B Resnick Halliday Krane: Venturimeter formula mistake?

AI Thread Summary
The discussion centers on a potential mistake in the Venturimeter formula, highlighting that if both fluids have the same density, the calculated velocity would be zero, which is illogical. Participants suggest that the formula implies a condition where the density of one fluid must be greater than the other for accurate results. They emphasize that the manometer only functions correctly without flow, as it reflects pressure differences rather than velocity. A clarification is made regarding the application of Bernoulli's equation, noting the need to relate pressures at different points accurately. Overall, the conversation focuses on the correct interpretation and application of fluid dynamics principles in the context of the Venturimeter.
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If this formula were to be correct, if we use the liquids in the Venturimeter and the tube to be of the same density, the velocity would come out to be zero which makes no sense. I calculated the formula and got a slightly different numerator. Am I correct?
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My guess would be that it means that if the two fluids has same density then for any speed the interface between them will never "settle" at a fixed height difference but simply make a "flow loop". Or in other words, the formula as it is given has an implied condition that ##\rho' > \rho##.
 
Filip Larsen said:
never "settle" at a fixed height difference but simply make a "flow loop".
Yep. To my intuition (not to be totally relied on) tells me that filling the whole apparatus with the same liquid, you just have two paths with the same pressure difference at each end. The manometer only works when you have no flow through it; it just ends up lop sided according to the pressure difference over the venturi tube.
 
Filip Larsen said:
My guess would be that it means that if the two fluids has same density then for any speed the interface between them will never "settle" at a fixed height difference but simply make a "flow loop". Or in other words, the formula as it is given has an implied condition that ##\rho' > \rho##.
Yes but what mistake did I make in my derivation using the bernoullis equation then?
 
Your formula ##P_1 = P_2 + \rho'gh## should read ##P_a = P_b + \rho'gh##, where points ##a## and ##b## are as shown
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You'll need to relate the pressures at 1 and 2 to the pressures at ##a## and ##b##.

Bernoulli's equation holds for two points on the same streamline. Note that a streamline passing through points 1 and 2 (shown below) rises by a distance ##h_{12}## in going from 1 to 2. You'll need to take this into account in Bernoulli's equation if you use this streamline.

1732209316401.png

If you take all of this into account, you should be able to derive the equation for ##v_1## given in the textbook.
 
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