A Strategy for Bioremediation of Marine Shorelines by Using Several Nutrient Release Points - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 32

μ ∂ˆ
∂

ˆμ ∂

g

g ∇· μ ∇ ˆ dr = g dS − dS + ˆ ∇· μ ∇ gdr ,

n ∂ n

D ∂ D ∂ D D

g ∇· ˆ s dr = g ˆ s · n dS − ˆ ∇· g s dr ,

D ∂ D D

where g s =− v s g k . Then

( A ˆ, g ) = ˆ( − U ·∇ g −∇· μ ∇ g + ˃ g −∇· g s ) dr

D

ˆμ ∂

g

μ ∂ˆ
∂

+ g ˆ U · n dS + dS − g dS + g ˆ s · n dS .

∂ n n

∂ D ∂ D ∂ D ∂ D

D into four integrals over S T , S + , S − and
S B , and using conditions ( 2.6 )-( 2.9 ) and ( 2.12 ), we obtain that

Dividing the integrals over boundary ∂

( A ˆ, g ) = ˆ( − U ·∇ g −∇· μ ∇ g + ˃ g −∇· g s ) dr

D

provided that the function g satisfies the boundary conditions ( 2.20 )-( 2.23 ) (see
below). Thus, the Lagrange identity is fulfilled if

A ∗ g

=− U ·∇ g −∇· μ ∇ g + ˃ g −∇· g s .

On the other hand, multiplying Eq. ( 2.4 )by g and integrating the result over the
space-time domain D × ( 0 , T ) , we get

g N

drdt

T T T

g ∂ˆ
∂

drdt + gA ˆ drdt = Q i ( t )ʴ( r − r i ) .

t

i = 1

0 D 0 D 0 D

Integration by parts of the first integral, together with conditions ( 2.10 ) and

g ( r , T ) = 0, leads to

T T

g ∂ˆ
∂

ˆ ∂

g

0

drdt =− g ( r , 0 )ˆ ( r ) dr − drdt

t ∂ t

0 D D 0 D