Reaction-Diffusion Equations for an Autocatalytic Reaction

This Demonstration shows the behavior of a reaction-diffusion system in which an autocatalytic reaction takes place.
Consider the reaction scheme with rate of reaction , where and are the reactant concentrations and is the reaction rate constant. The reaction takes place in a capillary tube of length filled with a fluid separated by an impermeable membrane. Half of the tube's length contains reactant and the other half contains reactant . The membrane is removed at time and the reaction-diffusion process begins. The plots of functions and are shown for user-selected values of time , diffusivity , and reaction rate constant .

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DETAILS

The governing equations are
and
.
With the membrane at , the initial conditions are
,
and
,
with boundary conditions
at and .
Here and are the diffusion coefficients of and , respectively, is distance, and is time.
The system can be simplified by making the transformations
and
to obtain the system
,
,
which has the following analytical solution when [1]:
,
,
with
, where is a free parameter taken as .
Reference
[1] A. H. Salas, L. J. Martinez H., and O. Fernandez S., "Reaction-Diffusion Equations: A Chemical Application," Scientia et Technica, 17(46), 2010 pp. 134–137. www.redalyc.org/pdf/849/84920977041.pdf.
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