# Interdisciplinary Applied Mathematics

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p(r) = —I p(r + s) d3s. na /6 s<a/2

Figure 10.9 shows the local average density distribution in an 8.0a channel (Bitsanis et al., 1988). Clearly, even though the oscillations in the local fluid density are significant, the local average density shows very little oscillation. This explains the small diffusivity variation in the pore (see Figure 10.8) even though there is a significant variation in the local fluid density. From the above discussion, we can conclude that in confined nanopores the parallel diffusivity at a given position is determined primarily by the average density in the pore and not by the local density.

To investigate the effect of the wall structure on the diffusivity parallel to the pore wall, (Somers and Davis, 1992) performed diffusivity calculations by considering structured and smooth walls. Figures 10.10 (a) and (b) show the variation of the diffusivity D with the pore width for the structured and smooth walls, respectively. The results for the smooth-pore wall (panel

(b)) agree qualitatively with those reported in (Magda et al., 1985), and the results for the structured wall show some interesting differences from the results for the smooth wall; i.e., as the pore width increases, the diffusivity in the structured pore approaches the bulk value more slowly compared to

the smooth wall. This can be explained by the added fluid ordering in the structured pore. Figure 10.10 also shows the variation of the diffusivity with bulk transport of the fluid. For both the smooth and structured pore walls, the presence of Couette flow does not change the diffusivity noticeably for a shear rate less than 0.20Jtjvn,a’21 but the diffusivity increases considerably for shear rates higher than 0.20J tjma2. It is likely that shear thinning may be responsible for the observed increase in the diffusivity. Similar results have also    been    observed    by    (Bitsanis    et    al.,    1987).    In    addition,    the    inset

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