# Interdisciplinary Applied Mathematics

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This behavior suggests that the continuum flow theory is not valid for fluid

10

8 —

6 —

4 —

2 —

0

2 —

-6 —

8

10

0

0.05    0.1    0.15

Distance from the channel center (nm)

FIGURE 12.8. Shear    viscosity    across the channel    for    case    5    (W    = 0.95    nm,    as    =

+0.124 C/m2). Note that the shear viscosity, computed by using a linear, local constitutive relationship, diverges at z ~ 0.14 nm and becomes negative in the region 0.09 nm < z < 0.14 nm.

flow in such narrow channels. Specifically, the velocity profile shown in Figure 12.7 indicates that the shear stress cannot be related to the strain rate by a local, linear constitutive relationship. The shear stress across the channel can be computed by

Tzx(z)=    c(z)zqEext dz,    (12.6)

J 0

where z = 0 is the middle plane of the channel, c(z) is the ion concentration, and Eext is the external electric field along the channel length (x-direction). Figure 12.8 is a plot of the shear viscosity calculated by

+(z)

Tzxjz)

du/dzz

(12.7)

Figure 12.8 indicates that the shear viscosity, computed by assuming a local, linear constitutive relationship between shear stress and strain rate, diverges at z « ±0.14 nm and z « ±0.09 nm, and becomes negative in the region 0.09 nm < z < 0.14 nm. These unphysical results indicate that the continuum flow theory, which assumes that the shear stress can be related to    the    strain    rate    by    a    local    constitutive    relationship, is    not valid

for electroosmotic flow in a 0.95 nm wide channel.

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