The underwater Hero engine. Discussion.
This puzzle turns out to be quite simple. It attempts to deceive us by using jet propulsion, something like the nozzles of Hero's engine. Many people think that jets work by "pushing on the surrounding fluid". This is the "rocket fallacy". So they are tempted to think that the jet on the left pushes on the outside water, while the hole at the right takes in water without any effect on the wheel. But niether of these processes exert a force on the device, only the moving water inside can do that, exerting forces on the walls.
We immediately see that this can't work by applying Stevin's principle. After an imagined revolution of 180° this device would be in exactly the same state it began. Therefore no work was done and it has no energy change. Idea demolished.
Sweating the details.But the details are interesting. Let's imagine the device, as shown, immersed in water, initially in static equilibrium. Will anything initiate rotation? Why would it? Water will not flow unless there's a pressure difference. Pressure does depend on height, but the water pressure on the top of the right red piston is the same as the water pressure at the bottom of the left piston (Pascal's principle). Let's call this water pressure p The pressure on the trapped air is, on the right, p + P where P is the additional pressure due to the piston's weight. The air pressure on the top of the left piston is the same. This makes the pressure on the bottom of the left piston p + P + P = p + 2P. We have generated a result that contradicts our assumption of a static system. In fact, this strongly suggests that both pistons will move downward, just as we may have naively supposed from casual analysis.
Well, what if the pistons were moving and forcing water through the left nozzle and pulling water in on the right side? As water goes "around the bend" of the left jet chamber it exerts a force on the chamber that is upward and outward, just as the water going around the bend of a lawn sprinkler. So this exerts a clockwise torque on the system. In the chamber on the right, the incoming water exerts a force on the chamber walls that is also upward and outward, causing a counterclockwise torque that is equal and opposite to the clockwise torque of the jet at the left.
The left jet ejects water during this piston movement, but the right input takes in water at the same rate, since water in the chambers is nearly incompressible and preserves volume. So the movement of the pistons doesn't cause the whole device to move. The fact that the water exchange is with a jet on the left, but not on the right doesn't make any difference.
The floating balls (black) that close openings in the jet chamber are not a problem; they would work as describedeven if the device were turning round and round.
The ejected water does exert a force on the outside water, but that force does not act on or affect the spinning device.
So what forces do act on the device? Water inside exerts forces on the walls of the container as the water changes direction going around the "bend" of the enclosure. But these produce torques that add to zero. The wheel will not rotate, even if the pistons are moving.
Comparison with the Hero Engine.The very same thing happens in the Hero engine. The steam going around the bend in the nozzles experiences a force due to the right angle bend in the tubes, and the steam exerts an equal size and oppositely directed force on the tubesa force that has an outward radial component, and a tangential component. It is this tangential component that supplies the thrust (torque) that causes the device to rotate.
In plain language, the Hero engine and the water sprinkler rotate because both nozzles are ejecting material, and that material has net angular momentum. Our puzzle engine doesn't rotate because one arm ejects water while the other arm is taking in an equal amount of water. (It makes no difference that water on one side moves through a nozzle, and the water on the other side doesn't.) The pistons initially move, but reach their limit and then just sit there, at rest.
Hero's engine is so named because Hero of Alexandria was the first to describe it. We have little hard evidence about this man, not even his birth and death dates, though he was apparently writing about 62 CE. Devices that spin by ejecting material from orifices are called Eolipiles. Lawn sprinklers are in this category.
The Water Sprinkler.Which way would a water sprinkler turn if it were submerged in a tank of water and water were being "sucked in" by a water pump on the other end of the hose? This is the classic "reverse water sprinkler" problem, described and explained by Ernst Mach in his classic book "The Science of Mechanics" (first German edition 1883). Before you answer, consider that if it did begin to rotate, where would the angular momentum come from? Then ask yourself whether the ordinary lawn sprinkler conserves angular momentum. So what's the crucial difference between an ordinary lawn sprinkler and the reverse lawn sprinkler? What would an ordinary lawn sprinkler do if it were immersed in a pool of water? There's an experiment you can do yourself in your backyard pool (or even your bathtub). But before you do it, predict the outcome.
Physicists will also observe that there's no energy input, no momentum input, and no angular momentum input to our perpetual motion device puzzle. Hence no reason to expect it will magically acquire any of these.
One can get deeply into physics with problems of this sort. Richard Feynman experimented to settle the question of the reverse water sprinkler problem. It isn't an easy experiment, because of the viscosity of water. There a toy, the pop pop boat that takes advantage of water viscosity to propel a small tin boat using the heat from a candle or an alcohol burner.
Ernst Mach's classic book "The Science of Mechanics" (Open Court publishing company, English edition) can be downloaded in a free pdf file here. The discussion of the reverse sprinker is in chapter III, section III p. 388. But the one big difference between Hero engines, water sprinklers, pop pop boats and this perpetual motion puzzle is that the PM puzzle has no external energy source. The others do.
Final observation.After all this discussion, we realize that all this analysis should have been unnecessary. Our device may look similar to the water sprinkler or Hero engine, but it has a significant and crucial difference. We may be left with the feeling "Why did I ever suppose for even a moment that this silly thing might work?" Could it have been the nozzles that misled us?
References for further reading.Rueckner, Wolfgang. The puzzle of the steady-state rotation of a reverse sprinkler. Am. J. Phys. 83, April 2015. pp. 296-304. This thourough examination of the problem concludes that the reverse sprinkler will not rotate in steady state operation. It only rotates during the time the water flow is turned on or off.