flux/fickslaw.hh File Reference

Diffusive mass flux according to Fick's law. More...

#include <dumux/flux/fickslaw_fwd.hh>
#include <dumux/flux/cctpfa/fickslaw.hh>
#include <dumux/flux/ccmpfa/fickslaw.hh>
#include <dumux/flux/box/fickslaw.hh>
#include <dumux/flux/staggered/freeflow/fickslaw.hh>

Go to the source code of this file.

Description

Diffusive mass flux according to Fick's law.

Fick's law describes the diffusive flux of mass as proportional to it's concentration gradient in a given phase, caused by the Brownian molecular motion.
For a single phase system, the proportionality constant is the molecular diffusion coefficient $D_m$.

$$\mathbf{j}_{d} = - \varrho D_m \textbf{grad}\, X$$

Extending this to multi-phase, multi-component systems, Fick's law can be expressed as follows:

$$\mathbf{j}_{d,\alpha}^\kappa = - \varrho_\alpha D_\alpha^\kappa \textbf{grad}\, X_\alpha^\kappa$$

Here $D_\alpha^\kappa$ is the molecular diffusion coefficient of component $\kappa$ in phase $\alpha$.
In a porous medium, the actual path lines are tortuous due to the impact of the solid matrix. The tortuosity and the impact of the presence of multiple phases is accounted by using an effective diffusion coefficient $D_{pm,\alpha}^\kappa$.
The effective diffusion coefficient is then a function of tortuosity $\tau$, porosity $\phi$, saturation $S$ and the molecular diffusion coefficient $D_{m}$ ( $D_{pm,\alpha}^\kappa=f(\tau,\phi,S_\alpha,D_m)$).
Models to describe those effects are for example Millington-Quirk [45] or Constant-Tortuosity [12], [9].

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