Bernoulli’s Law: Definition, Equations, and Example Problems

Bernoulli’s Law – Bernoulli’s Law originates from a Dutch mathematician named Daniel Bernoulli, a figure born in a family who has a high dedication to science. The mother named Dorothea Falkner and father named Johann Bernoulli was a head of mathematics in Groningen. Until the emergence of Bernoulli’s law which was created by the second child of three siblings.

Long story short, Bernoulli’s success in publishing research results related to mechanical fluids was the beginning of the development of science. Bernoulli explained the basis of the kinetic theory of gases and their relationship to Boyle’s law. Through his research, Bernoulli explained the lift force that appears on the plane so it can make it fly in the air.

Bernoulli’s Law Inventor Biography

Daniel Bernoulli.

Bernoulli’s law was discovered by Daniel Bernoulli. He was a mathematician and physicist who was born in the city of Groningen, Netherlands and died in Basel, Switzerland. One of his thoughts which is important in the field of physics is the Bernoulli principle on the flow tube which is used to measure the velocity of fluid flow due to pressure.

1. Childhood

Daniel Bernoulli is the son of Johann Bernoulli, a mathematician who was born in the city of Groningen. His brother named Nicolaus (II) Bernoulli, his younger brother named Johann (II) Bernoulli, and his uncle named Jacob Bernoulli was also a mathematician. This situation gave rise to rivalry and envy in the family.

At first, his father wanted Daniel to become a trader or work in business. At the age of 13, Daniel studied logic and philosophy at the University of Basel. However, while in college he continued to study calculus from his father and older brother. Daniel also studied medicine and earned a doctorate in medicine for the applications of physical mathematics in medicine that he advanced.

As a result, Bernoulli studied philosophy and logic at the age of 13, then graduated with a bachelor’s degree in 1715, and won a master’s degree in 1716. From 1718 to 1720, Bernoulli had to return to medical education at the undergraduate and doctoral levels in Heidelberg, Strasbourg , and Basel. In fact, at that point, Bernoulli wanted to study mathematics, but Johann still disagreed.

Johann agreed to privately tutor Bernoulli in advanced mathematics and physics. In 1738, Bernoulli managed to publish his research results related to mechanical fluids in an article entitled ” Hydrodynamica “. In this article, Bernoulli explained the basic kinetic theory of gases and their relationship to Boyle’s law, and collaborated with Euler for the development of the Euler-Bernoulli equation.

He applied the idea of ​​conservation of energy to moving fluids based on the initial ideas he had learned from Johann long ago. Through his research, Bernoulli also formulated the Bernoulli Effect, which explains the lift force of an airplane.

2. Scientific Contribution

Daniel Bernoulli is one of the scholars who thinks that natural behavior can be understood through the concept of small particles. Bernoulli’s principle is one of the principles of physics created by Daniel Bernoulli. The application of this principle to the concept of pressure and height of fluid dynamics.

Bernoulli’s principle is the statement that the velocity of a dynamic fluid is inversely proportional to the degree of pressure it experiences during displacement. The faster the dynamic fluid moves, the less pressure it experiences. Conversely, the slower the dynamic fluid moves, the greater the pressure.

Bernoulli’s principle applies to both compressed and incompressible fluid flows. The formulation of this principle was carried out by Bernoulli using the basic operations of mathematics. One of its uses is the manufacture of airplane wings that are able to adjust to air speed and air pressure.

Definition of Bernoulli’s Law

The sound of Bernoulli’s law states that an increase in the flow velocity of a fluid can cause a decrease in fluid pressure simultaneously. Or it can also be interpreted as a decrease in the potential energy of the fluid. What’s interesting about Bernoulli’s law is that it can be applied to various types of fluid flow with a number of assumptions.

Please note that Bernoulli’s law can only be applied to liquids flowing, at different speeds through a pipe. This law essentially emphasizes that a pressure will decrease if the velocity of the fluid flow increases or increases, this law is taken from the book written by the inventor entitled Hydrodynamica.

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Another understanding of Bernoulli’s law is a fluid in an ideal composition that fulfills the characteristics or characteristics of flowing. Through soft flows and current lines, it is not thick to the point where it is incomparable, meanwhile, according to experts, there is no further explanation regarding Bernoulli’s law. However, this is not a problem because what needs to be understood in this case is not the understanding of experts.

But a few important things related to Bernoulli’s law, starting from fluid pressure that arises from anything, how the law of continuity sounds, the understanding of the sound of Stokes’ law, the ideal line to the principle of aircraft and fluid transport forces from the smallest pressure occurs at what number, as follows Some of the assumptions used in Bernoulli’s law.

  • The fluid in this case is incompressible.
  • The fluid has no inviscid or viscosity.
  • Fluid flow does not change with time.
  • The fluid flow is laminar, it is fixed and there are no eddies.
  • There is no loss of energy due to fluid and wall friction.
  • There is no loss of energy due to the turbulence that appears.
  • No heat energy is delivered to the fluid.

Bernoulli’s Law Equation

Bernoulli’s law equation is closely related to the pressure, velocity, and height of two points with a fluid flow that has a density. The emergence of the Bernoulli equation is obtained from the balance of mechanical energy or kinetic energy and potential energy together with the pressure that appears to produce the following implementation.

Pressure + E kinetic + E potential = constant

Where:

P : pressure (Pascal)
rho : density of fluid (kg/m3)
v : velocity of fluid (m/s)
g : acceleration due to gravity (g = 9.8 m/s2)
h : height (m)

This one Bernoulli equation can be written as below:

Figures 1 and 2 indicate the point or location where the fluid is observed. For example, as shown below, point 1 has a larger diameter than point 2. Bernoulli’s law can be solved for every two point locations in fluid flow.

How do we know where is the best location to choose a point location?

If we want to know a quantity at a location in the fluid flow, then we must make that location one of the location points. The second point is a location where we already know the quantities at that location, so we can find the quantity we want to find (at point 1) with the Bernoulli equation formula.

Principles of Bernoulli’s Law

The principle of Bernoulli’s law is one of the terms used in fluid mechanics that describes an increase in a fluid. This increase in fluid will cause a decrease in the flow pressure contained in the fluid flow. The principle of Bernoulli’s law is a simplification of the Bernoulli equation.

This principle was also explained again by mathematicians, who is none other than Daniel Bernoulli. This Dutch mathematician devised a form of equation that applies to both stagnant and unstoppable fluid flows. Bernoulli’s law explains knowledge and knowledge that can interpret something that is around.

Bernoulli’s Law Formula

1. Incompressible Flow

Is a fluid flow that has characteristics with no change in the amount of mass density or density of a fluid along the existing flow. Simple examples such as materials contained in incompressible fluid flows, including water, emulsions, all types of oil and others.

The form of the Bernoulli Equation for an incompressible flow is as follows:

 

with:

The above equation is valid for incompressible flow with the following assumptions:

  • Flow is steady ( steady state )
  • There is no friction ( inviscid )

In another form, the Bernoulli Equation can be written as follows:

 

2. Compressed Flow

Compressed flow is characterized by a change in the mass density or also called the density of the fluid along the flow. Examples of materials that include compressed fluid flow are air, natural gas and the like.

Compressed flow is a fluid flow which is characterized by a change in the density of the fluid along the flow. Examples of compressed fluids are: air, natural gas, etc. Bernoulli’s equation for compressed flow is as follows:

with:

Bernoulli’s law states that the amount of pressure is denoted by p, the kinetic energy per unit volume by (1/2 PV^2 ), the potential energy per unit volume or (ɋgh). All have the same value at every point along a current line, to carry out the discussion requires Bernoulli’s understanding to find the equation and write it down.

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Bernoulli’s Law in Everyday Life

1. Leaky Water Tank

When draining the water tank, the first thing to think about is of course how long it will take to wait for the water to run out. This question can be answered using Bernoulli’s understanding, this legal equation can be used to find out how fast the water is coming out of the small hole in the water tank.

Open the lid of the water tank which is at the top, if it doesn’t have a lid and there are parts with holes it means that these two parts will directly meet the atmosphere in the air. The pressure in that section both comes from atmospheric pressure. After that, you can find the formula related to how long it will take to wait for the water to run out.

2. Riding a Motorcycle

Usually someone who rides a motorbike without a jacket and only wears a shirt, the back of the shirt flies and swells. Conditions that indirectly show the application of Bernoulli, why is that? when riding a motorcycle fast, the airspeed at the rear becomes smaller.

The air pressure that appears behind the body will be greater, because it is this difference in air pressure that ultimately makes the air pressure push the clothes back. Until it flew and bloated abysmally.

3. Pressing the Water Hose

Conditions that are usually done when watering flowers, cleaning motorbikes and the like are by pressing the water hose. The goal is for the water flow to be faster and the shot distance to be farther, this condition is related to Bernoulli’s equation. The smaller the surface area of ​​an object, the greater the pressure.

4. Aircraft Lift

If you pay close attention, when the plane is about to take off, it can be seen that the shape of the wings changes to bend downwards. This condition is not done without reason, this is because the aircraft manufacturers take into account the use of Bernoulli carefully. Speed ​​and pressure are inversely proportional, at high speed the pressure will be low.

The airplane lift formula itself is as follows:

Meanwhile, when the plane is already at a certain height and maintains its speed, then the following formula will apply

Information:

Examples of Bernoulli’s Law Problems

Example Question 1

Water flows through the pipe as shown above. At point 1 it is known from the measurement of the water velocity v 1 = 3 m/s and the pressure P 1 = 12300 Pa. At point 2, the pipe has a height of 1.2 meters higher than point 1 and flows at a speed v 2 = 0.75 m/s. Using Bernoulli’s law, determine the pressure at point 2.

Discussion:

Bernoulli’s (Law) Equation Formula:

It is known that point 1 has no height (h 1 = 0), so:

Then, the magnitude of P 2  can be found by:

2 = 4.080 Pa

Example Problem 2

A pipe system for the fountain is installed as shown above. The pipe is buried underground and then the water flow is flowed vertically upwards with a smaller diameter pipe. Calculate how much pressure (P 1 ) is needed so that the fountain can work as it should.

Discussion:

We first write down the known quantities from the problem:

in 1 = ?
1 = ?

Before looking for the pressure value at point 1 (P 1 ), we must find the velocity value at point 1 (v 1 ) so that Bernoulli’s law formula can be applied.

Using the law of the conservation of mass:

So, the magnitude of v 1 is obtained  , namely:

Then, the Bernoulli equation formula can be used:

Since point 1 has no height (h 1 = 0), then:

The magnitude of P 1  can be found by:

For simplicity, we use the value of P 2 = 0, so:

This is an explanation of Bernoulli’s law, starting from the definition, formula, to examples of easy problems to do. sinaumedia not only provides scientific material, such as Bernoulli’s law, but also invites students to practice directly applying Bernoulli’s law.