Choosing Reference Points in Bernoulli's Equation for Fluid Flow

• Alettix
In summary, Bernoulli's equation states that the pressure, speed and volume are inversely related, and that the pressure is highest at the surface of the fluid and decreases as you descend. The small opening at the bottom of the tank has a much lower pressure than the surface of the water, and the water flowing out of the small opening has a much higher speed than the water flowing in. Reference point number 2 should be placed just outside the opening of the small pipe in order to maintain equilibrium.
Alettix
Hello!
I have encountered some trouble with choosing the right reference points when using Bernoulli's equation and I would be glad if you could help me sort it out with this made up example. :)

1. Homework Statement

There is a large, open, cylindrical water tank with a cross section area of ## A_1 = 2 m^2 ##. 1 meter down the water surface (## \Delta h = 1 m##) is a small pipe with cross section ## A_2 = 0.0003 m^2##. Determine the speed of the water flowing from the pipe.

Homework Equations

Bernoulli's equation: ## \frac{p_1}{\rho} + gh_1 + v_1^2/2 = \frac{p_2}{\rho} + gh_2 + v_2^2/2 ## (1)
Continuity equation: ##A_av_a =A_bv_b## (2)

The Attempt at a Solution

We can assume that the flow is laminar, the fluid incompressible and the viscosity very small, therefor we can apply Eq.(1). We choose to place reference point 1 at the surface of the water in the tank, and point 2 at the small pipes opening. Because ##A_1>>A_2## we can assume ## v_1 \approx 0##. We set ##h_2 = 0## and therefore ##h_1 = 1 ##. We do also know that ## p_1 = p_0##, where ##p_0## is the atmospheric pressure.
Solving for ##v_2## we have:
## v_2 = \sqrt{\frac{2(p_0-p_2)}{\rho} + 2g\Delta h} ## (3)

Now to my question: Where exactly should I put reference point number 2? If I put it just outside the opening of the smal pipe we have ##p_2=p_0## and consequently ##v_2 =\sqrt{2g\Delta h}##. However, if I put the point just inside the opening the pressure will be: ##p_2 = p_0 + \rho g\Delta h## which yields ##v_2 = 0##.

I think something here must be wrong. It isn't reasonable that the water just inside the small opening should be completely still, but outside it flow with a speed as large as in case of free fall. This result seem especially crazy if we consider that the continuity equation must hold in the small pipe. So where is the fault in my reasoning? Should point 2 be placed just outside or inside the opening of the pipe? Why?

Thank you for your help!PS: I really wanted to draw a figure, but there is something very wrong with paint on my computer right now.

Either point is OK, and both points give the same answer. Just inside the opening, the velocity is also equal to the velocity just outside, and the pressure is equal to the pressure just outside. Your mistake is assuming that ##p_2 = p_0 + \rho g\Delta h## just inside the opening. The system is not in static equilibrium, so this equation cannot be applied.

Chet

Alettix and CrazyNinja
Chestermiller said:
Either point is OK, and both points give the same answer. Just inside the opening, the velocity is also equal to the velocity just outside, and the pressure is equal to the pressure just outside. Your mistake is assuming that ##p_2 = p_0 + \rho g\Delta h## just inside the opening. The system is not in static equilibrium, so this equation cannot be applied.

Chet

Thank you very much Sir! This really clarified it!

What is Bernoulli's Equation?

Bernoulli's Equation is a fundamental equation in fluid mechanics that describes the relationship between pressure, velocity, and elevation in a flowing fluid. It states that as the speed of a fluid increases, the pressure decreases and vice versa.

How is Bernoulli's Equation used in science?

Bernoulli's Equation is used to analyze and predict the behavior of fluids in various situations, such as in pipes, pumps, and aircraft wings. It is also used in the design and optimization of engineering systems that involve fluid flow.

What are the assumptions made in Bernoulli's Equation?

The main assumptions made in Bernoulli's Equation are that the fluid is incompressible, inviscid, and irrotational. This means that the fluid has a constant density, no internal friction, and no rotation in its motion.

Can Bernoulli's Equation be applied to all fluids?

No, Bernoulli's Equation is only applicable to ideal fluids, which do not exist in the real world. However, it can still provide a good approximation for many real fluids, such as air and water, under certain conditions.

What are some practical applications of Bernoulli's Equation?

Bernoulli's Equation has numerous practical applications, including in the design of airplanes, cars, and ships. It is also used in the design of pipes and pumps for water distribution and in the study of blood flow in the human body.

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