8.3 Concentration of Matter for Molecular and Turbulent Diffusion
269
a z
b
z
h p+-------~~--~x
Fig. 8.5: Velocity distribution between parallel plates: a velocity distribution in a
fixed coordinate system, b velocity distribution in a coordinate system moving at
the mean velocity
After substituting Eq. (8.44) into Eq. (8.43) and using the 'chain rule' of differentiation we obtain:
(8.45)
As we mentioned in the previous section, the spreading along the direction
of flow due to shear effect dominates over molecular diffusion. Therefore, the
longitudinal diffusion term in Eq. (8.45) can be neglected, i.e.:
(8.46)
As the general solution of Eq. (8.46) can not be found, Taylor (1953) examined
the orders of magnitude of the particular terms in this equation and found that
Eq. (8.46) can be substantially simplified as follows:
(8.47)
It should be noted that the term 015/[17, which we are interested in, has
also been discarded. For a more detailed justification of the simplification
of Eq. (8.46), the reader should consult Taylor's (1953) paper, or Fisher et ai.
(1979) book.