Bernoulli's Principle

 

In the 18^th century Daniel Bernoulli noticed that fluids flow faster when forced through constrictions. We usually come across these kinds of phenomenon when we observe and compare rapids and meandering rivers.

Bernoulli reasoned that when a particular energy increases (kinetic) in a system the rest of the forms of energy must decrease.

To understand this principle the best example would be observing the lifting of an airplane. An airplane wing is shaped in such a way that the air passing over it must travel further and faster than the air passing underneath. This leads to a lower air pressure build up on the wing than the air below. This produces the lift force which keeps the airplane afloat.

 

Bernoulli's Principle Definition

 

This principle basically gives us a relation between velocity, pressure and height of the flowing non viscous fluid in a horizontal flow.

According to it, the speed and pressure of the flowing fluid are inversely proportional to each other, that is, if the velocity increases, it will lead to an automatic decrease in the pressure of the fluid.

or

The theorem states that for the streamline flow of an ideal liquid, the total energy (sum of pressure energy, potential energy and kinetic energy) per unit mass remains constant at every cross-section, throughout the flow.

Hence according to the principle, for a horizontal flow if the velocity decreases then the pressure exerted by the fluid will decrease.

We have many forms of Bernoulli equation according to the flow of the liquid. According to it if we add all the energy components of the flowing fluid along the streamline then we will get a constant value for it throughout the flow of line. Even the sum of the potential and kinetic energy follows a constant value. We can also say that Bernoulli principle is the result of Newton's second law of motion.

Bernoulli's Principle Equation

This theorem is a consequence of the principle of conservation of energy, applied to ideal liquids in motion. As per the theorem statement that is for the streamline flow of an ideal liquid, the total energy (sum of pressure energy, potential energy and kinetic energy) per unit mass remains constant at every cross-section, throughout the flow.

Consider a tube AB of varying cross-section and at different heights. Let an ideal liquid (an ideal liquid is incompressible and non-viscous) flow through it in a streamline. Since the liquid is flowing from A to B, p1>p2.
Now A1v1r = A2v2r = m (according to the equation of continuity)
Here A1>A2 so v1<v2
The force on the liquid at A=ρ1A1 and the force on the liquid at B=ρ2A2
Now, the work done per second on the liquid at

section A = r1A1v1

= ρ1v1

Where v1 is velocity and v₁ is volume of liquid per sec.
(here, Work donesec= Force × DistanceTime = force×velocity)
Now, the work done per second on the liquid at

section B = ρ2A2v2

= ρ2v2

since v₁ = v₂ = v (equation of continuity)

Net work done per second on the liquid by the pressure energy in moving from A to B = ρ1v−ρ2v
The net work done per second, in turn, increases the P.E. per second and also increases the K.E. per sec, from A to B. This is in accordance with the law of conservation of energy.

p1v−p2v = (mgh2−mgh1) + (12mv22−12mv21)

or p1v+mgh1+12mv21 = p2v+mgh2+12mv22

or p1vm + gh1+12v21= p2vm + gh2+12v22

or p1ρ + gh1+12v21 = p2ρ + gh2+12v22

pρ + gh + 12v2 = constant

Pressure energy per unit mass (p/e) + potential energy per unit mass (gh) + kinetic energy per unit mass is constant for Streamline flow of an ideal liquid.
The Bernoulli equation is different for adiabatic as well as isothermal processes.

Bernoulli Effect

• We have already studied the Bernoulli's law. The use of this law has been seen as Bernoulli Effect. If we take the example of the air plane then the air flowing under the wing is faster than the air flowing above the wing. Hence the air pressure is higher under the wing than over the wing. This results in air plane lift. And this is also called Bernoulli Effect.
• The Bernoulli Effect is same as the Bernoulli law or statement or whatever we call it.

Bernoulli's Principle Examples

1. The most common example of this principle is the air plane lift. The wings of the plane are designed in such a manner that the air flowing under the wings moves at a speed which is greater than that of the upper part. Hence it results into a difference in pressure such that the pressure exerted is more upwards than in downwards direction. This results in a lift.
2. It is used to make Venturi meter. It is a device that is used to measure the rate of flow of a fluid through a pipe. It has three parts: a short converging pipe, a throat and a diverging part as well.
3. It is used in the manufacturing of the orifice meter which has the same function as above but is available at a cheaper rate.
4. It is used in pilot tube. It is a device used for measuring the velocity of flow at any point in a pipe or a channel. It is based on the principle that if the velocity of flow at a point becomes zero the pressure there is increased due to the conversion of the kinetic energy into pressure energy.
5. The carburetor available in engine has a venture which operates on the principle of Bernoulli.
6. If we want to calculate the maximum drain rate through the hole in the tank then we can use the above principle of Bernoulli.
7. It is also used in open channel hydraulics.
8. Used in the Bernoulli grip in order to create the non contact adhesive force between surface and gripper.
9. The spoiler of the race car is shaped in a way to obtain the maximum speed during the race.
10. Race cars also use this principle to keep the wheel pressed to the racing track.