Do we know how a constant level oiler works?

The Basic Principle (Based on Bernoulli’s law)

When rotary equipment is under rest, the constant level oiler very much resembles a simple, mercury barometer. As long as the neck of the filler bottle (reservoir) is immersed in the pool of lubricating oil, preventing air intrusion, the column of liquid in the bottle is supported by the atmosphere. The pressure in the air/vapor space at the top of the bottle, plus the head of fluid, exactly balances ambient air pressure (p).

If depletion of lubricant causes the level (x) to drop below the bottleneck, it allows air to enter. As air bubbles up into the bottle, it displaces lubricant, which pours down into the pool below.

 When the pool surface rises sufficiently to again seal the bottle from air, lubricant flow stops. The vapor space and new, lower, liquid height have re-equilibrated with ambient pressure. Figures are self-explanatory.