I don't think it's going to work. I had another idea if the stove were a box, and there was no entrance or exit, and you have a tube like secondary air going into the stove (the air from the tube never touching the inside of the stove, that is, not to give air to the inside of the stove but to draw heat away from the stove), and then an L shape so it exits at the top. The L-shape would be a chimney, so it would extend for some distance, but the heat of the stove
should create air movement in the tube, as well as the draw from it being a chimney. But then you could make the tube line the entire inside of the stove, so the heat of the stove would cause movements of air in the tube, which is isolated from the inside of the stove. What I'm trying to say, is to make a pressure difference somehow, so that the air insulation lining the inside of the stove inverts the temperature there, so the stove is cold, and the tube liner draws heat away from the stove. I noticed on a pressure cooker, submerged in
water, right next to the surface it was cold, and any distance away the water was hot in the sink filled with water to cool the pressure cooker. But my idea is that smoke rings are persistent, so if you can get the stove to burn like smoke rings, it could possibly burn on only the air that was in it to begin with. And, if you can get it into smoke rings, movement would be smoke snakes that travel through the rings, but the usual way to create a snake is to have two holes, to form a figure 8 around the two holes. So the stove may not need to be completely closed but have two holes somewhere. I would also add, that once the snake forms a figure 8, it would have to travel to a center and curl up into a ball before travelling on, but, it would have to be made so that it can keep moving and not disintegrate when it touches the stove walls.
Basically what I'm thinking is that suppose the original idea of two holes is correct near the base of the barrel, and everything else is closed. Both holes are open. Air is coming through the two holes, and then a computer closes them simultaneously. Then air around the 2 closed holes, spins around both holes and a figure 8 forms across the two holes, with the 2 holes of a figure 8 drawing the two small stove holes. Then it can form a snake, but the problem about forming a snake is that you can never get it to continue forever, as perfect symmetry, the snake will touch the edges (I made a cellular automata that did the same thing) and disintegrate. So the snake will have to be stabilised by opening one or the other of the holes but not both, because if you open both, a new snake will form destroying the old one. So probably it would work but it would need a sensor, and a learned neural net, so I'll try to work on my cellular automata, by adding only one dark square at any given time, to see if it will stabilize, but never adding two simultaneously exactly at the same time after the first. But it also would have to be a square box, as a stove with a hole on the left and a hole on the right, probably. I think the sensor would be the heat of the outside of the stove, if it's cold it's working, if it's getting hotter, change something.