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Nondimensionalization of diffusion equation

  1. Apr 23, 2017 #1
    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 :
    upload_2017-4-23_11-22-49.png

    with the conditions :
    upload_2017-4-23_11-22-40.png
    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 :
    upload_2017-4-23_11-24-30.png
    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
     

    Attached Files:

  2. jcsd
  3. Apr 23, 2017 #2

    hilbert2

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  4. Apr 23, 2017 #3
    Hi, I've tried what you've advised me, here are my results :
    upload_2017-4-23_16-42-1.png
    We therefore have:
    upload_2017-4-23_16-43-59.png

    For the conditions I found:

    upload_2017-4-23_16-42-42.png

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

    Attached Files:

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