# Interdisciplinary Applied Mathematics

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FIGURE 12.6. Comparison of water velocity profile across the channel for case 1 (W = 3.49    nm,    as    = +0.120    C/m2)    as    predicted    by    the    MD    simulation    and by

the continuum flow theory.

2002) that the viscosity of water increases dramatically in the near wall region. Such a dramatic increase in viscosity seems to be related to the high electric field strength (Hunter, 1981), layering of the fluid molecules (Lyklema et al., 1998), and the high concentration of ions near the channel wall. However, a comprehensive theory accounting for all the effects is not yet available.

The question of whether the continuum flow theory based on a constant viscosity can predict the flow behavior in the central part of the channel is an interesting one. We observe that if the predicted velocity is shifted down by about 22.4 m/s, the continuum prediction matches the MD velocity at a distance 6    away    from    the    channel    wall;    i.e.,    the    continuum    prediction

matches the MD simulation result very well in the central portion of the channel. This is equivalent to saying that if the velocity at a position 6 away from the channel wall is given as the boundary condition to the Stokes equation (12.1b), then the continuum flow theory based on a constant viscosity can still be used to predict the velocity in the central part of the channel. Figure 12.6 also indicates that the no-slip boundary condition is applicable to the case studied. However, the no-slip plane is not located at the center of mass of    the    innermost    layer    of    the    channel    walls    (i.e.,    layer    I in Fig

ure 12.2), but is located at approximately 0.14 nm from the channel wall, where the water concentration is almost zero. In Section 16.3, we describe a multiscale approach that can be used to calculate the velocity profile in the entire channel.

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