Interdisciplinary Applied Mathematics

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2000; Travis et al., 1997). For example, Figure 10.14 shows the velocity distribution for Poiseuille flow in a 4-fluid-molecule-diameter slit channel for three systems (see Section 10.2 for a description of the potentials used in the three systems) with different interaction potentials between fluid-wall and fluid-fluid (Travis and Gubbins, 2000). For each system considered, the velocity profile obtained from MD simulations is no longer parabolic and deviates significantly from the Navier-Stokes prediction. Specifically, for system A (we will discuss only the result for system A, since the results for system B and system C are similar to that of system A), the velocity decreases in the region 0.75 < z < 0.97 as we approach the channel center, and    there is a local    maximum    for    the    velocity located    at    z    «    0.95.

The corresponding strain rate profile is shown in Figure 10.15, and the strain rate is zero at z « 0.97 and z « 0.2. The fluid viscosity inside the channel, calculated by

M*) = Tji$,    (Ю.4)


is shown in Figure 10.16. Note that in equation (10.4), Txz(z) is the shear stress at position z, and a local, linear constitutive relationship between the shear stress and the strain rate, on which the classical Navier-Stokes equation is based, is assumed. We observe that the viscosity calculated by equation (10.4) is negative in the region 0.75 < z < 0.97 and 0 < z < 0.2 and diverges at z « 0.97 and z « 0.2. This indicates that the viscosity in such a narrow channel cannot be described by a local, linear constitutive relation. Therefore, the classical Navier-Stokes equation is not valid for the analysis of fluid flow in a 4.0-fluid-diameter slit channel.

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