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I'm an engineer, so don't expect too much :)

I'm modelling with FEM the convection-diffusion-reaction equation which is derived by the continuity equation.

$\frac{\partial c}{\partial t} +\nabla \cdot j = R$

$c$ - scalar quantity like concentration

$R$ - source term

The flux $j$ is splitted in an additive manner into an advectivon and diffusion flux

$j = j_{diff}+j_{adv}=-D\nabla c+\mathbf{v}\cdot c$

$D$ - Diffusion matrix/coefficient

$\mathbf{v}$ - velocity

https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation#Derivation

As far as I understood the $j_{diff}$ term is obtained by using Fick's first law? Since this is a linear relationship I'm wondering if there are nonlinear laws which describe the macroscopic level. In case I couldn't express well what I exactly mean, feel free to ask, then I try to correct myself accordingly.

Best,

Max

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  • $\begingroup$ Hi and welcome to Physics SE. I think it might help if you could explain clearly what $c$, $j$, $R$ and $D$ are, and perhaps provide a link or reference for where that first equation comes from. $\endgroup$ – Time4Tea Nov 7 '18 at 19:44
  • $\begingroup$ Hey @Time4Tea, I've edited the post $\endgroup$ – Maxi Köhler Nov 7 '18 at 19:51
  • $\begingroup$ Sorry but, what's the question then? Are you asking for higher order diffusion formulae? $\endgroup$ – FGSUZ Nov 7 '18 at 21:41
  • $\begingroup$ "First, diffusive flux arises due to diffusion. This is typically approximated by Fick's first law" - Wiki page from the post. I'm wondering about the word approximated, this implies that there may be different laws to approximate the diffusive term in a more sophisticated way? $\endgroup$ – Maxi Köhler Nov 7 '18 at 22:11
  • $\begingroup$ So ye kind of higher order diffusion formulae $\endgroup$ – Maxi Köhler Nov 7 '18 at 22:20

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