Environmental Engineering Reference
In-Depth Information
Applying the divergence theorem [ 18 ], it is possible to rewrite some integrals as
U
·∇ ˆ
dr
=
∇· (
U
ˆ)
dr
=
U
·
n
ˆ
dS
,
D
D
D
μ ∂ˆ
∇· μ ˆ
dr
=
μ ˆ ·
n dS
=
dS
,
n
D
D
D
∇· ˆ s dr
=
ˆ s ·
.
n dS
D
D
D into the four integrals over S T ,
S + , S and S B , and applying Eqs. ( 2.6 )-( 2.9 ) and observation ( 2.12 ), we obtain the
mass balance equation:
Finally, dividing each integral over boundary
N
ˆ
dr
=
Q i (
t
)
˃ˆ
dr
U n ˆ
dS
ʶˆ
k
·
n dS
+
ʽ s ˆ
k
·
n dS
.
t
i
=
1
D
D
S +
S T
S B
(2.13)
0at S B , the total mass of the nutrient
increases due to the discharge processes ( Q i (
Since k
·
n
>
0at S T and k
·
n
<
t
)>
0), and decreases because of
0), advective outflow through S +
the chemical transformations (
˃>
( U n
>
0),
superficial evaporation (
0).
We now show that the dispersion problem ( 2.4 )-( 2.11 ) is well posed. Indeed, the
model operator is:
ʶ>
0) and sedimentation ( v s >
A
ˆ =
U
·∇ ˆ −∇· μ ˆ + ˃ˆ +∇· ˆ s .
(2.14)
D ˆ
Defining the inner product in L 2 (
D
)
as
(
A
ˆ,ˆ) =
A
ˆ
dr we obtain the
expression
2 dr
(
ˆ,ˆ) =
ˆ
·∇ ˆ
+
˃ˆ
ˆ ∇· μ ˆ
+
ˆ ∇· ˆ s dr
.
A
U
dr
dr
D
D
D
D
The divergence theorem allows modifying some integrals in the last equation:
1
2
2 U
ˆ
U
·∇ ˆ
dr
=
ˆ
·
n dS
,
D
D
 
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