Bernoulli’s principle

Bernoulli’s equation

Narrated video. Transcript:

Water flows through a pipe that changes in width and climbs uphill. At each point it has a pressure, a speed and a height. In 1738 Daniel Bernoulli found how these three are linked.

Look at the liquid between two sections. At B, lower down, the pressure is P 1, the area A 1, the speed v 1 and the height h 1. At D, higher up, they are P 2, A 2, v 2 and h 2.

In a short time Δt, the liquid at B moves forward by v 1 Δt. The pressure behind it pushes it along and does work P 1 A 1 v 1 Δt, which is P 1 times the volume ΔV.

At D, the liquid ahead pushes back, doing work minus P 2 ΔV. By continuity the volume is the same at both ends, so the net work done by pressure is P 1 minus P 2, times ΔV.

What did this work do? Effectively, a small mass Δm, equal to ρ ΔV, has been moved from B to D. Its kinetic energy changed by half Δm times v 2 squared minus v 1 squared.

Its gravitational potential energy changed by Δm g times h 2 minus h 1. By the work–energy theorem, the work done equals the total change in energy.

Divide everything by ΔV and rearrange: P 1 plus half ρ v 1 squared plus ρ g h 1 equals P 2 plus half ρ v 2 squared plus ρ g h 2.

So along a streamline, P plus half ρ v squared plus ρ g h stays constant. This is Bernoulli’s principle. Each term is an energy per unit volume.

It holds for steady flow of a liquid that is incompressible and has no viscosity, so no energy is lost to friction. Real liquids lose a little energy as heat, so it is an approximation.

Set v to zero and the equation becomes P plus ρ g h equals a constant: the pressure–depth rule for a liquid at rest. And in a level pipe, where the liquid is faster, its pressure is lower.

Speed of efflux and the venturimeter

Narrated video. Transcript:

Make a hole in the side of a tank of water, a depth h below the surface. Water jets out. How fast? The speed of a liquid flowing out of an opening is called the speed of efflux.

Apply Bernoulli’s equation between the open surface and the hole. Both are open to the air, so both are at pressure P a. The tank is wide, so the surface moves down very slowly; take its speed as zero.

The equation becomes ρ g h equals half ρ v squared. So v equals the square root of 2 g h. This is Torricelli’s law: the water leaves as fast as a stone dropped from the height of the surface.

Worked example: a tap is 5 metres below the water level in a rooftop tank. The water comes out at the square root of 2 times 9.8 times 5, about 9.9 metres per second.

If the tank is closed and the air above the water is at a higher pressure P, the extra pressure pushes the water out faster: v equals the square root of 2 times P minus P a over ρ, plus 2 g h.

Now a level pipe with a narrow throat. This is a venturimeter. By continuity, the water speeds up in the throat. In a level pipe, faster flow means lower pressure.

Open tubes on the wide part and on the throat show the pressure as a height of water. The tube on the throat stands lower. The difference measures P 1 minus P 2.

For example: the pipe’s cross-section is four times that of the throat, and the water enters at 0.8 metres per second. In the throat it moves at 3.2. P 1 minus P 2 is half ρ times v 2 squared minus v 1 squared: 4800 pascals, a 49 centimetre difference in the tubes.

Working backwards, measuring the height difference tells you how fast the liquid is flowing. That is how a venturimeter measures flow.

The same low pressure in fast-moving air is used in a perfume sprayer, a Bunsen burner and the carburettor of a motorcycle engine. Fast air across the top of a tube sucks liquid or gas up into it.

Dynamic lift: wings and spinning balls

Narrated video. Transcript:

Hold a strip of paper under your lips and blow across the top of it. The paper rises. The fast-moving air above it is at lower pressure than the still air below.

An upward force on a body moving through a fluid, like this, is called dynamic lift. It is what keeps an aeroplane in the air.

Look at a cross-section of an aeroplane wing. It is shaped specially, rounded in front, curved on top and tapered at the back. This shape is called an aerofoil.

As the plane moves, air flows past the wing. Watch the streamlines: they crowd together above the wing, so the air above moves faster than the air below.

By Bernoulli’s principle, faster air means lower pressure. The pressure below the wing is higher than above it, and this pressure difference pushes the wing up.

The same effect is dangerous in a storm. Strong winds blow fast over a tin roof, the pressure above drops, and the still air inside the house can push the roof off.

Now throw a ball without spin. The air streams past both sides symmetrically, and there is no sideways force.

Give it spin. The rough surface drags air around with it. On one side the dragged air moves with the passing air, so the air there is faster; on the other side, slower.

The faster side has lower pressure, so the ball is pushed towards it. This sideways lift on a spinning ball is called the Magnus effect.

With backspin the force is upward, and the ball floats longer. With topspin it is downward, and the ball dips sharply. Spin bowlers and footballers use it to make the ball swerve.

Try it: water jet from a tank

Interactive simulation. Things to try:

• The jet speed is v = √(2gh), where h is the depth of the hole below the surface.
• Raise the water level: the jet gets faster.
• Add gas pressure above the water (a closed tank): the jet is faster again.
• The jet is a projectile: R = v √(2y/g). Find the hole height that gives the longest range.

Try it: a venturimeter

Interactive simulation. Things to try:

• The tubes show the pressure as a height of water. The throat tube is lower: faster water, lower pressure.
• Make the throat narrower and watch the head difference h grow fast: it depends on v².
• Check: P₁ − P₂ = ½ρ(v₂² − v₁²) = ρgh.

Blowing between two sheets

Hold two sheets of paper hanging parallel, a few centimetres apart, and blow hard between them. What happens?

1. They fly apart, because the air pushes them outwards
2. They move towards each other, because the fast air between them is at lower pressure
3. Nothing, because the air does not touch the outer sides
4. They swing forwards in the direction you blow, with no sideways motion

Speed of efflux

A small hole is made in the side of an open water tank, 1.0 m below the water surface. With what speed does the water come out? (g = 9.8 m/s²)

1. 3.13 m/s
2. 4.43 m/s
3. 9.8 m/s
4. 19.6 m/s

Summary

• Bernoulli’s equation: P + ½ρv² + ρgh = constant (along a streamline)
• Torricelli’s law (open tank): v = √(2gh)
• Closed tank, gas at P: v = √(2(P − Pₐ)/ρ + 2gh)
• Venturimeter (level pipe): P₁ − P₂ = ½ρ(v₂² − v₁²)

• Bernoulli’s equation is the work–energy theorem for a flowing liquid. It holds for steady, incompressible flow with no viscosity.
• In a level pipe, where the speed is higher, the pressure is lower.
• Liquid leaves a hole at depth h with the same speed as a body falling freely through h.
• Dynamic lift: faster air over an aerofoil gives lower pressure above a wing; a spinning ball swerves towards its faster side (the Magnus effect).