Interdisciplinary Applied Mathematics

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In Figure 10.22 (left) the flowrate normalized by the theoretical prediction for no-slip Poiseuille flow is plotted, taken from (Cheng and Giordano, 2002). Specifically,


«the°ry =

(where h, w, L denote the height, width, and length of the microchannel) is used in the normalization. We see that for all the liquids tested, the flowrate increases except    for    water.    In    particular,    hexadecane    (the    fluid    with    the

largest molecular size) exhibits the largest deviation from the no-slip theory. This is in agreement with the results in (Pit et al., 2000), for the capillary hexadecane flow, although the slip length values reported in (Cheng and Giordano, 2002), are much smaller. In general, the experimental evidence given in (Cheng and Giordano, 2002), indicates a monotonic increase of the slip length with the molecular size but for channel height h < 300 nm; above this value the no-slip theory seems to be valid according to the results of (Cheng and Giordano, 2002). This dependence on the molecular structure is shown in Figure 10.22 (right), indicating also that there is some weak dependence of the slip length on the channel height in the slip regime. However, this effect may be associated with the uncertainties in measuring the very small values of h.

Other experiments with larger microchannels for pressure-driven flows revealed boundary slip for water, in contrast to the aforementioned results of (Cheng and Giordano, 2002). For example, in (Tretheway and Meinhart, 2002), microPIV (300-nm diameter fluorescent polystyrene spheres) was used to measure velocity profiles of water in a 30 x 300 pm channel. The channel surfaces were treated with a 2.3 nm OTS layer. The velocity profiles were measured in a 25 x 100 pm plane to within 450 nm of the channel wall. A slip velocity of about 10% of the maximum velocity was measured, which corresponds to slip length of about 1 pm. This is a very large value for the slip length, of the order of magnitude that is typically encountered in polymer flows. For the untreated glass surface, which is hydrophilic, noslip conditions were observed. Similar reults were also reported in (Choi et al., 2003), in smaller hydrophobic microchannels of 0.5 pm and 1 pm height. The channels were 500 pm wide and 9 mm long, while the surfaces were coated with OTS layers to make them hydrophobic. In both the experiments of (Tretheway and Meinhart, 2002) and (Choi et al., 2003), roughness was negligible. The slip length was found to depend linearly on the shear rate with b = 30 nm at a shear rate of 105 s-1 for hydrophobic surfaces, while for hydrophilic surfaces b = 5 nm at the same shear rate. The corresponding slip velocity was of order 3 mm/s for the hydrophobic case and 0.5 mm/s for the hydrophilic case.

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