# Bernoulli’s theorem: equation, applications and solved exercise

The Bernoulli’s theorem , which describes the behavior of a moving fluid, was enunciated by the mathematician and physicist Daniel Bernoulli in his Hydrodynamics . According to the principle, an ideal fluid (without friction or viscosity) circulating through a closed pipeline will have a constant energy in its path.

The theorem can be deduced from the principle of energy conservation and even from Newton’s second law of motion. Furthermore, Bernoulli’s principle also states that an increase in the velocity of a fluid implies a decrease in the pressure to which it is subjected, a decrease in its potential energy, or both at the same time.

Daniel Bernoulli

The theorem has many different applications, both in the world of science and in people’s everyday lives.

Its consequences are present in the elevator of airplanes, in the chimneys of homes and industries, in water pipes, among other areas.

## Bernoulli’s Equation

Although Bernoulli deduced that pressure decreases as flow increases, the truth is that it was Leonhard Euler who actually developed Bernoulli’s equation as it is known today.

Anyway, Bernoulli’s equation, which is nothing more than the mathematical expression of his theorem, is as follows:

2 ƿ ƿ / 2 + P + ƿ g ∙ z = constant

In that expression, v is the velocity of the fluid through the section considered, ƿ is the density of the fluid, P is the pressure of the fluid, g is the value of the acceleration of gravity and z is the height measured in the direction of gravity

In the Bernoulli equation, it is implied that the energy of a fluid consists of three components:

– A kinetic component, which is the result of the speed at which the fluid travels.

– A potential or gravitational component, due to the height at which the fluid is.

– An energy under pressure, which is what the fluid has as a result of the pressure to which it is subjected.

On the other hand, Bernoulli’s equation can also be expressed like this:

2 ƿ ƿ / 2 + P 1 + ƿ g ∙ z 1 = v 2 ƿ 2/2 + P 2 + ƿ g ∙ z 2

This last expression is very practical for analyzing the changes experienced by a fluid when any one of the elements that make up the equation changes.

### Simplified form

On occasions, the change in the term ρgz of the Bernoulli equation is minimal compared to that experienced by the other terms, so it is possible to overlook it. For example, this happens in the currents that an airplane in flight experiences.

On these occasions, Bernoulli’s equation is expressed as follows:

P + q = P

In this expression q is dynamic pressure and is equivalent to av 2 2 ƿ / 2, and P is what is called total pressure and is the sum of static pressure P and dynamic pressure q.

## applications

Bernoulli’s theorem has many different applications in fields as diverse as science, engineering, sports, etc.

An interesting application is found in the design of chimneys. The chimneys are built at a high height, in order to obtain a greater pressure difference between the base and the chimney outlet, thanks to which it is easier to extract combustion gases.

Obviously, Bernoulli’s equation also applies to the study of the motion of liquid flows in pipelines. From the equation, it is concluded that a reduction in the transverse surface of the tube, in order to increase the velocity of the fluid flowing through it, also implies a decrease in pressure.

The Bernoulli equation is also used in aviation and in Formula 1 vehicles. In the case of aviation, the Bernoulli effect is the origin of the aircraft lift.

Aircraft wings are designed with the aim of achieving greater airflow in the upper part of the wing.

Thus, in the upper part of the wing, the air velocity is high and therefore the pressure less. This pressure difference produces a vertically directed force (lifting force) that allows the planes to be held aloft. A similar effect is obtained on the ailerons of Formula 1 cars.

## Exercise solved

A stream of water flows at 5.18 m / s through a tube with a cross section of 4.2 cm 2 . The water drops from a height of 9.66 m to a lower level with a height of zero, while the transverse surface of the tube increases to 7.6 cm 2 .

a) Calculate the velocity of the water flow at the lowest level.

b) Determine the pressure at the lower level knowing that the pressure at the upper level is 152000 Pa.

#### Solution

a) As the flow must be conserved, it is true that:

higher level = Q lower level

1 . S 1 = v 2 . S 2

5.18 m / s. 4.2 centimeters 2 = v 2 . 7.6 cm^ 2

Compensation, you get:

2 = 2.86 m / s

b) Applying Bernoulli’s theorem between the two levels, and considering that the density of water is 1000 kg / m 3 , we obtain that:

2 ƿ ƿ / 2 + P 1 + ƿ g ∙ z 1 = v 2 ƿ 2/2 + P 2 + ƿ g ∙ z 2

(1/2) 1000 kg / m 3 . (5.18 m / s) 2 + 152000 + 1000 kg / m 3 . 10 m / s 2 . 9.66 m =

= (1/2). 1000 kg / m 3 . (2.86 m / s) 2 + P 2 + 1000 kg / m 3 . 10 m / s 2 . 0 m

By deleting P 2, you get:

2 = 257926.4 Pa