2 A Strategy for Bioremediation of Marine Shorelines …
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Applying the divergence theorem [18], it is possible to rewrite some integrals as
D
U · ∇φ dr =
D
∇ · (Uφ) dr =
∂ D
U · nφ d S,
D
∇ · μ∇φdr =
∂ D
μ∇φ · n d S =
∂ D
μ
∂φ
∂n
d S,
D
∇ · φ s dr =
∂ D
φ s · n d S.
Finally, dividing each integral over boundary ∂ 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:
∂
∂t
D
φ dr =
N
i=1
Q i (t) −
D
σ φ dr −
S +
U n φ d S −
S T
ζ φk · n d S +
S B
ν s φk · n d S.
(2.13)
Since k · n > 0 at S T and k · n < 0 at S B , the total mass of the nutrient
increases due to the discharge processes (Q i (t) > 0), and decreases because of
the chemical transformations (σ > 0), advective outflow through S + (U n > 0),
superficial evaporation (ζ > 0) and sedimentation (v s > 0).
We now show that the dispersion problem (2.4)–(2.11) is well posed. Indeed, the
model operator is:
Aφ = U · ∇φ − ∇ · μ∇φ + σ φ + ∇ · φ s .
(2.14)
Defining the inner product in L 2 (D) as (Aφ, φ) =
D
φ Aφdr we obtain the
expression
(Aφ, φ) =
D
φU · ∇φ dr +
D
σ φ
2 dr −
D
φ∇ · μ∇φ dr +
D
φ∇ · φ s dr.
The divergence theorem allows modifying some integrals in the last equation:
D
φU · ∇φ dr =
1
2
∂ D
φ
2 U · n d S,