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Solve Fick's second law of diffusion

  1. Aug 5, 2012 #1
    I'm curious how to solve Fick's second law of diffusion [tex]\frac{∂c}{∂t}=D \frac{∂^2c}{∂x^2}[/tex]For conditions:[tex]c(x,0)=0[/tex][tex]c(0,t)=A[/tex][tex]c(\infty,t)=0[/tex]Physically this means:
    -c(x,t) is the concentration at point x at time t.
    -Initially there is no concentration of diffusing species.
    -At x=0 for all t the is a constant concentration "a".
    -As x goes to infinity for all time, the concentration is 0.
    -D is the diffusivity, assume it is a constant.

    The solution is:[tex]c(x,t)=A erfc(\frac{x}{2\sqrt{Dt}})[/tex]
    What method was used to arrive at that solution?
  2. jcsd
  3. Aug 5, 2012 #2
    Hi !

    May be this formula was obtained thanks to the Laplace method for PDE resolution.
    Normally we would have to use the double Laplace transform (relatively to x AND t), which would be rather arduous.
    But the PDE and boundary conditions are simple enough to use the usual single Laplace transform (relatively to t only).

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  4. Aug 7, 2012 #3


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    Possibly try a similarity solution?
  5. Aug 10, 2012 #4
    Transport Phenomena by Bird, Stewart, and Lightfoot show how to solve this (singular perturbation boundary layer problem) using similarity solutions. Look for the analogous viscous flow startup problem.
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