# Interdisciplinary Applied Mathematics

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FIGURE 10.13. Velocity profile obtained from MD simulation of Poiseuille flow in an 11-fluid-diameter channel. The solid line is a quadratic fit to the velocity profile. The temperature of the fluid is set to 2.5, the average fluid density is 0.8, and a constant force of 0.1 is applied on each fluid molecule to generate the flow. All the variables are measured in reduced units.

ture) do not vary appreciably over the length and time scales comparable to the molecular free path and molecular relaxation time. However, as shown in Section 10.2, the fluid density near the solid-liquid interface can vary significantly over intermolecular distances. While these local density oscillations may not necessarily mean the breakdown of the continuum theory, it is important to understand in detail how the continuum theory works for fluids in confined nanochannels.

During the last several years, researchers have used MD simulations to test the accuracy of Navier-Stokes equations in nanochannels (Koplik et al., 1989; Koplik et al., 1987; Travis and Evans, 1996; Travis et al., 1997; Bitsa-nis et al., 1987; Travis and Gubbins, 2000; Pozhar, 2000). In many of these simulations, a    Poiseuille    flow    with    a    constant    force    on    each    fluid molecule

is used as a prototypical problem. The continuum Navier-Stokes equations predict a parabolic velocity profile across the channel for the Poiseuille flow. The velocity profiles in slit channels as narrow as 10 molecular diameters indicate that the deviation between continuum and MD predictions is very small    (Travis    et    al.,    1997).    Figure    10.13    shows    the    velocity    pro

file obtained from MD simulation of Poiseuille flow in an 11-fluid-diameter channel and    its    quadratic    fit.    Clearly,    the    deviation of    the    velocity    profile

from the Navier-Stokes equation is small. However, if the channel width is smaller than 10 fluid diameters, the deviation of the MD velocity from the continuum prediction becomes more significant (Travis and Gubbins,

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