Resnick Halliday Krane: Venturimeter formula mistake?

[KnightTheConqueror]

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?

 

[Filip Larsen]

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 $\rho' > \rho$.

 

[sophiecentaur]

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.

 

[KnightTheConqueror]

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?

 

[TSny]

Your formula $P_1 = P_2 + \rho'gh$ should read $P_a = P_b + \rho'gh$, where points $a$ and $b$ are as shown

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.

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