Bernoulli’s Equation: Roots and Uses

FROM THE LECTURE SERIES: Understanding the Misconceptions of Science

By Don Lincoln Ph.D., University of Notre Dame

Bernoulli’s equation dates back to 1738 when it was published by Daniel Bernoulli, a Swiss physicist. However, in 1752, Leonhard Euler rewrote the formula in a more modern format, adding energy conservation to it. The principle explains the laws of motion of fluids and is widely used in pipes with varying diameters.

Schematic and Bernoulli equation written on a school blackboard.
Bernoulli’s equation is used to explain fluid pressure and speed. (Image: Ivica Drusany/Shutterstock)

Bernoulli’s equation was introduced by Swiss physicist Daniel Bernoulli in his book “Hydrodynamica”. It states how fluid speed and pressure are related: the higher the speed, the lower the pressure. Later on, in 1752, another Swiss physicist called Leonhard Euler, who worked on energy conservation, wrote the equation in a more modern way. There is a constant amount in the formula, which is based on energy conservation law.

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Energy Conservation

Euler worked on energy conservation, which refers to the fact that energy cannot be created or destroyed. The famous theory explains that energy can only change forms: from potential to kinetic, from electrical to chemical, even from mass to energy, and back. However, the total amount of energy can never change and is a constant number. Energy has more than one type.

Cartesian impulse conservation law experiment with metallic balls.
Energy is conserved; i.e., it cannot be created or destroyed. (Image: Iakov Filimonov/Shutterstock)

This is a transcript from the video series Understanding the Misconceptions of Science. Watch it now, on The Great Courses Plus.

Kinetic and Potential Energy

The total energy is made up of two types of energy: kinetic and potential. Simply put, kinetic energy refers to the moving energy, and potential energy is non-moving energy. In different situations, each general energy category can have different types, as well. Energy types can change and turn into each other. For example, when one holds a ball in hand, above the ground, the energy is potential. When the ball is dropped, the same energy turns into kinetic.

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Bernoulli’s Equation Elements

Bernoulli’s final equation is closely related to energy conservation. Thus, one can write it not in terms of before and after an occurrence, but rather situation one and situation two.

If the pressures in two locations of a medium of constant density are compared, the equation is as follows: the pressure of location one plus the density of the medium, times gravity, times its vertical location from some arbitrary point, plus one-half times the material’s density, multiplied by the velocity of the material at that location, squared. The result is equal to the same amount but at the other location.

If the equation is to explain an object flying in the air, the mass is difficult to assume. Therefore, it is easier to consider a specific volume of air. Dividing the specific volume and the mass will result in the density of the matter. The Greek letter rho is used to show density in Bernoulli’s equation. If the energy equations are divided by the volume, rho will replace mass, and the results will be very similar to the original equations. This is how Euler wrote the equation in a more modern way. Perhaps, a tangible example of the equation’s application would be liquid movement in a pipe.

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Water Speed in a Pipe with Changing Diameter

Suppose water flows through a horizontal pipe, which narrows at the midpoint. Water is an incompressible liquid, which means squishing water will not change its volume. When water passes through a pipe with changing diameter, unlike gases, it cannot be compressed. Considering the entrance and the exit point of the pipe as points one and two, one can assume the density will be the same at both points. Thus, water’s velocity cannot remain constant.

A diagram and statement of Bernoulli's equation.
Using Bernoulli’s equation, one can calculate water pressure relative to its speed. (Image: Fouad A. Saad/Shutterstock)

Every second, a certain volume of water will go through the inlet: the area of the inlet, multiplied by some length along the pipe. If the narrow part of the pipe is twice as long as the wider part, water should pass both in the same amount of time – and it does. Water passes the narrow part with double speed; hence, a pressure difference results. This is what Bernoulli’s equation expects and describes. As these two spots are at the same depth, h_1 and h_2 are equal, and the rho-gh terms do not apply anymore.

As Bernoulli’s equation deals with pressure and speed, many people, including scientists, have tried to explain airplanes flying through it. However, the attempt has led to a common scientific mistake in planes’ flight description, ignoring all the other factors, especially the main factor that makes planes fly.

Common Questions about Bernoulli’s Equation

Q: What does Bernoulli’s principle state?

Bernoulli’s principle is a physical principle formulated by Daniel Bernoulli, Swiss physicist. The concept is that as the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases. Bernoulli’s equation was generated based on this principle when Leonhard Euler wrote it in a more modern way.

Q: What is Bernoulli’s equation formula?

Bernoulli’s equation is written as P + ½ ρ v2 +ρ g h = constant. Each figure represents a factor in the Bernoulli principle, and the sum in the system will be the same, regardless of what changes in the formula. The formula contains the following elements: Pressure + ½ density * square of the velocity + density * gravity. acceleration* height = constant. The ‘constant’ is a result of the energy conservation principle.

Q: What is H in Bernoulli’s equation?

In Bernoulli’s equation, there are two points considered, simply labeled 1 and 2, in many cases. Normally, the equation is used for pipes. hL represents all minor (valves, elbows, etc.) and major (pipe friction) losses between 1 and 2.

Q: When can you use Bernoulli’s equation?

Bernoulli’s equation describes how the pressure of the fluid (gas or liquid) decreases as the speed increases. Its main application is in the pipe industry, where it helps find pressure in pipes according to diameter changes.

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