
z
,
t
) ,
Ω × (
0
, + ) ,
(6.134)
L
(
u
,
u r ,
u z ) | ∂Ω =
0
,
u
(
r
, θ ,
z
,
0
)=
u t (
r
, θ ,
z
,
0
)=
0
is
t
u
=
W f τ (
r
, θ ,
z
,
t
τ )
d
τ ,
(6.135)
0
where f τ =
. Therefore, Theorem 2 in Section 6.1 is also valid in cylin-
drical coordinate systems.
f
(
r
, θ ,
z
, τ )
Proof. By the definition of W ψ (
r
, θ ,
z
,
t
)
,the W f τ (
r
, θ ,
z
,
t
τ )
satisfies
2 W f τ
1
τ
0
W f τ
+
B 2
A 2
=
Δ
W f τ +
t Δ
W f τ ,
(6.136a)
t
t 2
∂Ω =
L W f τ ,
W f τ
r ,
W f τ
0
,
(6.136b)
z
t = τ =
W f τ t = τ =
W f τ
0
,
f
(
r
, θ ,
z
, τ ) .
(6.136c)
t
 
Search WWH ::




Custom Search