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I am making an "air bearing" as a model of a zero length fibre torsion balance. As part of the initial design I need to ensure I can actually levitate the float. As a simple approximation to get a rough estimate I considered the air bearing to be a chamber where pressurised air is tapped in, and there is a single hole of radius r in the top of the chamber where the air exits, causing the float of mass m and radius L to levitate. I feel like it should be a simple problem but i cannot estimate the elevation height or the maximum mass that can levitated. Any help would be appreciated.

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  • $\begingroup$ Are you assuming that the system is 2D, and infinitely long into the paper? $\endgroup$
    – Chet Miller
    Commented Oct 22, 2018 at 11:50
  • $\begingroup$ No the float and chamber are cylindrical with a radius L, the diagram is a cross section at full radius if you like $\endgroup$
    – jjp1996
    Commented Oct 22, 2018 at 12:34
  • $\begingroup$ I still don't fully understand the geometry, but, in the design of bearings, lubrication theory is typically employed. $\endgroup$
    – Chet Miller
    Commented Oct 22, 2018 at 14:27

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For a back-of-the-envelope estimate of the maximum possible mass that could be levitated, you can assume that all of the force of the air attempting to escape the chamber is used to levitate the mass. If the chamber is at pressure $P$ and the hole has a radius $R$, then the maximum total upward force you would expect is $P\pi R^2$. As such, any mass greater than $\frac{P\pi R^2}{g}$ will not be levitated.

As for the levitation height, you would need to know the details of the air flow at various distances from the hole; such a calculation is heavily dependent on other details, like the Reynolds number of the flow, and as such, I don't know that there is a simple way to estimate it outside of using computational fluid-dynamics simulation software.

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  • $\begingroup$ I think lubrication theory can be used. $\endgroup$
    – Chet Miller
    Commented Oct 22, 2018 at 14:25
  • $\begingroup$ @ChesterMiller Given that the leading-order equations include $\frac{dp}{dz}=0$, I'm not sure that this is helpful in calculating the variation in pressure with gap height. $\endgroup$
    – probably_someone
    Commented Oct 22, 2018 at 14:48
  • $\begingroup$ Lubrication theory is how bearings, including air bearings, are designed. $\endgroup$
    – Chet Miller
    Commented Oct 22, 2018 at 14:52
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Let Q be the volumetric throughput rate of air through the bearing, and neglect the compressibility of the air. Then the average radial air velocity in the gap is: $$v=\frac{Q}{2\pi r h}$$ where h is the levitation height. From lubrication theory, the radial pressure gradient is related to the radial velocity by $$\frac{dp}{dr}=-\frac{12\eta }{h^2}v=-\frac{6\eta Q}{\pi rh^3}$$where $\eta$ is the air viscosity. The equation can be integrated from $r=r_0$ to arbitrary r to get p as a function of r. Once p is known as a function of r, one can integrate again from $r=r_0$ to r = L to get the levitation force for a given air gap. One can then solve for the air gap h that matches the weight of the float.

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