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Your hands coming together rapidly displaces air, pushing it in a focused direction towards the flame. If you watch on high speed cameras, the flame is extinguished before the clap. The wick has no air to combust the fuel anymore.

It's the same as blowing out a candle. You are separating the burning gases from the fuel faster than it can burn more. At the point that there are no more burning gases touching the fuel, the flame is out.

Hang on. My understanding was that blowing out a candle suffocated it (your breath was all CO2). Is that understanding incorrect?

The flame in your example need 3 things: fuel, oxygen and heat.

The fuel comes from the candle wax or lighter fluid or whatever, the oxygen comes from the air and the heat initially comes from the flint in a lighter or the friction of striking a match etc.

When you blow, or move air rapidly in some manner, on a flame you move the heat off the fuel for long enough for the reaction to stop.

No, the air you breathe out is only about 5% carbon dioxide. It still contains about 15% oxygen, compared to about 20% oxygen in regular air

Yes that understanding is incorrect. Your breath is not all CO2 and contains enough oxygen for combustion.

The CO2 doesn’t help, but it’s not the reason it goes out. A candle can be extinguished by a fan or wind blowing regular air with normal oxygen richness just the same.

Yeah, I guess I was kinda stupid there. You also blow on fires when they are first starting, so I should have realized that it doesn’t necessarily put it out.

Think of the concept of blowing gently on a small fire to make it grow. They steady stream of fresh o2 from your breath gives it more oxygen than the air around the fire.

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Wicked flame sources such as candles depend on convection to operate. Once lit, gasses in the candle's flame rise and cooler oxygenated air rushes in from the sides and below to replace it. As this new air approaches the flame it rapidly heats up, reacts with the fuel in the wick, ignites and rises, continuing the cycle. This is why the flame's colour transitions from blue (initial ignition) through orange and finally yellow (where the last of the evaporated fuel is consumed). This is also why a candle flame extends up from the wick.

Rapid air movement from blowing or clapping drastically increases the amount of air flowing past the wick, which elongates the flame and lowers the concentration of heat around the wick. Less heat near the wick reduces the rate at which its fuel vapourizes, which makes less of it available to react with the air.

The air itself absorbs heat. Slow moving air absorbs more heat per unit volume, so it more readily reaches the fuel's ignition temperature and sustains the reaction. Faster moving air absorbs less heat per unit volume, making ignition more difficult.

This is a long-winded way of explaining how blowing or clapping your hands cools the area around the wick to the point that ignition ceases.

A slight increase in airflow velocity beyond what simple convection produces actually improves ignition by making more oxygen available at the point of ignition. This is why blowing on a fire causes the embers to glow brighter and increase in temperature. But there's a tipping point beyond which the heat drawn away from the seat of the flame overcomes the more efficient ignition due to extra oxygen from the additional airflow.