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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.

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collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg