We left off with some useless equations for incompressible flow. Well, I guess they'd be useful if anyone wants to build a water pump out of a Tesla turbine. The author also provided an equation for compressible flow:

Where K= ratio of specific heats, which is 1.4 for air. M is the mach number, but it is unclear to me how to estimate the Mach number. Mach is the ratio of flow velocity to the speed of sound, however In one part of the paper he uses the flow velocity at the inside radius (radius of the intake hole), in another part he uses the outside radius (MUCH higher Mach speed). Also, I have no idea what Pr is. Presumably it is the static pressure, but at what point? (intake or exhaust?) He doesn't actually use compressible flow for his application in the paper, so without an example to follow I don't know enough about fluid dynamics to figure it out. If there are any aerodynamics gurus reading this that can shed some light for me that'd be awesome!

In the meantime, I'm going to take a different route. Ideally, the air isn't really flowing past the discs much at all, but is stuck to them and slowly migrates out radially as it picks up speed. If we assume this "no slip" ideal case, and take our reference frame to be the discs themselves, then we essentially have an acceleration gradient. Gases subjected to high speed rotation show up in another application, gas centrifuges for isotope separation. Turns out this is a fairly well studied/understood topic, and I found a paper with just the equation I need. I'll skip the derivation here, look to the paper if you are interested. This is the simplest set of assumptions, what the paper calls "Iso-thermal rigid body rotation":

Ps(r) is the static pressure at radius "r", Pw is the pressure at the wall of the cylinder, M is the molar mass of the gas (.028964 kg/mol for air), Vw is the wall velocity (Vw=omega * radius of the cylinder), R0 is the general gas constant (8.31447 J/mol), and T0 is the average gas temperature. In a centrifuge, this initial equation must be modified to account for the length of the cylinder, as the effect of drag from the end caps diminishes towards the middle. In our case however, our two "end caps" are actually the discs, which will be VERY closely spaced, so the rigid body rotation assumption should actually be a fairly good approximation. I hope.

The paper from which this equation came notes that this only holds true down to the Knudsen flow range, where molecules stop interacting and pushing on each other. Normal flows due to pressure gradients don't really work the same here. I have been unable to find anything in the literature describing Knudsen or molecular flow in a Tesla disc turbine, so I don't really know if this will continue to describe my pump's operation below this range. However, molecular drag vacuum pumps are similar in construction to Tesla pumps, and are specifically designed with very small channel dimensions, so that the flow remains molecular into relatively high pressure ranges. This indicates that the function of the pump may actually improve at very low densities.

Next I'll plug in some real numbers and get an idea of what size to shoot for in the prototype.