Interdisciplinary Applied Mathematics

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C* « 2.5 for sebacine-amyl ester within a thickness of 12 nm, and 100 nm for nitrobenzene. This high viscosity was attributed to an ordered boundary phase. However, careful experiments by (Israelachvili, 1992b) using the

FIGURE 1.16. Results for silicone oil. Ratio of the Poiseuille number obtained experimentally over the Poiseuille number obtained theoretically (equation (1.8)). Channels with different sizes (cross-sections) are tested as shown in the legend. (Courtesy of H. Bau.)

atomic force microscope (AFM) showed that water films as thin as 2 nm (corresponding to approximately ten molecular diameters) have continuumlike diffusion characteristics. We have already seen in Figure 1.7 that large density fluctuations occur below ten molecular diameters, and thus in that region it is possible to have C* ^ 1. Water is a seemingly simple fluid, but its numerical modeling is quite difficult; its static and dynamic behavior deserves a separate study, as we present in Chapter 11.

The work of Bau and collaborators (see (Bau and Pfahler, 2001), and references therein) included experiments with different types of liquids in microducts with diameter from 0.48 to 40 p,m. Specifically, systematic experiments with isopropyl alcohol, silicone oil, and distilled water were conducted. A typical result for silicone oil is shown in Figure 1.16, where C* is plotted against    the    Reynolds    number. For    the    small    channels    and    for    low

Reynolds number, C* < 1 is observed, but for larger channels, C* > 1 is observed. In general, the deviation of C* from the theoretical value C* = 1 (using continuum theory) is of order 20% for all cases tested. A history of the research efforts in establishing the no-slip condition as well as conceptual models of slip in liquids are discussed in Section 10.5.

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