Environmental Engineering Reference
In-Depth Information
6.6.4 Green Function of the Dual-Phase-Lagging
Heat-Conduction Equation
By using Eq. (6.123), the
u
(
r
,
θ
,
z
,
t
)
in Eq. (6.135) reads
t
t
1
M
mnk
β
mnk
m
,
n
,
k
e
α
mnk
(
t
−
τ
)
u
=
W
f
τ
(
r
,
θ
,
z
,
t
−
τ
)
d
τ
=
0
0
Ω
r
,
θ
,
z
,
τ
)(
θ
+
θ
)
r
·
f
(
cos
n
θ
cos
n
sin
n
θ
sin
n
J
n
(
k
mn
r
)
k
mn
r
)
z
)
θ
d
r
d
z
d
·
J
n
(
Z
k
(
z
)
Z
k
(
sin
β
mnk
(
t
−
τ
)
d
τ
t
r
;
θ
,
θ
;
z
z
;
t
r
,
θ
,
z
,
τ
)
=
(
,
,
−
τ
)
(
τ
,
G
r
f
d
Ω
d
(6.137)
0
Ω
where
e
α
mnk
(
t
−
τ
)
M
mnk
β
mnk
=
m
,
n
,
k
k
mn
r
)
z
)
(
θ
−
θ
)
G
J
n
(
k
mn
r
)
J
n
(
Z
k
(
z
)
Z
k
(
cos
n
sin
β
mnk
(
t
−
τ
)
(6.138)
is called the
Green function of the dual-phase-lagging heat-conduction equation in
a cylindrical domain
. The Green function is clearly boundary-condition dependent.
When
f
(
r
,
θ
,
z
,
t
)=
δ
(
r
−
r
0
,
t
−
t
0
)
, the solution of PDS (6.134) reduces to
u
=
G
(
r
,
r
0
;
θ
,
θ
0
;
z
,
z
0
;
t
−
t
0
)
,
=(
,
θ
,
)
=(
,
θ
,
)
(
,
θ
,
θ
,
where
r
r
z
,
r
0
r
0
z
0
. Therefore the Green function
G
r
r
0
;
0
;
z
0
,
)
δ
(
−
,
−
)
z
0
;
t
t
0
is the solution from the source term
r
r
0
t
t
0
.
Remark 1
.The boundary condition
L
0 in PDS (6.114) encompasses
27 combinations. Since PDS (6.114) has nontrivial solutions for any nontrivial
(
u
,
u
r
,
u
z
)
|
∂Ω
=
ϕ
,
or
f
, there exist a total of seven cases
C
3
+
7
of nontrivial solutions.
C
3
+
C
3
=
ψ
Therefore, PDS (6.114) actually contains 27
×
7
=
189 PDS. Their solutions can be
written in the general form
1
W
ϕ
(
τ
0
+
∂
B
2
W
u
=
r
,
θ
,
z
,
t
)+
)
ϕ
(
r
,
θ
,
z
,
t
)
(
2
k
k
mn
+
λ
∂
t
t
+
W
ψ
(
r
,
θ
,
z
,
t
)+
W
f
τ
(
r
,
θ
,
z
,
t
−
τ
)
d
τ
,
(6.139)
0
where the structure of
W
ψ
(
r
,
θ
,
z
,
t
)
is available in Eq. (6.123). Since Eq. (6.123)
exhibits the structure of
W
ψ
(
r
,
θ
,
z
,
t
)
, we can write
W
ϕ
(
r
,
θ
,
z
,
t
)
and
W
f
τ
(
r
,
θ
,
z
,
t
−
τ
)
using the structure in Eq. (6.123) even for PDS with
ψ
(
r
,
θ
,
z
)=
0. There-
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