Generalized Liquid Undulation Grapher (GLUG!)

It seems here in France they only sell whole and 2% milk, no skim, so I've been relying on almond/soy milk for my morning coffee. Trouble is, those come in a container style that causes me no end of irritation:

The spout is so tiny that the liquid blocks it, causing the system to oscillate between pouring out liquid and sucking in air. This is especially bad when the container is nearly full, and I usually wind up spilling some.

To try to get something good out of this bad design, I wondered if I could predict what the rate of oscillation is between the two states. I figured I'd simplify things a bit, and just consider the bottle completely upside down:

We have a volume of air, a volume of liquid, an inner diameter D, and an opening diameter d. This situation is similar to that of an orifice plate, which uses Bernoulli's principle to determine the mass flow rate through an opening:

C_d is a constant called the coefficient of discharge, which depends on the shape of the opening, 𝜌 is the density of the liquid, and Δp is the difference in pressure across the opening. The air pressure inside starts off at normal atmospheric pressure when the bottle is right-side up, but as the liquid runs out, the volume of the air increases, proportionally decreasing the pressure:

The other pressure inside the bottle is the weight of the liquid:

For the flow equation, we want the difference between these two pressures and the outside air pressure, p_0. We can switch the mass flow from above to a volume using the density, which gives a differential equation once we plug everything in:

This is a mess, and when I tried to put some real numbers in, I realized the weight term is orders of magnitude smaller than the other pressures – This makes sense looking at the video, since each "glug" is only a small amount of liquid. More enlightening might be to find the amount of liquid that pours before the pressures are equal. Setting the terms under the radical to zero gives 8.7 microliters, which makes me less sure of my "this makes sense" assertion above. Either I've made an algebraic error, or (more charitably) this is not an accurate model of the system. I suppose I'll just have to add this to the collection of unsatisfying results!