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I love the way candles flicker. It's a great effect and I almost want to see it replicated in an actual lightbulb. I was curious if there is any way to express that mathematically? I'm not that familiar with physics but I go in assuming that most things can be expressed in math, just very complex in some cases!

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You could model the whole process at ever increasing physical fidelity until you got the effect you wanted, but some how I don't think that is what you're looking for. I recall reading somewhere that the behavior of the smoke column goes from deterministic and simple to chaotic as you turn up the total heat under it, so the flame may exhibit a similar transitions. Possibly you can get something "good enough" by tapping into the output of some other chaotic system (damped over-driven oscillator or whatever). –  dmckee Oct 2 '11 at 15:04

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If you want to determine physical causes of flame shape evolution, you would have to consider all processes taking place (chemical, phase changes, fluid dynamics, heat, ...); they are described using known differential equations, i.e. in one point in space. The exact global solution then requires those equations to hold simultaneously (coupling between processes) in all points of your domain, but such (analytical) solution it is only obtainable in very special cases. Approximate global solution is found numerically, by discretizing your problem in space and time, and requiring those differential equations to hold only in finite number of points in some approximate (though mathematically precisely defined) manner.

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