# Interdisciplinary Applied Mathematics

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u(z) = [ F (s) — ds = F (z)u(z) — [jL-^-u(s)ds.    (16.33)

J s = 0    ds    J 0    ds

Equation (16.33) can be used to compute the velocity near the channel wall in large channels. Note that no derivatives of the MD velocity in the small channel are needed. Instead, one needs to calculate the derivative of the function F(z), which is obtained by integrating the ion concentration, and it is much easier to obtain good statistics for ion concentrations in MD simulations. In principle, equation (16.33) can be applied in the region from the no-slip    plane to    the    center    of the small    channel    (i.e.,    point        in

Figure 16.8). However, equation (16.33) is used only in the region within S’ from the channel wall. There are two reasons for this:

1. Evaluation of the function F(z) is difficult as we approach the center of the small channel because the integration term in the denominator is close to zero.

2. Since the viscosity variation is important only near the channel wall, we can use a constant viscosity in the region S’ away from the no-slip plane instead of embedding the MD velocity.

In the simulations presented here, S’ is taken to be 0.64 nm, since MD simulations indicate that the viscosity variation beyond this length scale is small. As mentioned earlier, the no-slip plane is typically located at 0.14 nm from the channel wall. Hence, the region in which the velocity is obtained from the embedding technique is S = 0.78 nm from the channel wall for the larger channel.

Figure 16.9 shows the velocity profile across a 10.0-nm channel (case 6; see Table 12.2 for details) obtained by using the embedding multiscale approach. The velocity within the S distance from the channel wall is embedded using equation (16.33) and the MD velocity of a 2.22-nm channel. The velocity in the central portion of the channel is computed using a constant viscosity of 0.743 mPa-s. Although there is considerable noise in

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