Engineering
Physics
BernoulIis Equation and Equation of Continuity
Question

A non - viscous incompressible fluid is pumped steadily into the narrow end of a long tapered pipe and emerges from the wide end. The pressure at the input is greater than at the output. A possible explanation is:

the fluid speed is the same at the two ends

the fluid is flowing downhill

the fluid is flowing uphill

the fluid speed increases from input to output

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Solution
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This scenario involves Bernoulli's principle for incompressible fluids: P+12ρv2+ρgh=constant. The pipe's cross-sectional area increases from input to output. For steady flow, the continuity equation A1v1 = A2v2 applies. A larger area at the output means the fluid speed decreases. If the height (h) were constant, a decrease in speed (v) would cause an increase in pressure (P), but here the input pressure is greater. Therefore, the height must be changing. The only way for the input pressure to be higher despite a speed decrease is if the fluid is flowing uphill, increasing the ρgh term and allowing P to drop from input to output.

Final Answer: the fluid is flowing uphill