# Nondimensionalization of diffusion equation

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1. Apr 23, 2017

### DzoptiC

1. The problem statement, all variables and given/known data
We let a dye diffuses into an environment of dimension L. We inject that dye into a box by one face, at t = 0 on x = 0. The linear density c follows that equation :

with the conditions :

2. Relevant equations / questions
a. nondimensionalize the equations and the conditions
b. reveal a term homogeneous to time, and its signification
c. compare the characteristic lenghts of these equation systems

3. The attempt at a solution
By nondimensionalize this equation, I found this :

But I think it's wrong... I use the "formal way" to nondimensionalize the equation as shown in the Khan academy video on youtube.
May I ask for help ?
Thanks a lot

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2. Apr 23, 2017

### hilbert2

I think you should start by finding out the numbers $\alpha_1 , \alpha_2 , \alpha_3$, $\beta_1 , \beta_2 , \beta_3$, $\gamma_1 , \gamma_2 , \gamma_3$ so that the variables

$\tilde{x}=L^{\alpha_1}m_0^{\alpha_2}D^{\alpha_3}x$
$\tilde{t}=L^{\beta_1}m_0^{\beta_2}D^{\beta_3}t$
$\tilde{c}=L^{\gamma_1}m_0^{\gamma_2}D^{\gamma_3}c$

become dimensionless. $L$ is any characteristic length of the system you want to choose.

3. Apr 23, 2017

### DzoptiC

Hi, I've tried what you've advised me, here are my results :

We therefore have:

For the conditions I found:

I'm not quite sure about the integral term though..