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I left out what holds it down, for the same reason I left out what set it in motion, I wanted to keep things simple. The wheels in the non-rising case are in a frictionless track that prevents the airfoil from rising. (If I’m not allowed to stipulate that something is “frictionless” for the purpose of analysis, then the analysis becomes a little bit more complicated but overall it doesn’t really change. In the throwing the ball up and then letting it fall analysis presented above the implied stipulation is that there is no friction.) ---- Whether the change in the upward “buoyant force” on the moving airfoil is enough to make it rise by itself, or if it is only a part of all the factors contributing to lift (and even if it is only a tiny, tiny part) it is still a factor in lift and there must be a decrease in another form of energy (even if it only a tiny, tiny decrease in another form of energy) for energy to be conserved. And the point of my post here, and my issue, is that I cannot find that decrease in another form of energy.

I may in fact be using the term “buoyant force” wrong if it technically only applies to fluid statics and not also to fluid dynamics. If there is a different term I should be using, please let me know. However, the upward and downward pressures from the surrounding fluid on an airfoil do definitely affect lift. How could they not? Like any submerged body, airfoil or not, if the pressure from the fluid pushing down on the top plus the weight of the body is greater than the pressure from the fluid pushing up on the bottom then it will fall, if they are equal it will remain in place, and if it is less it will rise. And when an airfoil moves, the pressure from the fluid on the top and on the bottom decreases (Bernoulli’s principle). And it has been experimentally shown that the air moves faster over the top of the airfoil than it does under the bottom, which means (again, Bernoulli’s principle) the pressure pushing down on it decreases more than the pressure pushing up on it decreases. And so, a horizontally moving airfoil has more upward “buoyant force” on it from the surrounding fluid than when it is at rest. And this is definitely a factor in lift. ---- I agree that simple is better. In my original post there is nothing to do with engines or fuel. And when engines were first mentioned I should have simply responded by saying “There are no engines. The airfoil is simply stipulated to have been set into motion.” In trying to address it so we could then move on, it became a distraction. ---- I don’t know how to state the issue more simply that I did in the original post. It’s not a super complicated question, but there is enough to it that it takes a couple of paragraphs to state it. I wish I could reduce it to a couple of sentences, but I can’t. Perhaps there is something specific that I didn’t make clear to you, and you can let me know and I’ll be happy to try to be clearer.

Okay. I’m not exactly sure what the issue is here when it comes to defining the “isolated system.” When I throw a ball up into the air, and as it rises it slows down, energy is conserved because as the kinetic energy decreases there is an equal increase in gravitational potential energy and then when the ball falls back down energy is conserved because as gravitational potential energy decreases there is an equal increase in kinetic energy as the ball speeds up. This, to me, is a perfectly fine analysis of how energy is conserved when a ball is thrown up into the air and then comes back down. I don’t see the need analyze whether the thrown ball is truly in a closed system or not. It can just be stipulated that it is. So, fine, the rising and non-rising airfoils are not in the whole universe but are stipulated to be in isolated systems. ---- It could be that these hypothetical airfoils have hypothetically perfect engines and so no heat is produced and so all of the chemical potential energy lost becomes an equal amount of kinetic energy. Or, it could be these hypothetical airfoils have realistic engines and so the chemical potential energy lost becomes an equal amount of thermal and kinetic energies. Either way, once the airfoil is set into horizontal motion that is where this question begins.

(I missed the edited part of this response.) Are you saying that when the airfoil rises and gravitational potential energy increases, the other form of energy that correspondingly decreases is “lifting potential energy”? And are you saying that as the airfoil moves horizontally and the upward buoyant force on the airfoil increases “lifting potential energy” is created? If so, I disagree. The upward buoyant force on the airfoil is a force and not energy. ---- Sure, a complete analysis of the conservation of energy of a rising airfoil needs to include an analysis of all aspects contributing to lift. And if you’d like we can discuss the other factors that contribute to lift and how energy is conserved in each aspect. However, one of those aspects is an increase in the upward buoyant force and it is that aspect that I am stuck on (not the others).

Not all potential energy in the universe in gravitational. Gravitational potential energy is one form of potential energy. And it is part of the conservation of energy analysis of a rising airfoil. ---- Yes, it will not rise if constrained. But if it is not constrained, and if the upward buoyant pressure (from the different decreases in pressures on the top and bottom) is sufficient enough then it will rise. And this means an increase in gravitational potential energy. And for energy to be conserved this must come with a corresponding decrease in another form of energy. Yes, I did ask you all where I went wrong. I believe I understand the law of conservation of energy fairly well. In this case the isolated system could be the whole universe. (But I don’t see how the rest of the universe, and all of the rest of the various dynamics occurring out there, effects the analysis of the rising versus non-rising airfoils.)

Conservation of energy is one of the most fundamental principles of physics. ---- The engines on an airplane (unlike that of a rocket) are not setting the body into vertical motion but into horizontal motion. They convert chemical potential energy (in the form of jet fuel) into an equal amount of kinetic energy (in the form of horizontal motion). However, due to the different decreases in pressures on the top and bottom of the airfoil, this horizontal motion also means there is an increase in the upward buoyant pressure on the airfoil. And this increased upward buoyant pressure (if it is sufficient enough) will cause the airfoil to rise. And as the airfoil rises there is an increase in gravitational potential energy. And if energy is to be conserved this increase in gravitational potential energy must come with a corresponding decrease in another form of energy.

I’m having trouble with the conservation of energy analysis of a rising versus a non-rising airfoil. (The non-rising airfoil is held down in place and prevented from rising.) ---- When an airfoil moves horizontally the surrounding fluid moves more quickly over the top of the airfoil and less quickly under the bottom of the airfoil. This means the downward pressure on the top of the airfoil decreases more and the upward pressure on the bottom of the airfoil decreases less. If the (decreased) upward pressure on the bottom of the airfoil becomes greater than the weight of the airfoil plus the (even greater decreased) downward pressure on the top of the airfoil then it will rise. If it rises then there is an increase in gravitational potential energy. If it is prevented from rising then there is no increase in gravitational potential energy. So, for energy to be conserved, when an airfoil rises there must be a decrease in another form of energy that does not decrease when the same airfoil prevented from rising. And so, when an airfoil rises its motion must slow down so that there is an equal decrease in kinetic energy while if it is prevented from rising its motion must not slow down in this same way. And so (the only way I can think of to end up with this difference), when an airfoil rises there must be an increase in drag that does not occur when the same airfoil is prevented from rising and that increase in drag must slow the airfoil down by the precise amount where the decrease in kinetic energy equals the increase in gravitational potential energy. And if all of this is correct, then the conservation of energy analysis is resolved. But I don’t think I’m right. Two airfoils of equal masses (and so equal increases in gravitational potential energies) will produce two different amounts of drag if they are shaped differently and so will slow down by two different amounts and produce two different decreases in kinetic energies. Unless, it could be that, yes, two differently shaped airfoils will produce two different amounts of drag (in both the rising and non-rising cases), but if they are allowed to rise then there is also an additional amount of drag, and this additional amount of drag is the same no matter the shape of the airfoil (and is precisely what is needed to slow the airfoil down to match the increase in gravitational potential energy). While this is possible it doesn’t seem likely. If two airfoils are shaped differently, and if there is additional drag on them as they rise, common sense suggests that that additional drag will vary depending on the shape of the airfoil (and so energy cannot be conserved in both cases). What am I missing? ---- I understand that there are other factors that contribute to lift (angle of attack, conservation of momentum, and so on) and a full analysis of lift is complicated and sometimes disputed. But even if all of the other factors were included and the above aspect of lift only contributed a small amount to the overall lift for a particular airfoil, it still is a contributing factor and there still must be some slowing in the rising case that does not occur in the non-rising case for energy to be conserved and this slowing must be the same for airfoils of equal masses but of different shapes. ---- ?