Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 44

(a) (b) (c)

FIGURE 3.1 Distribution of molecules of two different substances, inside a box at three

different moments: initially kept apart by a diaphragm (a), during mixing (b), and when spatial
gradient has disappeared (c).

mixing (stage (b) in Figure 3.1). After some time the proportion of white and dark
fluid is the same in both the box halves. At this stage, the probability of a white
molecule moving from the right side of the box to the left side is equal to the
probability of another white molecule moving the opposite way and, therefore, there
is no net exchange (stage (c) in Figure 3.1).
A macroscopic view of the mixing is represented in Figure 3.2. In the initial
condition (stage (a) in Figure 3.2), a black and a white side is observed (stage (b)
in Figure 3.2); during the mixing process a growing gray area occurs, corresponding
to the mixing zone; and after complete mixing, a homogeneous gray fluid is observed
(stage (c) in Figure 3.2).
The velocity of each elementary portion of fluid, like the velocity of any other
material point, is defined using two consecutive locations, as shown in Figure 3.3:

r r r

t + ∆ t t

v dx − x

== =

dx
dt

dx
dt

u = lim

∆

i

u (3.1)

i

∆ t

t → 0

Knowing the velocity of each individual molecule, it will be possible to fully
characterize transport. But, according to Heisenberg's uncertainty principle, it will
never be possible to know the place and the velocity of each molecule simultaneously.
Consequently, the fluid has to be considered a continuum system, for which a velocity
is defined.

(a) (b) (c)

FIGURE 3.2

Macroscopic view of the fluid composed by the molecules represented in

Figure 3.1.