## The Bernoulli Equation Overview

Before we can look into the various applications of the Bernoulli equation, let's recall the principle it's based on: the **Bernoulli principle**.

**The Bernoulli principle** states that the pressure exerted by a moving fluid is inversely proportional to its velocity in a horizontal flow.

This was proven by and named after the Swiss mathematician Daniel Bernoulli. In other words, the total mechanical energy of a flowing liquid or gas remains unchanged at any point along a streamline, assuming that the fluid in question is incompressible and has zero viscosity. This total energy consists of the static pressure energy, gravitational potential energy due to elevation, and the kinetic energy of the fluid in motion. All the respective energy terms are easily recognizable in the Bernoulli equation defined below.

### The Bernoulli Equation

We can use this principle and the *law of conservation of energy* to derive **Bernoulli's equation**:

$$ P_1 + \rho g y_1 + \frac{1}{2} \rho v^2_1 = P_2 + \rho g y_2 + \frac{1}{2} \rho v^2_2.$$

Here, \(P\) is the static pressure measured in pascals (\(\mathrm{Pa}\)), \(\rho\) is the fluid density \(\left(\frac{\mathrm{kg}}{\mathrm{m^3}}\right)\), \(g\) is the acceleration due to gravity \(\left(\frac{\mathrm{m}}{\mathrm{s}^2}\right)\), \(y\) is the elevation of the fluid measured in meters (\(\mathrm{m}\)), and \(v\) is the velocity of the fluid \(\left(\frac{\mathrm{m}}{\mathrm{s}}\right)\). The subscript simply indicates the respective values in two particular points within a closed system, as visualized in Figure 1 below.

### The Continuity Equation

Another important relation to consider when dealing with the Bernoulli equation, and fluid mechanics in general, is the *continuity equation, *derived using the principle of conservation of mass. Mathematically, it can be expressed as

$$ A_1v_1=A_2v_2 $$

where \(A_1\) and \(A_2\) are the cross-sectional areas of the pipe in two different locations, corresponding to the velocity \(v\) of the fluid at each point. In other words, for an incompressible fluid in streamline flow, the mass of the fluid passing through the different cross-sections remains equal. This relation is visualized in Figure 2 below.

## Application of Bernoulli's Equation in Real Life

The Bernoulli equation has numerous applications in our everyday life, explaining the movement of gases and liquids in pipes, instruments, medical equipment, and many others. Some examples of real-life scenarios are:

**Swimming**- To achieve the most effective swimming, one can extend their feet, point their toes, and adjust the angle of their hands. This will result in the maximum lift force and minimum drag force.**Magnus effect**- A rotating ball drags air with it due to friction, so the speed of the airflow decreases on one side of the ball while increasing on the other.**Airplane wings**- The shape of an airplane wing is more curved on the surface; hence the surrounding airflow velocity is higher at the top of the wing than at the bottom.**Atomizers**- A device used to emit liquids in a fine spray, consisting of two tubes connected perpendicular to one another. As the air gets pushed through the horizontal tube, the velocity of the airflow increases above the vertical tube and decreases the air pressure.**Measuring blood pressure**- A sphygmomanometer compresses the artery, creating a difference in cross-sectional areas, which can be used to measure blood flow rate and determine blood pressure.**Venturi meter**- An instrument that measures fluid flow rate through a pipe utilizing the pressure difference created by different cross-sectional areas.

More in-depth explanations of each of these examples can be found in the later sections, as well as other StudySmarter articles!

## Application and Limitations of Bernoulli's equation

Before we dive deeper into some of the applications mentioned above, we must acknowledge that there are certain limitations to consider when applying Bernoulli's equation to a system.

### Limitations of Bernoulli's equation

It assumes

*uniform velocity*. In real life, that's not the case. Let's consider a fluid flowing through a pipe, for instance. In the central region, the fluid will have its maximum velocity. The velocity will decrease as it approaches the walls of the pipe.It assumes

*zero viscosity*. If that were the case, the fluid would flow indefinitely. In practice, however, the fluid is slowed down by the intermolecular interactions which create internal friction.It assumes complete

*conservation of energy*. Although the assumption in itself is correct, some of the energy is converted into heat, therefore doesn't contribute to the total energy of the system consisting of the pressure, kinetic, and potential energies. Similarly, some energy can be lost due to*shear force*(the fluid getting pushed into different directions).It ignores any potential contribution of

*centrifugal force*due to a curved path of the fluid.

## Application of Bernoulli's Equation in Medicine

Bernoulli's equation can often be applied to medicine, as transporting liquid medication, oxygen, and bodily fluids is a crucial aspect when treating a patient. A very distinct example is the procedure of measuring blood pressure. A medical tool called the sphygmomanometer is inflated around an arm to temporarily stop the blood flow in an artery. As the pressure from the cuff is slowly released, the artery expands and blood rushes back into the arm.

There are two sources of potential energy to consider here: the energy stored in the artery walls as capacitance and the heart acting as a pump, and driving the blood forward. Both of these are converted into kinetic energy as the blood moves, meaning the total energy of the system remains unchanged as predicted by the *Bernoulli principle. *

As the cuff compresses or decompresses the artery, we can think of it as a pipe with varying diameters. The artery changes its cross-sectional area, so the velocity also changes. It's a direct application of the *continuity equation.*

## Application of Bernoulli's Equation in Venturi Meter

Determining the flow rate of a fluid within a system requires an accurate and efficient tool. A simple and dependable instrument, relying directly on the Bernoulli principle, is a **venturi meter**.

**A venturi meter** is an instrument used to measure the flow rate of a fluid flowing through a pipe.

It consists of three parts:

A converging section;

A throat;

A diverging section.

The aim is to create a pressure difference within the pipe by altering its cross-sectional area. A manometer can then be used to measure the pressure of the flowing fluid, and knowing the diameter of each section, we can simply calculate the pressure of the fluid in the other parts of the pipe.

It doesn't have any moving parts or a drastic impact on the flow rate, making it an ideal solution for measuring airflow in cars, or natural gas in pipelines.

Let's apply the Bernoulli equation to a practical problem involving a venturi meter.

A venturi meter has two different cross-sectional areas in points \(1\) and \(2\), \(8.0 \, \mathrm{cm^2}\) and \(4.0 \, \mathrm{cm^2}\) respectively.

The water level in the tubes above these points has a height difference of \(30 \, \mathrm{cm}\), as visualized in Figure 4. Calculate the velocity at points \(1\).

Answer:

Starting from the Bernoulli equation, we can derive an expression for the velocity. Here, points \(1\) and \(2\) are at the same elevation, which means \(y_1=y_2=0\), so we can get rid of the potential energy term:

$$ P_1 + \bcancel{\rho g y_1} + \frac{1}{2}\rho v^2_1 = P_2 + \bcancel{\rho g y_2} + \frac{1}{2} \rho v^2_2.$$

Now we can rearrange the expression to have all the alike terms on the same sides

$$ P_1-P_2=\frac{1}{2} \rho v^2_2 - \frac{1}{2}\rho v^2_1$$

which can be simplified into

$$ \Delta P = \frac{1}{2} \rho \left ( v^2_2 - v^2_1 \right ). $$

Now we can use the continuity equation

$$ A_1v_1=A_2v_2 $$

to obtain an expression for velocity at point \(2\):

$$ v_2=\frac{A_1 \, v_1}{A_2}.$$

This can be plugged into the pressure difference equation as follows:

\begin{align} \Delta P & = \frac{1}{2} \rho \left ( \left ( \frac{A_1 \, v_1}{A_2}\right )^2 - v^2_1 \right ) \\ \Delta P & = \frac{1}{2} \rho \left ( \frac{A_1^2 \, v_1^2}{A_2^2} - v^2_1\right ) \\ \Delta P & = \frac{1}{2} \rho v_1^2\left ( \frac{A_1^2}{A_2^2}-1\right ).\end{align}

The pressure difference term can be expressed using the expression for pressure exerted by fluids:

$$ \Delta P = \rho g h $$

where the height difference between the two water levels in the tubes (\(h_1\) and \(h_2\)) is provided in the problem. Now we can set these two \(\Delta P\) terms equal and rearrange to get the velocity term:

\begin{align} \bcancel{\rho} g h & = \frac{1}{2} \bcancel{\rho} v_1^2\left ( \frac{A_1^2}{A_2^2}-1\right ) \\ 2gh&= v_1^2\left ( \frac{A_1^2}{A_2^2}-1\right ) \\\sqrt{v_1^2}&=\sqrt{\frac{2gh}{\left ( \frac{A_1^2}{A_2^2}-1\right ) }} \\ v_1&=\sqrt{\frac{2gh}{\left ( \frac{A_1^2}{A_2^2}-1\right ) }}.\end{align}

Let's plug in our values to find the velocity at point \(1\):

\begin{align} v_1&=\sqrt{\frac{2gh}{\left ( \frac{A_1^2}{A_2^2}-1\right ) }} \\ v_1&=\sqrt{\frac{2\left(9.8 \, \frac{\mathrm{m}}{\mathrm{s^2}}\right)(0.30 \, \mathrm{m})}{\left ( \left( \frac{8.0}{4.0} \right )^2-1\right ) }} \\ v_1&=1.4 \, \frac{\mathrm{m}}{\mathrm{s}}.\end{align}

## Application of Bernoulli's Equation in Civil Engineering

Civil engineering deals with the construction and maintenance of infrastructure all around us. It often involves the application of the Bernoulli principle and equation. A very distinct example is an airplane wing.

The shape of an airplane wing is built in a way to consider the Bernoulli principle. Because the wing is more curved on the surface, the surrounding airflow velocity is higher at the top of the wing than at the bottom. As the aircraft moves, the counter flow of air below the wing creates a more significant dynamic pressure than above it. Upon reaching a certain speed, the lift force becomes more significant than the force of gravity and the plane takes off the ground. On the other hand, for the plane to land back on the ground, the speed must be reduced.

Let's look at an example problem involving this lift force!

A typical small airplane has a mass of \(6.00\times10^3 \, \mathrm{kg} \) and its wings have an area of \(100\,\mathrm{m^2}\). If the speed at the upper surface of the wing is \( 50.0 \, \frac{\mathrm{m}}{\mathrm{s}}\), and on the lower surface it's \( 40.0 \, \frac{\mathrm{m}}{\mathrm{s}}\), what is the lift force acting on the wing? Use \(1.29 \, \frac{\mathrm{kg}}{\mathrm{m}^3}\) for the density of air.

Answer:

We can follow the same logic for the pressure difference equation, as completed above, in the venturi meter example to derive an expression for the force.

There is no elevation, so the potential energy terms cancel out

$$ P_1 + \bcancel{\rho g y_1} + \frac{1}{2}\rho v^2_1 = P_2 + \bcancel{\rho g y_2} + \frac{1}{2} \rho v^2_2.$$

leaving us with the following expression after simplifications:

$$ P_1-P_2 = \frac{1}{2} \rho \left ( v^2_2 - v^2_1 \right ). $$

We know that

$$P=\frac{F}{A}$$

so, we can multiply both sides by \(A\) to obtain

$$ (P_1-P_2)A = \frac{1}{2} \rho A \left ( v^2_2 - v^2_1 \right ). $$

Lift force is

$$ F_L=\Delta F = F_1 - F_2,$$

therefore, we can use the fact that

$$\Delta F=\Delta PA= (P_1-P_2)A$$

to replace the left-hand-side with \(F_\mathrm{L}\)

$$ F_\mathrm{L} =\frac{1}{2} \rho A \left ( v^2_2 - v^2_1 \right ).$$

Now we can plug in our given values, to calculate the lift force acting on the plane:

\begin{align} F_\mathrm{L} &=\frac{1}{2}\left(1.29 \, \frac{\mathrm{kg}}{\mathrm{m}^3} \right ) (100\,\mathrm{m^2}) \left( \left(50.0 \, \frac{\mathrm{m}}{\mathrm{s}}\right )^2 - \left(40.0 \, \frac{\mathrm{m}}{\mathrm{s}}\right)^2\right ) \\ F_\mathrm{L} &=5.81\times10^4 \, \mathrm{N}.\end{align}

## Application of Bernoulli's Equation - Key takeaways

- The Bernoulli principle states that the pressure exerted by a moving fluid is inversely proportional to its velocity in a horizontal flow.
- The Bernoulli equation can be expressed mathematically as \(P_1 + \rho g y_1 + \frac{1}{2} \rho v^2_1 = P_2 + \rho g y_2 + \frac{1}{2} \rho v^2_2\).
- The continuity equation can be written as \(A_1v_1=A_2v_2\).
- Some of the limitations of the Bernoulli equation include assuming uniform velocity, zero viscosity, and complete conservation of energy, as well as ignoring any potential contribution of centrifugal force.
- Some real-life applications of the Bernoulli equation include the Magnus effect, airplane wings, and atomizers.
- Bernoulli's equation can often be applied to medicine, for example when measuring a patient's blood pressure.
- A venturi meter is an instrument used to measure the flow rate of a fluid flowing through a pipe.
- Bernoulli's equation can be used to determine the lift force of an airplane.

## References

- Fig. 1 - The Bernoulli equation applied to a pipe system, StudySmarter Originals.
- Fig. 2 - The continuity equation applied to a pipe system, StudySmarter Originals.
- Fig. 3 - Venturi meters (https://commons.wikimedia.org/wiki/File:Venturi_meter,_Alden_Research_Laboratory_-_HAER_077091pu.jpg#/media/File:Venturi_meter,_Alden_Research_Laboratory_-_HAER_077091pu.jpg) by Jet Lowe is licensed by Public Domain.
- Fig. 4 - Venturi meter example problem, StudySmarter Originals.

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##### Frequently Asked Questions about Application of Bernoulli's Equation

What are the applications and limitations of Bernoulli's equation?

Applications of Bernoulli's equation include medicine, civil engineering, sports, and limitations of Bernoulli's equation include assuming uniform velocity, zero viscosity, complete conservation of energy, as well as ignoring any potential contribution of centrifugal forces.

Why is Bernoulli's equation important?

Bernoulli's equation is important as it relates pressure, height, and velocity of a fluid flowing along a streamline.

For which type of fluid, Bernoulli's equation is applicable?

Bernoulli's equation is applicable to zero velocity incompressible fluids flowing along a streamline.

When can you not use Bernoulli's equation?

Bernoulli's equation can't be used when the assumptions about the fluid having uniform velocity, zero viscosity, and complete conservation of energy are not applicable.

Can you use Bernoulli's equation for air?

Yes, Bernoulli's equation can be used for air, for instance, when finding the lifting force on an airplane.

What is pressure in Bernoulli's equation?

The pressure in Bernoulli's equation accounts for the static pressure in a moving fluid.

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