# Interdisciplinary Applied Mathematics

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the chemical potential. This is shown in Figure 10.23, which is taken from (Andrienko et al., 2003), and is in nondimensional units (the molecular size a is employed in the nondimensionalization). When a thick film is obtained the corresponding slip length depends on temperature, while below the threshold for transition the slip length is independent of the temperature.

4. No-shear/no-slip patterning: This model was first considered in (Philip, 1972), in an attempt to explain slip in porous media. The main idea is to consider the liquid-solid interface segmented into aternating stripes of no-slip and no-shear and deduce an effective slip length from this static configuration. This model was advannced more recently by (Lauga and Stone, 2003)    who    extended    some    of    the exact    solutions    in    (Philip,    1972),

and hypothesized the existence of small bubbles attached to the wall as providers of the slip and thus the corresponding stress-free condition. As was already mentioned, there is direct experimental evidence by AFM of the existence of such nanobubbles; see, for example, (Tyrrell and Attard, 2002). In rough surfaces or surfaces with tiny cracks, air pockets may exist that act as stress-free local boundaries. Therefore, the proposed model is that of surface heterogeneities that lead to an effective or equivalent macroscopic slip.

The two basic configurations, a longitudinal and a transverse one, considered in the works of (Philip, 1972) and (Lauga and Stone, 2003), are

(a)

(b)

h

^ H

FIGURE 10.24. Longitudinal (a) and transverse (b) models of no-shear stripes.

shown in the sketch of Figure 10.24 for a capillary. Semianalytical Stokes flow solutions can be obtained for these geometries, and the effective slip length be can be obtained in terms of the ratios 6 = h/H and L = H/R and the capillary radius R. This effective slip length is defined indirectly from the flow rate as follows: Let us assume that the partial slip condition is applied everywhere, then the velocity profile is

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