Total work done in pumping a layer water through a bottleneck

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Pumping water through a bottleneck requires more force compared to a constant radius due to the increased pressure needed to overcome the constriction. The work done can still be calculated using the formula Work = Force x Distance, but the force will vary based on the changing radius. Even if the final height of the water remains the same, the dynamics of flow through a bottleneck affect the overall work required. In ideal or laminar flow, wall collisions are disregarded, simplifying the calculations. Understanding these factors is crucial for accurately determining the total work done in such scenarios.
JustSomeGuy80
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I don't know much about physics but I know that Work = Force x Distance. Does it take more force to pump an x amount of water through a bottleneck (a shrinking radius from a to b and the constant radius from b to c) as opposed to pumping that same amount of water through a constant radius? If so, how do I factor that into the equation?
 
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If the final height of the water is the same, then the same amount of work against gravity should have been done...ideal flow, or laminar flow, disregards collisions against the bottle walls.
 

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