What is a Bernoulli’s Theorem : Derivation & Its Limitations

Bernoulli’s theorem was invented Swiss mathematician namely Daniel Bernoulli in the year 1738. This theorem states that when the speed of liquid flow increases, then the pressure in the liquid will be decreased based on the energy conservation law. After that, Bernoulli’s equation was derived in a normal form by Leonhard Euler in the year 1752. This article discusses an overview of what is a Bernoulli’s theorem, derivation, proof, and its applications.

What is Bernoulli’s Theorem?

Definition: Bernoulli’s theorem states that the whole mechanical energy of the flowing liquid includes the gravitational potential energy of altitude, then the energy-related with the liquid force & the kinetic energy of the liquid movement, remains stable. From the energy conservation principle, this theorem can be derived.

Bernoulli’s equation is also known as Bernoulli’s principle. When we apply this principle to fluids in a perfect state, then both the density & pressure are inversely proportional. So the fluid with less speed will use more force compare with a fluid that is flowing very fast.

Bernoulli’s Theorem Equation

The formula of Bernoulli’s equation is the main relationships among force, kinetic energy as well as the gravitational potential energy of a liquid within a container. The formula of this theorem can be given as:

p + 12 ρ v2 + ρgh = stable

From the above formula,

‘p’ is the force applied by the liquid

‘v’ is the liquid’s velocity

‘ρ’ is the liquid’s density

‘h’ is the container’s height

This equation provides huge insight into the stability among force, velocity, and height.

State and Prove Bernoulli’s Theorem

Consider a slight viscosity liquid flowing with laminar flow, then the whole potential, kinetic, and pressure energy will be constant. The diagram of Bernoulli’s theorem is shown below.

Consider the ideal fluid of density ‘ρ’ moving throughout the pipe LM by changing cross-section.

Let the pressures at the ends of L&M are P1, P2 & the cross-section areas at L&M ends are A1, A2.

Allow the liquid to enter with V1 velocity & leaves with V2 velocity.

Let A1>A2

From the continuity equation

A1V1=A2V2

Let A1 is above A2 (A1>A2), then V2>V1 and P2>P1

The mass of liquid entering at the end of  ‘L’ in ‘t’ time, then the distance covered by the fluid is v1t.

Thus, the work done through force over the fluid end ‘L’ end within’ time can be derived as

W1= force x displacement = P1A1v1t

When same mass ‘m’ goes away from the end of ‘M’ in time ‘t’, then the fluid covers the distance through v2t

Thus, work done through fluid against the pressure because of ‘P1’ pressure can be derived by

W2=P2A2v2t

Network done through force over the fluid in ‘t’ time is given as

W = W1-W2

= P1A1v1t- P2A2v2t

This work can be done on the fluid by force then it increases its potential & kinetic energy.

When kinetic energy increase in fluid is

Δk = 1/2m(v22-v12)

Similarly, when potential energy increases in the fluid is

Δp = mg (h2-h1)

Based on the relation of work-energy

P1A1v1t- P2A2v2t

= 1/2m(v22-v12) – mg (h2-h1)

If there is no liquid sink and source, then the fluid mass entering at ‘L’ end is equivalent to the fluid mass leaving from the pipe at the end of ‘M’ can be derived like the following.

A1v1 ρ t = A2v2 ρt = m

A1v1t = A2v2t = m/ρ

Substitute this value in the above equation like P1A1v1t- P2A2v2t

P1 m/ ρ – P2 m/ ρ

1/2m(v22-v12) – mg (h2-h1)

i.e, P/ ρ + gh + 1/2v2 = constant

Limitations

Bernoulli’s Theorem limitations include the following.

• The fluid particle velocity in the middle of a tube is utmost and reduces slowly in the direction of the tube because of friction. As a result, simply the liquid’s mean velocity must be in use due to the particles of the liquid velocity is not consistent.
• This equation is applicable to streamline the supply of a liquid. It is not suitable for turbulent or non-steady flow.
• The external force of the liquid will affect the liquid flow.
• This theorem preferably applies to non-viscosity fluids
• Fluid must be incompressible
• If the fluid is moving in a curved lane, then the energy because of centrifugal forces must be considered
• The flow of liquid should not change with respect to time
• In unstable flow, a little kinetic energy can be changed into heat energy & in a thick flow; some energy can be vanished because of shear force. Thus these losses must be ignored.
• The effect of viscous must be negligible

Applications

The applications of Bernoulli’s Theorem include the following.

Moving Boats in Parallel

Whenever two boats are moving side by side in a similar direction, then the air or water will be there in between that moves quicker compare with when the boats are on the remote sides. So according to Bernoulli’s theorem, the force between them will be decreased. Therefore because of the change in pressure, the boats are pulled in the direction of each other due to attraction.

Airplane

Airplane works on the principle of Bernoulli’s theorem. The wings of the plane have a specific shape. When the plane moves, the air flows over it with a high speed as contrasted with its low surface wig. Because of Bernoulli’s principle, there is a difference in the flow of air above & below the wings. So this principle creates a change in pressure because of the flow of air on the up surface of the wing. If the force is high than the mass of the plane, then the plane will rise

Atomizer

Bernoulli’s principle is mainly used in paint gun, insect sprayer, and carburetor action. In these, due to the motion of the piston within a cylinder, high speed of air can be supplied on a tube that is dipped in the fluid to spray. The air with high speed can create less pressure on the tube because of the rise in the fluid.

Blowing of Roofs

The trouble in the atmosphere due to rain, hail, snow, the roofs of huts will blow off without any harm to another part of the hut. The blowing wind forms a low weight on the roof. The force under the roof is bigger than low pressure because of the difference in pressure; the roof can be raised and blown off through the wind.

Bunsen Burner

In this burner, the nozzle generates gas through high velocity. Because of this, the force within the stem of the burner will decrease. Thus, air from the environment runs into the burner.

Magnus Effect

Once a rotating ball is thrown, then it moves away from its normal path within the flight. So this is known as the Magnus effect. This effect plays an essential role in cricket, soccer, and tennis, etc.

Thus, this is all about an overview of Bernoulli’s theorem, equation, derivation, and its applications. Here is a question for you, what are the