Combining 1D advective diffusive transport equation in porous media for two solutes (1 Viewer)

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1. The problem statement, all variables and given/known data

I am trying to derive the partial differential equation for the change in the ratio (r) of two solute (C1, C2) with time during 1D flow of a reacting advecting-diffusing fluid moving through a porus media. I can define the partial differential equations for the individual solutes and derive the quotient rule to combine them to obtain dr/dt but I cannot work out how to successfully combine the equations. Note: this is not actually a home work question but I have posted else where and had no response.

2. Relevant equations

The problem;

The change in the concetration of C1 in space (x) and time (t) of a fluid moving at velocity (v) with a flux J1 of C1 to the fluid is defined (similarly for C2) as

dC1/dt=D*d2C1/dx2-v*dC1/dx+J1

where D is a longitudinal dispersivity coefficient

and the ratio of C1 to C2 in the fluid (r) is

r=C1/C2

3. The attempt at a solution

The form of the chain rule to combine them is

C2*(dr/dt)=(dC1/dt)-r*(dC2/dt)

from which

C2*(dr/dt)=[D*d2C1/dx2-v*dC1/dx+J1]-[(C1/C2)*(D*d2C2/dx2-v*dC2/dx+J2]

But I dont think I am correctly simplifing the equations from here on... can someone please please help? I want the final expression in the form dr/dt= It would be so helpful. Or if someone could tell me if I have this totally wrong that would also really help!
 

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