- #1

nikokampman

- 2

- 0

## Homework Statement

I am trying to derive the partial differential equation for the change in the ratio (r) of two solute (C

_{1}, C

_{2}) 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.

## Homework Equations

The problem;

The change in the concetration of C

_{1}in space (x) and time (t) of a fluid moving at velocity (v) with a flux J

_{1}of C

_{1}to the fluid is defined (similarly for C

_{2}) as

dC

_{1}/dt=D*d

^{2}C

_{1}/dx

^{2}-v*dC

_{1}/dx+J

_{1}

where D is a longitudinal dispersivity coefficient

and the ratio of C

_{1}to C

_{2}in the fluid (r) is

r=C1/C2

## The Attempt at a Solution

The form of the chain rule to combine them is

C

_{2}*(dr/dt)=(dC

_{1}/dt)-r*(dC

_{2}/dt)

from which

C

_{2}*(dr/dt)=[D*d

^{2}C

_{1}/dx

^{2}-v*dC

_{1}/dx+J

_{1}]-[(C

_{1}/C

_{2})*(D*d

^{2}C

_{2}/dx

^{2}-v*dC

_{2}/dx+J

_{2}]

But I don't 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!