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According to the Bernoulli's pipe equation, whencross sectional area decreases,velocity of water flow increases. However, when the tap is slightly loose,a small cross sectional area is produced in pipe, water flow at low velocity.

Isn't that mass flow rate should be constant for both case.Then why are they result in two different velocity, one with high velocity, another with low velocity?

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The Bernoulli equation omits viscous friction which is the dominant factor in a valve or tap modulating water flow rate through the pipe.

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