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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