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##
Chemical reactions between two
species in a turbulent flow

The following semilinear diffusion equations describe a second-order chemical
reaction between two species in case of a turbulent flow:
where .
are the diffusion coefficients and
are the concentrations of both species.
and
are the constant mean and fluctuating velocities of the flow and
is the chemical reaction rate constant. By making the transformations
and
one get the new equations
where .
Note that for large time the shear dispersion may be approximated by constant
dispersion coefficients
(i=1,2), where the coordinates are moving with the mean flow velocity.

After making a statistical approach (similar to the turbulence theory
of plasma) we assume

where
are the statistical averages of the concentrations taken over the ensemble
of the flow. and
are the fluctuations in the concentrations. It can be shown that the averages .
In case of a homogeneous turbulence we can assume .
It follows .
If
(uniform flow) then we get
Here
and is called *segregation parameter*. Its values are varying between
-1 and 0.
depends on the Reynolds,. Schmidt and Damköhler numbers (in abbreviations:
Re, Sc, and Da). More on this model the reader can find in:

**A late reference:** H.I. Abdel-Gawad and A. M. El-Shrae (2000)
*An Approach to Solutions of Coupled Semilinear Partial Differential
Equations with Applications* MMAS 23, 845 - 864.

**Nächste Seite:**Stationary
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collected and worked out by Wolfgang Sprößig, TU-Bergakademie
Freiberg