Environmental Engineering Reference
In-Depth Information
Solution.
In the interest of specifying the boundary conditions, we consider this
problem in a cylindrical coordinate system. Let
u
be the temperature in
Ω
. Based
on the given boundary conditions,
u
should be independent of
θ
so
u
must satisfy
⎧
⎨
2
u
2
u
∂
1
r
∂
u
r
2
∂
1
r
2
+
r
+
z
2
=
0
,
0
<
r
<
a
,
0
<
z
<
h
,
∂
∂
∂
(7.25)
|
u
(
0
,
z
)
| <
∞
,
u
(
a
,
z
)=
0
,
⎩
u
(
r
,
0
)=
0
,
u
(
r
,
h
)=
f
(
r
)
.
Assume
u
. Substituting it into PDS (7.25) yields the eigenvalue problem
of the Bessel equation
⎧
⎨
=
R
(
r
)
Z
(
z
)
1
r
R
(
R
(
2
R
r
)+
r
)+
λ
(
r
)=
0
,
(7.26)
⎩
R
(
|
R
(
0
)
| <
∞
, |
0
)
| <
∞
,
R
(
a
)=
0
.
where
is the separation constant. Its eigenvalues and eigenfunctions are available
in Table 4.1 and Section 4.4.2 ,
λ
J
0
μ
n
r
a
=
μ
n
a
λ
=
λ
,
(
)=
,
R
n
r
n
where
μ
n
are the positive zero points of
J
0
(
x
)
.
The
Z
(
z
)
satisfies
Z
(
2
n
Z
z
)
−
λ
(
z
)=
0
.
Its general solution reads
Z
n
(
z
)=
A
n
ch
λ
n
z
+
B
n
sh
λ
n
z
,
where
A
n
and
B
n
are constants. Applying
Z
n
(
0
)=
0 yields
A
n
=
0, so the solution of
PDS (7.25) can be written as
∞
n
=
1
B
n
J
0
(
λ
n
r
)
shλ
n
z
.
u
(
r
,
z
)=
(7.27)
Note that
{
J
0
(
λ
n
r
)
}
is orthogonal in
[
0
,
a
]
with respect to the weight function
r
.
Thus applying the boundary condition
u
(
r
,
h
)=
f
(
r
)
yields
a
0
ρ
1
M
n
sh
B
n
=
f
(
ρ
)
J
0
(
λ
ρ
)
d
ρ
,
(7.28)
n
λ
n
h
a
a
2
2
J
1
(
λ
n
a
rJ
0
(
λ
n
r
where the normal square
M
n
=
(see Section 2.5).
Equations (7.27) and (7.28) form the solution of PDS (7.25).
)
d
r
=
)
0
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