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I am trying to figure out the energy balance at the end of a hose pipe with water pouring out. Applying Bernoulli's equation inside the pipe we have three energy terms: Kinetic energy, "pressure" energy and gravitational potential energy. This total energy is conserved unless its converted to heat or whatever through pressure losses.

Consider the situation just inside and just outside the end of the hose pipe. Assume the pipe is horizontal so we can ignore the gravitational component. Let say just inside the end of the hose we have 50psi and 2 feet per second. Then consider the situation just outside the end of the hose. We still have 2 feet per second because the fluid stream has a constant diameter (its not a nozzle) and is virtually incompressible. However, we now presumably have atmospheric pressure in the fluid stream, i.e, 14.7psi.

So what happened to the energy contained in the pressure difference, i.e, the difference in energy between the water at 50psi and at 14.7psi? Or is the pressure inside the fluid stream suspended in the air still at 50psi? Or does the water expand very slightly to account for the difference between 50psi and 14.7psi and the energy is released by the water doing work against the air to perform this expansion? (And maybe the expansion is so small that it is not visible so the fluid stream appears to be a constant diameter?).

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When the water flows out of a pipe to the atmosphere, the higher pressure will gradually decrease along the streamline. The pressure gradient means that there will be an accelerating force acting on the water. As the pressure drops, speed of water increases as predicted by the Bernoulli theorem. The water jet streaming out of higher pressure region actually gets contracted even if it pours just from a hole (take a look at mild water leak from kitchen tap, or from a hole in bottle of water). The volume expansion of the water and connected work is completely negligible here.

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