Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 50

(u 2 ) x 2 + ∆ x 2

∆ x 2

x 2

(u 1 ) x 1

(u 1 ) x 1 + ∆ x 1

∆ x 3

∆ x 1

(u 2 ) x 2

x 1

x 3

FIGURE 3.5 Cubic type elementary control volume. Velocity components are represented
on every surface of the volume.

corresponding surface. Keeping the orientation of the normal in mind, we can
write the fluxes for surfaces perpendicular to the x 1 axis:

βϕ φ

∂
∂

−− ∂
∂

βϕ φ u − u ∆∆ xx

1 1 23

x x

1 1

x +∆ x x

1 1 1

Doing a similar calculation for other directions, dividing the whole equation by ∆ V

and letting ∆ x i converge to zero, we obtain Equation (3.8):

+−

∂
∂

β

+ ∂
∂

β u

x

∂
∂

∂
∂

φ

i

= ϕ ( SS ) (3.8)

o i

t x x

i j j

In this equation the Einstein convention † for summation is used and sources and
sinks are calculated per unit of volume. Equation (3.7) and Equation (3.8) are general
and hold for any property. They can also be used to derive continuity and momentum
equations in the next sections.

† In the Einstein convention, a doubled index represents a summation of the terms obtained replacing
that index by each dimension of the physical space (3 in a three-dimensional space).