# Argument for stability of journal bearing via inertial and viscous forces?

As I understand a journal bearing, a rotor is levitated by hydraulic forces from the lubricant. While these are lots of studies on this subject, I can't manage to find a paper that coherently breaks down the forces into components based on qualitative arguments.

What's more, the idea seems to violate intuition. Let's restrict our view to fully laminar flow, just for simplicity. Literature claims that this is stable, like most Reynolds numbers, so there should be no problem there. The outer cylinder is taken to be stationary, so the no-slip condition on the outside holds the fluid in place there. Obviously the fluid matches the speed of the surface of the inner cylinder there as well. If we just drew a linear line between the velocity of the inner and outer cylinder we would arrive at a contradiction, because both of these values are constant (non-zero velocity and then zero), and the flow rate of fluid over the channel must be constant. This seems to be telling me that the fluid is moving faster in the center of the channel where the channel is more narrow.

Consider one of the many illustrations of this:

The problem: Isn't the fluid inertial force pointing to the bottom right in the left image above? We're squeezing the fluid through a very narrow channel in the bottom right, so the fluid pressure should drop by Bernoulli Law.

If fluid pressure is pulling the shaft down, and gravity is pulling the shaft down, what is holding the shaft up? While some non-neutral frictional force exists, it seems to be going more in the right than upward. My initial reading is that the fluid inertial forces would be perpendicular to the friction force at the narrowest point. This can't mathematically form a complete basis to hold up the shaft as far as I can tell.

Could anyone give something like a free-body diagram for a journal bearing, with the different kinds of fluid forces accounted for as well?

Is my interpretation of the fluid inertial forces correct? Or is the more deeper and more nuanced principle at work?

• Looking into this... Step 1: Bernoulli's law is specifically derived for inviscid flow. A journal bearing is very much a case of viscous flow. Bernoulli comes from conservation of energy, In this case we have energy being added by the rotating shaft and constantly dissipated by viscosity. – user3823992 Dec 11 '14 at 17:08