# Bernoulli's Equation Applied to Two Free Surfaces

## Main Question or Discussion Point

Consider two cylindrical containers of the same diameter. Fill them with a liquid to different levels. We all know that the liquid can be "siphoned" from one container to the other using a thin hose, which we will consider to be thin and of uniform diameter.

Bernoulli's Equation can be used to derive the flow rate (which is essentially the same as Torricelli's Theorem).

However, consider applying Bernoulli's Equation to the free surface of each container.

P1 + pgh1 + 1/2 p v1^2 = P2 + pgh2 + 1/2 p v2^2

Because the cylinders have the same diameter, v1 = v2.

Because a free surface is at atmospheric pressure, P1 = P2.

Bernoulli's Equation then reduces to h1 = h2, which makes no sense.

Does anyone have any insight to why Bernoulli's Equation does not work in this case?

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Does anyone have any insight to why Bernoulli's Equation does not work in this case?
It would help if you drew a diagram and identified your symbols.

The three points of interest in the flow are

1) The free surface in the first vessel. Here the velocity is zero. Call it A

2) The knuckle of the connecting pipe. Call it B This can be calculated from Bernoulli.

3) The discharge point into the second vessel. Call it C The velocity here is the same as at B

It would help if you drew a diagram and identified your symbols.
The three points of interest in the flow are
I understand that, but my question is why Bernoulli's Equation give nonsensical results when applied to the liquid surface on each side.

Diagram?

What is v1 and v2? Or rather where are they taken?

Diagram?

What is v1 and v2? Or rather where are they taken?
In the diagram attached, Bernoulli's Equation applied to points 1 and 2 gives h = 0, which makes no sense.

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That is not a syphon, that is two vessels connected by a pipe at the bottom.

Look at my diagram and think about the value of velocity at A

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That is not a syphon, that is two vessels connected by a pipe at the bottom.

Look at my diagram and think about the value of velocity at A
I'm familiar with your analysis. My question is why does applying Bernoulli's Equation at points 1 and 2 in my diagram give h = 0. Applying Bernoulli's Equation in this way yields the same result whether it's a siphon, or if the vessels are connected by a pipe at the bottom.

Because the cylinder's have the same cross sectional area, as I said in the original post.

I know you said that but you are not correctly applying Bernoulli's equation.

You will understand much more quickly if you stop accusing the equation of being wrong and look at your own work.

At the start of the process (vessel 1 full, vessel 2 empty) v1 = 0, v2 can be calculated by Bernoulli and h1 is not equal to h2.

Since the vessels are of limited capacity the velocity changes through the discharge.

At the end of the process v1 = v2 = 0 and h1 = h2 as you have already observed.

At the start of the process (vessel 1 full, vessel 2 empty) v1 = 0, v2 can be calculated by Bernoulli and h1 is not equal to h2.
OK. Why can't we start the process with initial conditions as they are in my diagram, and why does the application of Bernoulli's Equation to points 1 and 2 yield the non-sensical result that h = 0 in that case? I'm not accusing Bernoulli's Equation of being wrong, I'm simply trying to understand its range of application.

Why can't we start the process with initial conditions as they are in my diagram, and why does the application of Bernoulli's Equation yield the non-sensical result that h = 0 in that case?
Of course you can, but you don't have the information to fill in the Bernoulli equation part of the way through the process so you cannot solve it.

What you have written down is not correct for your diagram.

Of course you can, but you don't have the information to fill in the Bernoulli equation part of the way through the process so you cannot solve it.

What you have written down is not correct for your diagram.
OK, let's work through this:

v1 = v2 because the cylinders are identical.

P1 = P2 because both points are at the atmosphere.

So we get h = 0.

What did I do wrong?

OK, let's work through this:
You have shown a picture.

Is this your initial state, ie is there initially some water in each vessel and a tap in your pipe?

So do you start the process by opening the tap?

You have shown a picture.

Is this your initial state, ie is there initially some water in each vessel and a tap in your pipe?

So do you start the process by opening the tap?

Sure. The initial state is as shown, and the process starts by opening a tap.

So at the instant of opening the tap, can you describe the situation, that is put values into Bernoulli's equation?

So at the instant of opening the tap, can you describe the situation, that is put values into Bernoulli's equation?
It doesn't matter how the cylinders are connected or whether or not there are valves. We can start with one empty and one full, and then apply Bernoulli's equation at the time the situation is as depicted in my diagram.

It doesn't matter how the cylinders are connected or whether or not there are valves.

Of course it matters or I wouldn't have introduced a valve.

At the instant of opening the valve the velocity at both ends of the tube is zero.

As a result, Bernoulli will give you the result that the difference in fluid levels equals the static pressure difference between the fluids in each vessel.

You have three variables to insert into each side of Bernoulli. You can reduce this to two (or even one) if you can make one (or two) of the height, static pressure or velocity equal zero, or some other known value such as atmospheric pressure.

The problem arises because as soon as flow starts the static pressure drops and you therefore don't know either.

You keep repeating that the velocities are equal because the vessels are the same size. The velocity is quite independent of the vessel size.

You keep repeating that the velocities are equal because the vessels are the same size. The velocity is quite independent of the vessel size.
Points 1 and 2 are at the surface in each container. If the containers have the same diameter, then v1 = v2.

To solve this you obtain an expression for v using Bernoulli.

If h is the head difference between the higher free surface and the lower free surface, producing the flow then by Bernoulli

h = v2/2g

You can estimate the time to equalise levels by integrating the equation

Where A is the area of the upper free surface and t is time

You can obtain an expression for Q in terms of t from the velocity and connecting pipe diameter.

To complete the task you may need to consider the reverse pressure build up in the second vessel, due to increasing head

To solve this you obtain an expression for v using Bernoulli.

If h is the head difference between the higher free surface and the lower free surface, producing the flow then by Bernoulli

h = v2/2g
When you apply Bernoulli's Equation to two points -- the free surface on each side -- it
yields h = 0. Clearly this is a misapplication of Bernoulli's Equation, but why? What is wrong with using those two points?

Thanks for trying to help.

One last time it doesn't yield h = o.

One last time it doesn't yield h = o.
One last time, yes it does.

See image.

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Gold Member
Here's the thing: it should predict zero velocity. By applying Bernoulli's equation at the two free surfaces, you are basically asking the equation to predict any motion that would occur directly between two surfaces at the exact same hydrostatic conditions. Of course it will predict no flow. If you ignore the atmospheric pressure, there is no head between the two surfaces.

The problem is that your fluid motion is occurring at the bottom of the tank. What you need to do is relate the pressure at one end of the connecting pipe to the height of the water in that tank and then do the same with the other tank. Then you have a pressure differential and can determine the flow rate. The surfaces will certainly move at the same speed since one is losing liquid at the same rate the other is gaining it and the tanks are identical. Keep in mind that the flow rate will change depending on the height, so a differential equation would need to be solved to get the time history of the heights of the surfaces (or the flow rates).

Not once have you asked me how I arrived at my equations.

You also said that you understood them and they are standard stuff.

It is clear that you have a fundamental misunderstanding about the Bernoulli equation and its application.

I have drawn just the full vessel at two times.

At t1 just as the emptying process begins and again at t2 when the top parcel of fluid has travelled some distance down the vessel.

At t1 it is important to realise that the velocity is zero. Thus v1 = 0

I have filled in the other conditions for the Bernoulli equation.

At t2 the same parcel of fluid has now lost some potential energy but gained some kinetic energy. The pressure energy has not altered and is still atmospheric. Consequently it is now travelling with velocity v2

Finally I have substituted into the Bernoulli equation which leads to the conclusion that the loss of head (h1 - h2) supplies the energy to drive the fluid at velocity v2.

Can you see this ?

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