A Strategy for Bioremediation of Marine Shorelines by Using Several Nutrient Release Points - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 28

Fig. 2.1 View of domain D
from above

Fig. 2.2 Cross-sectional
area of domain D

S + ∪ S − ∪ vector to the boundary ∂ D = S T ∪ S B of domain D , ∂/∂ n is the derivative

t is the unit vector directed upward in the
Cartesian coordinate system (Fig. 2.2 ). We observe that

in the normal direction, and k = ( 0 , 0 , 1 )

0on S + ∪ S − k · n = and U · n = 0on S T ∪ S B . (2.12)

Also note that the boundary conditions ( 2.6 )-( 2.9 ) are general (i.e., not only for
horizontal free and bottom surfaces S T and S B ), and hence, the dispersion model can
take into account free surface wave motion and marine topography.
First of all we show that the solution of dispersion model ( 2.4 )-( 2.11 ) satisfies
the mass balance equation. Indeed, integrating Eq. ( 2.4 ) over domain D we get

∂
∂

D ˆ

+ · ˆ − ∇· μ ∇ ˆ + ˃ˆ + ∇· ˆ s dr

dr U dr dr dr

t

D

D D D

N

= Q i ( t )ʴ( r − r i ) dr .

i = 1

D