Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 46

of the velocity [ LT ] and the area [ L ] of the surface. The transport produced by

− 1 2

the velocity per unit of area is

=⋅ r φβ

adv

u

u

where is the velocity relative to the velocity of the surface. This flux is a vector
parallel to the velocity. The flux across an elementary area dA not perpendicular to

the velocity is given by

β v r d ( Φ= ⋅ ) ( u n dA )

adv

n

where is the external normal to the elementary surface dA . In the case of a finite

surface , the total flux is the summation of the elementary fluxes across the

elementary areas composing it. This summation can be represented by the integral
over that surface:

A

β rr

∫ ∫

Φ = d ( Φ ) = ( u n dA ⋅ ) (3.3)

adv adv

A A

This is the generic definition of the advective flux of a property B across a surface A .

3.2.3 D F

IFFUSIVE LUX

Diffusive flux is the net transport associated with the Brownian movement of mol-
ecules in the case of a laminar flow or due to both molecular and turbulent movement
in the case of a turbulent flow, depending on the property gradient. Diffusion
transports the property down the direction of the gradient, as determined by Fick,
who stated that the diffusive flux per unit of area is given by

r

r

φ =− ∇ ϕβ ( ) (3.4)

dif

where ϕ is the diffusivity and the quantity inside the parentheses is the gradient of

β , the specific value of B (concentration in case of mass).

Both

gradient and diffusivity can vary spatially, implying the calculation of
the flux on elementary surfaces:

β

r

r

d ( φ ) =− ∇ ϕ β ( ) n dA

dif

and its integration along the overall surface:

r

r

∫ ∫

Φ dif = d ( φ ) = − ∇ ϕ β ( ) n dA (3.5)

dif

A A

n

where is the normal to the elementary surface.