Bernoulli's equation and conservation of energy

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Bernoulli's equation can be derived from the principle of conservation of energy by analyzing a system of fluid between two cross-sections. The equation equates the work done by pressure at each end of the fluid volume to the changes in kinetic and potential energy of the fluid. The focus is on the fluid at the endpoints, as it is this mass that experiences changes in energy, while the fluid in between remains unchanged. Understanding this derivation involves considering the entire fluid system's movement and the energy transformations occurring at the boundaries. This explanation clarifies how Bernoulli's equation relates to energy conservation in fluid dynamics.
broegger
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Can anyone explain to me how Bernoulli's equation arises from conservation of energy?
 
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Why is it that you only account for the kinetic and potential energy change in the blue volumes.. What about the fluid between them?
 
The blue volumes are assumed to be the same volume of fluid as it proceeds in time.

You're looking at two different time slices
 
Originally posted by broegger
Why is it that you only account for the kinetic and potential energy change in the blue volumes.. What about the fluid between them?
All the action takes place in those end volumes. Nothing changes for the fluid between them.

Here's how to understand this derivation. Think of the entire chunk of fluid between the two cross-sections (A1 & A2) as "the system" to be studied. We want to study what happens when that system moves such that each end sweeps out a given volume of fluid. What Bernoulli's equation does is equate the work done on the system (done by the pressure at each end) to the change of kinetic and potential energy of the system.

To answer your question again, note that the net effect, as far as calculating the change in energy goes, is to move a mass of fluid from one end to the other. This is the only mass that changes kinetic and potential energy---the rest of the fluid doesn't change.

Make sense?
 

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