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Why does the same amount of fluid pass through every cross section of a pipe in steady flow?
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fluid mechanics
Figure 1: Schematic representations of (A) a differential manometer, (B) a Torricellian barometer, and (C) a siphon.
In fluid mechanics, the observation that the same amount of fluid passes through every cross-section of a pipe during steady flow is a direct consequence of the continuity principle.
This principle is rooted in the law of conservation of mass. In a steady flow—where the velocity of the fluid at any fixed point remains constant over time—mass cannot be created or destroyed within the pipe. Therefore, for any defined volume within the system, the mass entering the volume must equal the mass exiting it. If the flow rate were different at various cross-sections, mass would either accumulate or be depleted within the pipe, which would violate the fundamental requirement that mass is conserved in a closed system.12