# Interdisciplinary Applied Mathematics

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density profile.

While all the results presented above on the calculation of density profiles in nanochannels were based on MD simulations, the density profiles can also be calculated using analytical methods. (Fischer and Methfessel, 1980) as well as (Bitsanis et al., 1988) have used the Yvon-Born-Green (YBG) theory (McQuarrie, 1973) of inhomogeneous fluids with the Fischer-Methfessel approximation for the fluid pair-correlation functions (Fischer and Methfessel, 1980) to calculate the density distribution of Lennard-Jones atoms confined in a nanoscale channel. In order to obtain an equation for number

FIGURE 10.6. Fluid density distribution in smooth-slit channels of different widths. The results    with    reservoir    are    shown    as    a    line,    and    the    results    with    no

reservoir are shown as circles. A reservoir is introduced to simulate the Couette flow. (Courtesy of H. T. Davis.)

density one has to approximate the pair correlation function, which is the Fischer-Methfessel approximation. This is the only approximation needed, and in this respect the YBG theory is superior to the free-energy theories. The YBG theory has been shown to predict the fluid densities near smooth walls with good accuracy.

##### 10.3 Diffusion Transport

Diffusion transport is typically important in most nanofluidic systems. This can be understood by calculating the Peclet number, Pe = UL/D, which measures the ratio of the bulk transport (convection) to the diffusion transport. In most nanofluidic systems, the characteristic length L ranges from a nanometer to a micrometer, and the bulk velocity ranges from a micrometer per second to a millimeter per second. For a fluid with a diffusivity of D = 1.0 x 10~9 m/s2, the Peclet number ranges from 10~6 to 1, indicating that diffusion either dominates the transport or is as important as the bulk transport.

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