The Air Hockey Table

Another possible departure from the ideal conditions assumed in the analysis is that the film may become thin enough that the continuum-based hydrodynamic analysis may break down in various ways. First of all, for a sufficiently thin film, either the no-slip conditions or the bulk-phase continuum approximation itself may break down. In either case, the resistance to further decrease in the gap width would be decreased compared with that which underlies the predictions (5-121) and (5-122). However, as we will discuss in more detail in the next chapter, this is not expected to be a factor when the fluid is a small-molecule (Newtonian) liquid until the film thickness is less than about 10-100 A, and in many cases, the surface roughness discussed in the preceding paragraph will have a longer length scale than this. 

In the present section, we consider only the problem of calculating the height of a stationary disk as a function of the pressure p R in the air reservoir below the table surface. For this purpose, we consider a flat disk of radius R and total massM, levitated (or suspended) an (unknown) height d . R above a horizontal porous surface through which air is blown from a reservoir that is held at a pressure p R. A sketch of the configuration is shown in Fig. 5-10. Although the air motion and the air pressure distribution in the thin gap between the disk and the tabletop may be quite complicated, and there may be some significant 

pa is the ambient pressure, which acts on the upper surface of the disk and opposes/ ) . We thus consider reservoir pressures that exceed this minimum value. It is convenient to parameterize these pressures in terms of a dimensionless pressure p R,  

Finally, to complete specification of our problem, we need to specify the normal inflow velocity V across the porous tabletop. For the present purposes, we assume that V is linearly related to the pressure differences across the porous tabletop  

to determine the levitation height of the disk, d, we must first determine the pressure distribution in the thin air gap below the disk and then apply a vertical force balance on the disk. The analysis in the thin gap can be carried out by means of the dimensionless thin-film equations, (5-61), (5-65), and (5-66). These equations are scaled with respect to the characteristic length scale along the gap, lc, which in this case can be taken as the disk radius lc = R a characteristic velocity scale along the gap uc and a characteristic pressure Pc = [(puc/R) (1/e2)], where the thin-gap parameter e = [(d/R) 1], 

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A similar implementation was introduced in Delettre et al. (2010). This platform is able to move an object on an air-hockey table based on a novel traction principle. The object is pulled in and moved indirectly by an airflow which is... 

Figure 5-10. The configuration for a levitated air hockey puck above the surface of a porous table top through which air is being blown at a superficial velocity V .

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