Everything is about the correct proportions: just as too much or too little oxygen would kill us so too much or too little airflow will kill a fire (without an oxidizer present).
Other than scale candles and log fires do work on slightly different principals. When it comes to the candle you will note that neither the wax nor the wick are themselves on fire. Instead the combustion is occurring slightly above the candle. After you light the wick it does burn for a bit but it liquifies and then evaporates the hydrocarbons that form the wax of the candle. The wick provides a channel for the melted wax to be drawn up to the top where it can then evaporate. These hydrocarbons are then escaping in an upward stream and are then being combusted by the chain reaction of the flame once they’ve mixed with a little air containing O2 (and relatively low CO2 levels). When you blow on the candle you are moving and dispersing that stream of hydrocarbons and at some point it is no longer aligned with the combustion chain reaction and it is no longer dense enough or hot enough for those vapors to keep igniting. At the same time you are moving away the heat source that triggers the melting and evaporation of further hydrocarbon fuel.
With a log fire the fuel for the combustion reaction is coming from the breakdown of the cellulous fibers which is the main component of the wood. This cellulous doesn’t melt but skips straight to a gaseous state and has to fight its way to gaseous pockets in the wood until these gaseous pockets build up enough pressure that they can eventually erupt from the wood. It takes some time and a lot of heat to get this process going. Left to it’s own devices the combustion of these gases occur at various levels above the logs, as and when, they are able to find enough oxygen to react with. As the fire matures the burning penetrates the surface of the wood and you get a coal reaction occurring with the other residual materials. When you are blowing on the log fire you are bringing in fresh oxygen right to the surface of the wood and therefore you are allowing the combustion reaction to occur much closer to and within the surface of the logs/coals. This means that the logs/coals can be better heated by the both the combustion of the volatile gases being released from the lumber and combustion of the remaining dry material in the coals. This leads to quicker breakdown of the remaining cellulous material.
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