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Has anyone worked out a physically justified equation (perhaps parametrized) for the characteristic (2D outline) shape of a candle flame? Just one half suffices, as it is clearly symmetric about a vertical (in the absence of wind).


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            (Image from this link.)
Motivation: To make realistic candle flames in computer graphics. We recognize the shape when we see it. And when it is not quite right, we notice the discrepancy. I want to get it right.

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  • $\begingroup$ I really like this question. This is going to be a very difficult question to answer. Does a colour-blind person see the same shape? Does a dog, or a cat see the same shape? (Some animals' eyes are sensitive to different parts of the electromagnetic spectrum.) $\endgroup$ – Fly by Night Nov 12 '16 at 23:08
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    $\begingroup$ @FlybyNight: Let us assume: humans :-), and the 92% of humans not colorblind. $\endgroup$ – Joseph O'Rourke Nov 12 '16 at 23:10
  • $\begingroup$ Sorry to be a pain, but do all humans see things the same? I've read that some people have a third type of photoreceptor, i.e. more than just rods and cones. I think that the heat equation is the way to look at this question. How does heat propagate from the source? You'll get some kind of contour plot which shows the dissipation of energy. Some of which will manifest itself as visible to some people. touchophthalmology.com/articles/… $\endgroup$ – Fly by Night Nov 12 '16 at 23:16
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    $\begingroup$ Regardless of colour vision, the edges are fuzzy, so there is not one clear outline. $\endgroup$ – Arthur Nov 12 '16 at 23:20
  • $\begingroup$ @FlybyNight: The point you are raising is legitimate but not relevant to the vast majority. I am content to hit 92% of people. $\endgroup$ – Joseph O'Rourke Nov 12 '16 at 23:21
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I Googled "Heat equation applied to a burning candle" and found

Analysis of Burning Candle

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  • $\begingroup$ Just as I expected, some fluid dynamics is involved. $\endgroup$ – Berrick Caleb Fillmore Nov 12 '16 at 23:36
  • $\begingroup$ That (nice) paper does not directly address my question: I am seeking a planar equation of one side of the flame shape (one side because the other is symmetric). This is not evident from the physics of the fluid-mechanics convection, etc. $\endgroup$ – Joseph O'Rourke Nov 12 '16 at 23:37
  • $\begingroup$ @JosephO'Rourke That's a the best I could find. Like I said: what is visible? I love your question, and please don't think I'm being difficult. I truely believe that different people see different things, You might be able to get a computer program to find a boundary, but only because you told it what to look for, and told it what a boundary is. Digital cameras are poor imitations of the human eye. The answer to your question is the solution to the differential equations in the paper I linked to. The sad fact is that analytic solutions are out of our grasp :-( $\endgroup$ – Fly by Night Nov 12 '16 at 23:44

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