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The figure below shows a water column that is separated into $3$ parts: Grid1, Grid2 and Grid3. Their thickness are $\operatorname{H1}$, $\operatorname{H2}$, and $\operatorname{H3}$. The concentration $(\operatorname{C1}, \operatorname{C2}, \operatorname{C3})$ is defined at the center of each grid.

If the grids have uniform size, the diffusive flux $J$ can be calculated using Fick' Law. For example, the flux between Grid1 and Grid2 is:

$$J = -D\cdot\frac{{\mathrm{d}C}}{{\mathrm{d}z}} \approx -D\cdot\frac{{\operatorname{C2} - \operatorname{C1}}}{{0.5\cdot(\operatorname{H1}+\operatorname{H2})}}$$

However, since the concentration profile changes quickly near Grid1 and slowly downward, these three grids are coarser one by one. For example, $\operatorname{H1} = 1$cm, $\operatorname{H2} = 3$cm, $\operatorname{H3} = 20$cm. Can we still use the equation above? Given the large $\operatorname{H3}$ here, the diffusive flux between Grid2 and Grid3 will be low because the denominator is $0.5\cdot(\operatorname{H2}+\operatorname{H3})=11.5$, compared with $0.5\cdot(\operatorname{H1}+\operatorname{H2}) = 2$.

Fig.1

T X
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  • The basic equations don't change, the discretizations do. I'm not seeing how your approximation is valid as you let your mesh shrink. – A rural reader Feb 25 '21 at 01:09
  • @Aruralreader Could you say a bit more details? I didn't check if the mesh shrink is valid: just draw three boxes according to the shape of concentration profile (red line in the right panel). – T X Feb 25 '21 at 01:36

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