I was recently studying applications of Bernoulli Equation and came across the Ventury tube. This is diagram I have used to analyse the venturimeter. I understand how we obtain the first equation using bernoulli theorem which is

$$P_1 - P_2 =(1/2)ρ(v_2^2 - v_1^2) \tag{1}$$

and also the continuity equation

$$A_1 v_1 = A_2 v_2. \tag{2}$$ However, I am unable to process how to obtain the following third equation

$$ P_1 - P_2 = ρgh $$

where $h$ is difference in the heights of the liquid level in the two tubes and $ρ$ is density of fluid.

Lets say the atmospheric pressure at the top of each tube is $P$. Now since the fluid in the two VERTICAL tubes are at rest and not moving, their velocities are 0. Hence, if I proceeded by applying Bernoulli equation.

$$P_1 +ρ(v_1^2)/2 = ρg(h_1) + P \tag{3}$$ and

$$P_2 + ρ(v_2^2)/2 = ρg(h_2) + P \tag{4}$$

Here $P_1$ and $P_2$ are the pressures at the points in the tube and constriction respectively and the points are at the SAME HORIZONTAL LEVEL.

Subtracting (3) and (4) and even using (1) does not yield me

$$P_1 - P_2 = ρgh $$

rather gives me $0 = ρg(h)$ which makes no sense whatsoever.

What am I missing here and how do I obtain the right result ? enter image description here

  • $\begingroup$ It’s been a while and I haven’t yet received a suitable answer to this question. Can someone tell me if they’re having a problem with the way the question has been posed because I would really appreciate an answer to this question as it would help my understanding of fluids significantly. Please let me know so I can edit to the right demands/interests in hope of obtaining an answer. $\endgroup$ – Hola Jun 10 '18 at 17:48

Let the atmospheric pressure be P° Therefore P¹=P°+(rho)gh¹ P²=P°+(rho)gh² Subtracting we get the required equation P¹-P²=(rho)gh 🏴🏳️🏴

  • $\begingroup$ That is in static condition right? Here the liquid in the horizontal tube is moving. How can you apply that? $\endgroup$ – Hola Mar 27 '18 at 14:55
  • $\begingroup$ Umm nothing to do with it.The liquid is not moving in the vertical direction.We only need to see that the pressure below the first tube is P¹ which is shown by the height of liquid raised.🏴🏳️🏴 $\endgroup$ – Priyam Shah Mar 28 '18 at 17:33

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