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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:
\begin{eqnarray*}\partial _Tc_1+V\cdot \nabla c_1 + v\cdot \nabla c_1&=&\nabla\......la c_2 + v\cdot \nabla c_2&=&\nabla\cdot(D_1\nabla c_2)-kc_2c_1.\end{eqnarray*}
where $\nabla=\partial /pa X$$D_1, D_2$ are the diffusion coefficients and $c_1,c_2$ are the concentrations of both species. $V$ and $v$ are the constant mean and fluctuating velocities of the flow and $k$ is the chemical reaction rate constant. By making the transformations $t:=T$ and $x:=X-VT$ one get the new equations
\begin{eqnarray*}\partial _tc_1+v\cdot( \nabla c_1) &=&\nabla\cdot(D_1\nabla c_...... _tc_2+v\cdot (\nabla c_2) &=&\nabla\cdot(D_1\nabla c_2)-kc_2c_1\end{eqnarray*}
where $\nabla=\partial /\partial x$. Note that for large time the shear dispersion may be approximated by constant dispersion coefficients $D_i$ (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

\begin{eqnarray*}c_i(x,t)=<c_i(x,t)>+\delta c_i(x,t) (i=1,2)\end{eqnarray*}
where $<c_i>$ are the statistical averages of the concentrations taken over the ensemble of the flow. and $\delta c_i$ are the fluctuations in the concentrations. It can be shown that the averages $<\delta c_i>=0$. In case of a homogeneous turbulence we can assume $<v>=0$. It follows $<v\cdot \nabla c_i>=<v\cdot\nabla\delta c_i>$. If $v=0$ (uniform flow) then we get
\begin{eqnarray*}&&\partial <c_1>/\partial t=D_1\nabla ^2<c_1>-k<c_1><c_2>(1+\b......partial <c_2>/\partial t=D_1\nabla ^2<c_2>-k<c_1><c_2>(1+\beta).\end{eqnarray*}
Here $\beta:=<\delta c_1\delta c_2>/<c_1><c_2>$ and is called segregation parameter. Its values are varying between -1 and 0. $\beta$ 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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Nächste Seite:Stationary compressible fluidsAufwärts:Models of Mathematical PhysicsVorherige Seite:Oseen equations
collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg