# Interdisciplinary Applied Mathematics

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6= , pvvz    pvvz

where the thickness h of the gaseous film is assumed negligible. Upon substitution of typical values for water vapor in the above expression, we obtain a slip    length    of    a    few    microns,    which is    clearly    much    higher than    any    of

the available experimental data. We note that the model of de Gennes shows that the    slip length    increases    with    the    viscosity    and    thus    with    the

molecular weight, which is consistent with the measurements in (Cheng and Giordano, 2002).

3. Viscosity model: This model, proposed by (Vinogradova, 1999), is inspired by the slip mechanism in polymer melts. It provides a connection between the slip length and a decrease in viscosity within a very thin

8

.3

0.35

0.4

Temperature, T

0.45

0.5

FIGURE 10.23. Slip length versus temperature in prewetting transition for different values of the chemical potential p. (Courtesy of O.I. Vinogradova.)

boundary layer 6 close to a hydrophobic surface. Assuming a bulk viscosity Pb and a near-wall viscosity ps, then the slip length is

This expression shows that there are two mechanisms for obtaining a large slip length, i.e., either by increasing 6 or by increasing the viscosity ratio in the bulk and the surface. For example, for ръ/ps = 21 and 6 =10 nm, a slip length of b = 200 nm can be obtained, but a more realistic viscosity ratio is цъ/ps = 3, which corresponds to b = 20 nm.

The above arguments suggest that there may be another mechanism in place that produces thick films (i.e., large 6), and that is why in some experiments large values of the slip length have been reported. To this end, in (Andrienko et al., 2003), a new model that accounts for prewetting transition was developed. It takes into consideration the structure of the binary mixture in the region near the solid surface and allows for a temperature dependence of the thickness in the form 6 ж — ln(|TwT|), where Tw is the wetting temperature of the surface.

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