# Interdisciplinary Applied Mathematics

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To investigate the effect of the local fluid density on the diffusivity parallel to the pore (D|), the pore (h = 11.57) has been divided into five slices parallel to the solid-liquid interface, and the diffusivity Dц is calculated inside each slice. Figure 10.8 shows the density profile and the diffusivity in each slice. Clearly, even though there is a significant variation in the fluid density, the diffusivity in each slice is within the statistical error of those of the others. To understand this result in more detail, an empirical theory, local average density method (LADM), has been developed to describe the transport coefficient (e.g., diffusivity) of a fluid confined in a nanochannel

FIGURE 10.7. Correlation of the pore-averaged diffusivity parallel to the wall (Dу) with    the    average    fluid    density.    He re    h is    the    pore    width    (reduced    unit);

pave/Pbuik    is    the    average    density    of    the    occupied    pore    volume divided    by    the

density of the bulk liquid. (Courtesy of H.T. Davis.)

FIGURE 10.8. Diffusivity as a function of the distance from the pore walls. Here Di is the diffusion coefficient parallel to the pore walls averaged over the ith slice parallel to the interface. The pore width is 11.57<t. (Courtesy of H.T. Davis.)

(Bitsanis et    al.,    1988). In the    LADM    theory,    the    diffusivity    of    the    fluid    at

a position r depends on the local average density p(r) of the fluid instead of the local density p(r). The local average density at r is defined as the average density inside a sphere with its center at r and with diameter equal

FIGURE 10.9. Density and local average density profiles in an 8<r-wide slit pore. (Courtesy of H.T. Davis.)

to the diameter of the fluid molecules a, i.e.,

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