Diffusion law for continuous domains

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

• 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. – Time4Tea Nov 7 '18 at 19:44
• Hey @Time4Tea, I've edited the post – Maxi Köhler Nov 7 '18 at 19:51
• Sorry but, what's the question then? Are you asking for higher order diffusion formulae? – FGSUZ Nov 7 '18 at 21:41
• "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? – Maxi Köhler Nov 7 '18 at 22:11
• So ye kind of higher order diffusion formulae – Maxi Köhler Nov 7 '18 at 22:20