I'd never seen this sort of behavior in a flame (usually it's either steady burning, or that quick ~2hz oscillation with lots of smoke) but it's pretty clear that there are some balancing feedback loops here with a delay. I'd like to figure out what it was about the candle that led to this sort of behavior, and how it varies from the behavior of a candle in either of the other modes.
Here are a series of still images of the process in various points in the cycle:
In the fourth image you can see that the wick is folded over on itself, such that there is a length of the wick which is thick, connected to the remainder of the wick by a narrow section. I have a hypothesis that this area acts as an additional 'reservoir' of liquid wax, and allows an introduction of a delay as the liquid wax fills this wick reservoir and evaporates. I'm going to try and build a model of the candle to see if this is a plausible mechanism for oscillation, and identify the general state of parameters is which leads to this behavior.
B1 is the balancing feedback loop which works to keep the quantity of wax in the wick at a stable level. Loop R1 is a reinforcing feedback loop which works to increase the size of the flame, and loop R2 is a reinforcing feedback loop which works to evaporate all of the wax in the wick.
In normal operation, the balancing feedback loop dominates, stabilizing the amount of wax in the wick. This gives rise to a relatively constant evaporation and heat transfer rate, and a stable flame. In the oscillating case, it seems that the reinforcing feedback loops dominate, depleting the wick. When the wick is empty, the rate of flow into the wick is greater than the evaporation, and the wick fills up again.
In the next post on this topic, I'll develop a quantitative model of the candle, and see if our hypothesis about the candle's feedback loops actually gives rise to the behavior we see in the video.
In the mean time, here are some interesting references:
Heat and Mass Transfer Inside a Candle Wick: Mandhapati P. Raju.
Paper on heat flux in candle flames