Say for instance we had a humanoid creature. how would they achieve flight by taking in a gas then expelling it at a greater pressure from a point on their body like the feet, back, or hands?
This creature needs a matural electrolyzing unit in its back. Given that it must live in a very humid environment, it must collect said humidity while, inside the 'tank' a symbiotic variant of the geobacteria will separate hydrogen and oxygen. One is expelled, the other is stored.
When in danger, the creature's adrenalyne stimulates the chamber where the hydrogen is stored and ignites it, turning the backpack into a rocket pack for a long jump.
Earth's fauna proves beyond doubt that flight is possible only with a proper body conformation, there's no way around it. Hollow bones in an aerodynamic body, better if feathered to better soar.
If a humanoid wanted to fly, without being a bird, it should have ample, foldable wings between legs, and between arms and trunk, a tissue much less dense than a human's, so that it could float like a manta in the air. On land, it would be clumsy like a condor or a vulture.
This is not worldbuilding, at my university there were humanoids capable of lift by expelling gas after a beans based meal. :)
Ok, to answer the question, the main issue is the required power to have a proper lift. Humans/humanoids can't fly using their own power because their power-mass ratio. Birds and insects have a way better ratio and, notably, the flight mechanism tends to be more efficient for bigger dimension: insects fly easily stationary (actually, like the hummingbird, which is indeed very small), while the big birds tends to glide rather than flap their wings because it's energy efficient. If you don't want you humanoids to be weak on the ground because of hollow bones and/or with a bird-like shape because fly efficiency, you'll need a lot of power.
In order to lift a humanoid (~ 80 kg) you'll need ~ 800 newton of force just to levitate, if you also want to gain altitude you'll need more. The only thing that i can imagine that's powerful enough for that is solid rocket fuel. This could work this way:
Your humanoids don't have feet but have hoofs instead, really, really strong hoofs.
The place in which they live the ground is abundant of rust powder because of a near by iron mine, and they collect this rust in their hoofs just by walking. If you don't want to have the flight capability location-dependant you can have the same result extracting iron from blood, maybe they are vampire like creatures, but you will really need a lot of blood.
They don't dump manure and urine as we do, instead they have a chamber/organ that extract potassium nitrate before the dump, and they store also this nitrate in their hoofs.
Since they eat a lot of fruits, they also have sugar abundancy in their organism, and to avoid dangerous sugar concentration their digestive system dump the excessive cristallized sugar - guess where? - in their hoofs!
By walking they mix these components and this is more or less a proper rocket solid fuel, so they only need some ignition spark to start the propulsion and aim for the sky.
Of course, the process of accumulation is slow, so they can't fly often and/or for long time but still they can fly.
This is a bit outside of your question, but it's more or less feasible that an animal could achieve flight by imitating a hot air balloon.
It would need some large bladder of sorts, and a lot of heat. This heat could be chemically produced. Perhaps the biggest difficulty would be surviving its own heat.
It could also be a helium balloon, but biologic helium distillation seems like it would be very difficult. Hydrogen might be more feasible in that sense as it can be created by water electrolysis. (But it might require some sort of low conductivity tissue.)
In these ways, an animal or even a plant could fly for long periods of time.
You have two options: static or active
- Static: more or less what you get when you inflate a party balloon and let it loose. The internal pressure, as long as it last, will propel the balloon around. You need to have a large storage and a very light body to do this, which is really non practical for a living organism.
- Active: you need to imitate a reaction engine, sucking in air, compressing it, increasing its enthalpy by some mean and then releasing it. Sadly, combustion and living organisms do not match well together. This type might work for really small organisms: a mechanism similar to that used by the bombardier beetle (see note at the bottom) could maybe lift a tiny organism, but it would be a pulsed moved, not a continuous propulsion.
If instead you are not in air but in water, the static method can be used. See what octopuses, squids and cuttlefishes do.
How the bombardier beetle works:
The spray is produced from a reaction between two chemical compounds, hydroquinone and hydrogen peroxide, which are stored in two reservoirs in the beetle's abdomen. When the aqueous solution of hydroquinones and hydrogen peroxide reaches the vestibule, catalysts facilitate the decomposition of the hydrogen peroxide and the oxidation of the hydroquinone. Heat from the reaction brings the mixture to near the boiling point of water and produces gas that drives the ejection. The damage caused can be fatal to attacking insects. Some bombardier beetles can direct the spray over a wide range of directions.