Interdisciplinary Applied Mathematics

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10.5.1 Experimental and Computational Results

The first experimental work to investigate the validity of the no-slip boundary condition    at    a    solid    surface    was    conducted    by    Coulomb    (1784),    who

concluded that it was valid even at microscopic scales. About a century later, Helmholtz and von Piotrowski (1860) found evidence of slip between a solid surface and a liquid, and later, (Brodman, 1891) verified their results. However, Couette (1890) and others used glass tubes with grease inside and concluded that the no-slip boundary condition is valid. It is

FIGURE 10.18. Definition of slip length b. The inner boundary layer S is explained in the section on conceptual models of slip.

worth noting also that the experiments of Whetham (1890) that led to the acceptance of the no-slip condition were made on hydrophilic surfaces (Bonaccurso et al., 2003).

Navier    (1823)    was    the    first to model partial    slip    at    the    wall    for    liquids

well before Maxwell’s slip condition for gases (1879). Specifically, Navier’s boundary condition at the wall is

vs = bn ■ (Vv + VTv) .    (10.5)

An    interpretation    of    the    slip    length    b is    shown in    the    sketch    of Figure

10.18 for unidirectional flow over a flat wall. It is the distance behind the solid-liquid interface at which the velocity extrapolates to zero.

The validation of slip boundary conditions continued in the beginning of the twentieth century, focusing mostly on flow in capillaries. (Traube and Whang, 1928) reported a four- to five-fold increase in flowrate of water in a capillary treated with oleic acid. However, their results could be interpreted either as boundary slip or simply as surface-tension-induced capillary rise. In independent experiments with water flowing in capillaries treated with paraffin, (Ronceray, 1911) also studied how changes in the surface tension may affect the flowrate but concluded aganist the slip condition. The most systematic study, perhaps, of this effect was undertaken by (Schnell, 1956), who used water in glass capillaries (from 240 to of 800 p,m) treated with dimethyldichlorosilane (i.e., silicone) to make them hydrophobic. He found that for a small pressure drop in the capillary the flowrate was lower in

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