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Assume I have a given volumetric concentration $c$ of $N=10000$ particles in a qubic box of volume V:

$$c = \frac{N}{V}$$

The units of $c$ are

$$[c] = \frac{[\#]}{[m^3]}$$

Now if I introduce an infinitely small path along the $z$ direction: $dz$, so that the concentration in the infinitely small volume becomes:

$$dz \cdot c = \frac{N}{V}\cdot dz$$

If I want to reduce a 3D problem to a 2D problem in order to make some numerical studies , however, this is not very helpful since $dz \cdot c$ will approach zero as $dz$ goes to zero.

I realize that I can insert an arbitrary $dz$, say $dz = \Delta z = 1$ m. This basially means that the surface concentration is equal to the volumetric concentration. Is that correct?

If not, how do I get a surface number concentration out of the volumetric number concentration that I can actually do calculations with?

EDIT: So what I am trying to model in the end is a fluid flow through a pipe in three dimensions. At the inlet of the pipe, I inject particles which are given as molar concentrations. The next step is to study diffusion and convection phenomena of those concentrations. The reason why I want to do this in 2D first is simply that it saves a lot of computation time.

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  • $\begingroup$ I think it depends on what you're modelling. For example pressure is a 2D phenomenon and if you take an arbitrarily short timeslice the pressure would be zero. The pressure is non-zero because it's collisions per unit time. Can you give us an idea what you're trying to model. $\endgroup$ – John Rennie Jun 29 '13 at 8:25
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You can say that your particle surface distribution is

$$\sigma=\frac{N}{\ell^2}=\frac{c}{\ell} $$

but take care, because $\sigma$ and $c$ have not the same dimensions. They may have the same numerical value (up to dimensions) if $\ell=1$ but they represent different physical quantities. Your idea is kind of related to a physical and very intuitive motivation of Fubini's theorem

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  • $\begingroup$ ok, thanks. I think this will work ok, since I'm running the 2D case only to check if my code is stable over large domains. I will do the computations in 3D eventually. $\endgroup$ – seb Jun 29 '13 at 8:40

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