# Interdisciplinary Applied Mathematics

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For fluid flow in channels larger than 10 fluid molecular diameters, the classical Navier-Stokes equation can be used to analyze the flow. However, since the fluid can be highly confined in nanochannels, the fluid viscosity may be significantly different from the bulk value. Despite the fact that the fluid viscosity is an important parameter in determining the flow characteristics, only a few papers have discussed the viscosity of fluids in nanochannels systematically. (Bitsanis et al., 1990) have calculated the effective viscosity for Couette and Poiseuille flow in slit channels ranging from 2 to 9.5 fluid molecular diameters. The effective viscosity is defined in such a way that the Navier-Stokes equation using the effective viscosity can predict the macroscopic observables correctly, e.g., the surface shear stress in a Couette flow or the flowrate in a Poiseuille flow. Figure 10.17 shows the variation of the effective viscosity for Couette and Poiseuille flow with the pore width. The plot indicates that:

1. The effective fluid viscosity is flow-dependent;

2. At small pore width, the effective viscosity increases dramatically; and

FIGURE 10.14. Velocity profiles in a 4.0-fluid-diameter slit channel for three different systems. System A: WCA system, filled circles; system B: LJ system, open circles and system; C: WCA-LJ system, open triangles. See Section 10.2 for the definition of the three systems. (Courtesy of K.P. Travis.)

FIGURE 10.15. Strain rate profiles in a 4.0-fluid-diameter slit channel for three different systems. System A: WCA system, filled circles; system B: LJ system, open circles; and system C: WCA-LJ system, open triangles. (Courtesy of K.P. Travis.)

FIGURE 10.16. Shear viscosity in a 4.0-fluid-diameter slit channel for three different systems as calculated by equation (10.4). System A: WCA system, filled circles, system B: LJ system, open circles and system C: WCA-LJ system, open triangles. (Courtesy of K. P. Travis)

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