This section will introduce perhaps the most fundamental principle in our studies of fluid mechanics, Bernoulli’s principle. It is a direct application of conservation of mechanical energy density and proves useful for both fluid dynamics and fluid statics. Bernoulli’s equation and the continuity equation are usually the two most useful tools when approaching a fluid mechanics problem.

This OpenStax physics trailer provides real life examples of Bernoulli's equation in action.

https://www.youtube.com/watch?v=GuFHfQlgI1I

Pre-lecture Study Resources

Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.

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Learning Objectives

Summary

This section will introduce perhaps the most fundamental principle in our studies of fluid mechanics, Bernoulli’s principle. It is a direct application of conservation of mechanical energy density and proves useful for both fluid dynamics and fluid statics. Bernoulli’s equation and the continuity equation are usually the two most useful tools when approaching a fluid mechanics problem.

Atomistic Goals

Students will be able to...

  1. Match algebraic symbols to conceptual quantities in the continuity equation for fluid flow.
  2. Use proportional reasoning arguments to explore the relationship between velocity and cross-sectional area using the volume flow rate equation.
  3. Match algebraic symbols to conceptual quantities in Bernoulli's equation.
  4. Identify Bernoulli's equation as a statement of energy density conservation.
  5. Use proportional reasoning arguments to explore the relationship between quantities in Bernoulli's equation.
  6. Identify and reason with units and dimensions of quantities related to fluid dynamics.
  7. Differentiate between the volume flow rate and the velocity of a fluid.
  8. Algebraically analyze fluid flow using the continuity equation.
  9. Algebraically analyze fluid flow using Bernoulli's equation.
  10. Synthesize continuity arguments with Bernoulli's equation arguments to conceptually analyze a fluid in motion.
  11. Synthesize the continuity equation concepts with Bernoulli's equation concepts to make algebraic arguments about a fluid in motion.

BoxSand Introduction

Fluid Dynamics  |  Continuity and Bernoulli's Principle


How often are weather reports accurate?  Sometimes it feels like they can never get the forecast right.  Why is this?  Well, the dynamics of our atmosphere fall under the realm of fluid dynamics, which means predicting the future (predicting the weather) requires solving fluid dynamic equations.  Just like our early studies of kinematics, fluid dynamic equations require initial conditions, however, unlike kinematics, a very small change in the initial conditions can lead to a very different final answer in fluid dynamics.  Basically, the study of fluid dynamics is very complicated. 

Luckily we are able to make a few approximations which will allow us to quantify some aspects of fluids in motion.  Our first approximation is restricting ourselves to only study fluids that move in a certain way.  Consider two locations A and B within a tube of moving fluid as shown in the left image of the figure below.  Our model of the moving fluid will assume that any particle that passes through location A will follow a unique path to get to location B.  That unique path is represented by the dashed line between location A and B. 

 

This is a representation of different kinds of flow. There are two tubes that has some smaller area in the front part of the tube that expands into a greater area and both are identical. The first tube is shows Laminar flow where the flow of the fluid is direct from one part of the tube to the other without any interference. The second tube shows turbulent flow where there is no direct flow and there is a lot of interference of the motion of the particle.

 

Similarly, if we look at other locations such as C and D, a particle passing through each will also follow unique paths and no paths cross each other.  This type of fluid flow is called laminar flow (or smooth flow).  For comparison, the right image of the figure above shows turbulent flow where particles in the fluid that pass through each point follow chaotic paths that change in time, this makes the future position of each particle extremely hard to predict.

Furthermore, if we make another assumption and ignore viscous forces (internal friction between "layers" of fluid and external friction between the fluid and the walls of the tube) then the velocity profile (the speed at every location along the cross section of a cylindrical of the tube) is constant.  If this were not the case, then kinetic energy of the fluid near the surfaces of the tube would be lost to thermal energy because of the viscous forces (this would mean larger velocities in the center of the tube and velocities approaching zero at the edges).  Below is a simplified velocity profile comparison between inviscid flow (without viscous forces) and viscous flow (with viscous forces).

 

This is a representation of different kinds of flow. There are two tubes that has some smaller area in the front part of the tube that expands into a greater area and both are identical. The first tube shows velocity profile without viscous forces where the vector arrows are all equal in direction and magnitude where the only difference is the length of the arrows are shorter at a greater cross-sectional area. The second tube shows velocity profile with viscous forces where the vector arrows are greater at the center of the tube and smaller near the edges of the tube and the length of the arrows are collectively shorter at a greater cross-sectional area.

 

Our third assumption is that the fluid is incompressible, or in other words, that the fluid's density is a constant.

When combined, all three of these approximations allow us to make the following statement,  "the mass of fluid that enters a certain region in a given amount of time is equal to the mass of fluid that leaves another region in the same amount of time." This is known as the continuity of mass flow rate.  Mathematically we write this as...

$\sum{\dot{m}}_{in}$ $=$ $\sum{\dot{m}}_{out}$      where $\dot{m} = \rho \, v \, A$     and $v$ is the speed, $A$ is cross sectional area

What the the continuity equation above says is, "what goes in at some rate, must come out at the same rate".

If there we restrict our studies to only one type of fluid in a tube at a time, then density is a constant and we use volume flow rate, $Q$...

$Q = \frac{\dot{m}}{\rho} = v \, A$

Then our continuity equation becomes...

$\sum{Q}_{in} = \sum{Q}_{out}$ 

Bernoulli’s equation relates the speed, relative height, and pressure at any location within a moving or stationary fluid.  Bernoulli’s equation and the continuity equation are usually the two most useful tools when approaching a fluid mechanics problem.  Before defining Bernoulli’s equation, a simple derivation will help demystify its appearance.

Consider the figure below which shows a moving fluid inside a tube that changes height and radius.  We will consider an ideal fluid, (i.e. laminar flow, no viscous forces, and incompressible).  The pressure is measured at the center of each cross sectional area as indicated by the red $P_1$ and $P_2$.  The dashed blue line indicates the stream line that passes through the two locations we will consider. 

This is an image of a pipe with where the first part of the tube has a smaller cross sectional area than the second part and the second part of the tube is elevated higher. There is a dashed line of the trajectory of the particle that travels through the center of the tube.

Let’s apply conservation of mechanical energy to this system. 

$\sum{E}_1 + W_{nc} = \sum{E}_2$

Here we are ignoring any thermal energy since changes in temperature are typically very small in the scenarios we will study.

Some algebraic manipulations and we get our conservation of mechanical energy into the form below..

$\Delta KE + \Delta U^g = W_{nc}$

Each of the above terms have the following form…

$\Delta KE = \frac{1}{2} m \, \Delta v^2 = \frac{1}{2} \rho \, V \, \Delta v^2$     where $v$ is velocity and $V$ is volume.

$\Delta U^g = m \, g \, \Delta y = \rho \, V \, g \, \Delta y$   where $V$ is volume.

$W_{nc} = | \vec{F} | \, \Delta x = \Delta P \, A, \Delta x = \Delta P \, V$    where $A$ is cross sectional area and $P$ is pressure

In the expression for the non-conservative work, force is the force due to the horizontal pressure gradient within the fluid. 

Note that all the expressions contain the volume $V$, so let’s substitute each expression back into the conservation of mechanical energy and divide by volume.…

$\frac{1}{2} \rho \, \Delta v^2 + \rho \, g \, \Delta y = \Delta P$

Note that each term has units of energy per volume, thus this is a now a statement of conservation of mechanical energy density.  One more algebraic manipulation and we get the familiar form which we call Bernouill’s principle…

$P_{1} + \frac{1}{2} \rho \, v^{2}_{1} + \rho \, g \, y_{1} = P_{2} + \frac{1}{2} \rho \, v^{2}_{2} + \rho \, g \, y_{2} = constant$

Key Equations and Infographics

Now, take a look at the pre-lecture reading and videos below.

OpenStax Reading


OpenStax Section 12.1  |  Flow Rate and Its Relation to Velocity

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OpenStax Section 12.2  |  Bernoulli's Equation

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OpenStax Section 12.3  |  The Most General Applications of Bernoulli's Equation

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Additional Study Resources

Use the supplemental resources below to support your post-lecture study.

YouTube Videos

We will now begin to study the properties of moving fluids.  In general, this is a very complicated task which would require calculus and perhaps one or more courses that are solely focused on the field fluid dynamics.  In this section, we will apply a few approximations with regards to the moving fluid and study the consequences that arise, namely the continuity equation.

https://www.youtube.com/watch?v=bkq4Cn42EVk

https://www.youtube.com/watch?v=egIYa05Gn7E

This is a very good video lecture on continuity which brings in a connection with biology.

https://www.youtube.com/embed/wykn-JTnacE?rel=0

Bozeman Physics gives the usual clear description of continuity in this lecture.

https://www.youtube.com/watch?v=zgE0iNYw0JA

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Robert Ayton gives a very good lecture covering the Bernoulli Principle

https://www.youtube.com/embed/QWEq3xifCDw?rel=0

This 3D Animation shows Bernoulli's Principle in effect

https://www.youtube.com/watch?v=mmB-o5Auf-s

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Simulations


In this simple simulation, pay attention to the speed at which the tracked particles move through the pipes with different diameters.

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For additional simulations on this subject, visit the simulations repository.

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In this simple simulation you can see the effect of the Bernoulli Principle on water passing through a tube of different diameters. Think back to the simulation in the continuity section for when the Bernoulli Principle is not taken into account.

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For additional simulations on this subject, visit the simulations repository.

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Demos


For additional demos involving this subject, visit the demo repository

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Box Fall Experiment - Why does this box fall as I walk past? I do not touch it.

Box Fall Experiment

For additional demos involving this subject, visit the demo repository

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History


Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

Physics Fun

 

Other Resources


Flow rate and continuity is also covered in this section of Boundless

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NASA just so happens to have a page which covers mass flow rate and how it affects their lives

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This page from the University of Winnipeg is a short and sweet covering of flow rate and continuity

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This is the Boundless section for Bernoulli's Principle

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This website covers Bernoulli's Principle and has various animations

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Here's the Hyperphysics reference sheet for Bernoulli's Principle

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This is a fun activity involving a bird paper plain and connections to Bernoulli's Principle

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Resource Repository

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Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

Assumptions

Fluids is typically one of the more confusing topics for students, but it can be made easier by a couple steps.

1) Know if you're dealing with a static or dynamic fluid. This should be easy enough, just ask yourself if the water is flowing or stationary in the problem.

2) Finding the avalible equations.

For dynamic fluids, there are only two, Bernoulli's and Continuity. Bernoulli's is a beast, but usually some terms are 0 that you can pick out immediately. Keep in mind Continuity can be plugged into Bernoulli because they both have velocity in them, but this will only happen when water is flowing in a pipe and changes both pipe diameter and pressure/height of the pipe.

For static fluid problems, there is really only the defnition of pressure, pressure at depth, bouyant force and density. This is more complicated and can incorporate a couple extra tricks, but generally having all these written down and tracing what you know should give you a map to finding the correct answer. The other other tricks to this are drawing Free Body Diagrams with Buoyant force in them and realizing where pressures are equivalent (along any horizontal line in a fluid or at any point interacting with a piston in hydraulics problems.

Checklist

1. Read and re-read the whole problem carefully.

2. Visualize the scenario. Mentally try to understand what the problem is asking (think about the geometry, such as circles, rectangles, etc... that the problem might be asking about).

3. Draw a physical representation of the scenario, this may include a picture of an object with labeled dimensions.

4. Determine the volume flow rate $Q$ at each location of interest.

5. Set each location's volume flow rate equal to each other.    

6. Identify all the knowns and unknowns.

7.  Carry out the algebraic process of solving the equation.

8. Evaluate your answer, make sure the units are correct and the results are within reason.

Misconceptions & Mistakes

  • Although we often write the volume flow rate as $Q = v \, A$, remember that the $v$ in this case is a speed not a velocity.
  • When an tube that contains moving fluid changes height and diameter, the pressure at the higher location is not necessiarly lower than the bottom location.  As seen in Bernoulli's equation, it depends on buth the change in height and the change in cross sectional area.

Pro Tips

  • First determine the volume flow rate $Q$ for each location of interest.  Then apply continuity and set the volume flow rate at each location equal to each other.
  • Draw a physical representation labeling all dimensions, speeds, and pressures at the specific locations of interest.
  • If applying Bernoulli's principle results in 1 equation and 2 unknowns, try applying continuity to come up with a second equation.

Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

Physical

Physical Representations describes the physical phenomena of the situation in a visual way.

 The velocity of a fluid flowing through an enclosed area is dependent on the cross sectional area of the enclose area. Below, the velocity of the fluid changes with the geometery of the enclosure. 

This is an image of a tube with a greater area A one to a smaller area A two. The velocity of the fluid is greater moving through the smaller area denoted as v two at A two and the velocity of the fluid is lower moving through the greater area denoted as v one at A one.

The velocirty of the liquid flowing through the enclosed system below is dependent on the geometry, pressure, and potetntial energy.

This is an image of a tube where the first part of the tube has a greater cross sectional area than the second part. It shows the velocity vectors and pressure of the particles. The velocity and pressure is lower at the larger area of the tube and the velocity and pressure is higher at the smaller area of the tube.

Mathematical

Mathematical Representation uses equation(s) to describe and analyze the situation.

$\sum Q_{in} = \sum Q_{out}$

$\sum \rho V_{in} = \sum \rho V_{out}$

Heat ($Q$), Velocity ($V$), Density of fluid ($\rho$)

 

 

A representation with the words continuity of mass flow rate on the top. There is an equation that shows that the mass flow rate in to a given region is equal to the mass flow rate exiting the same region. This is also written in words below with a note that says the mass flow rate is equal to the density of the fluid times the velocity times the cross-sectional area.

A representation with the words continuity of volume flow rate on the top. There is an equation that shows that the volume flow rate in to a given region is equal to the volume flow rate exiting the same region. This is also written in words below with a note that says the volume flow rate is equal to the velocity multiplied by the cross-sectional area.

 

$P + \rho g y+ \frac{1}{2} \rho v^{2}= Const.$

$P_{1}+ \rho g y_{1} +\frac{1}{2} \rho v_{1}^{2}=P_{2}+ \rho g y_{2} +\frac{1}{2} \rho v_{2}^{2}$

The equation above reads: The pressure plus the potential energy per unit volume plus the kinetic energy per unit volume, is constant along a stream line .

Continuity: $\sum A_{1}v_{1}=\sum A_{2}v_{2}$

Pressure($P$), Density of fluid ($\rho$), Gravity ($g$), Height ($y$), Velocity ($v$), Area ($A$)

A representation with the words Bernoulli’s principle on the top. There is an equation that shows that along a laminar flow streamline, the pressure at one point, plus one half the density times the square of the velocity of the fluid at that point, plus the density multiplied by gravity and the vertical position, is equal to a constant. That means the summation of those same three terms is equal at another point along the streamline. This is also written in words below.

Graphical

Graphical Representation describes the situation through use of plots and graphs.

 The left graph describes the speed of a fluid with respect to the diameter of a pipe. As the size of the pipe increases, the speed of the fluid decreases. Similarly, the graph on the right depicts the speed with respect to the area of the enclosure.

This is a graph of speed versus radius of a pipe and shows a decreasing curve which shows that as the pipe radius increases, the speed decreases.   This is a graph of speed versus area of a pipe and shows a decreasing exponential curve which shows that as the area increases, the speed of the fluid exponentially decreases.

Descriptive

Descriptive Representation describes the physical phenomena with words and annotations.

 Continuity describes the flow rate of a fluid flowing through a confined space. The amount of fluid going in must match the amount of fluid flowing out. Therefore, as the geomety of the confined space change so must the speed of the fluid in order to maintain a constant flow rate. 

 Bernoulli's equation for flow is a conservation of energy density equation. One of the main principles is that as the velocity of the fluid increases the pressure decreases. Conversely, as the velocity decreases the pressure increases. As we can see from the equations above, the energy equation looks similar to the energy equation introduced earlier. However, now that we include the denisty of the fluid we are working with energy density. 

Experimental

Experimental Representation examines a physical phenomena through observations and data measurement.

Continuity is easily demonstrated by taking a hose and and observing the velocity of the water relative to the cross-sectional area that the water flow throughs. The following video demonstrates the principal of continuity. Observe, the difference in the water stream as the hose opening is covered. 

https://www.youtube.com/watch?v=Lon6Up4juH8

Grab a piece of paper and hold it by the corners and bring that short side near your lips letting the rest of the paper naturally drape down. With the short side of the paper near your lips, blow air toward the paper so some air flows above and below the paper. You should see the paper begin to rise similar to an airplane wing. The following video will show you how Bernoulli's principle applies.

https://www.youtube.com/watch?v=8vqMotb6m3c

Practice

Use the practice problem sets below to strengthen your knowledge of this topic.

Fundamental examples

1.  A faucet is capable of filling a $1 \, L$ bottle in $10$ seconds.  If the diameter of the piping in the house is $12.7 \, mm$, what speed does the water flow inside the pipes?

 

2.  A $1 \, cc$ syringe attached to a $25$ gauge needle is shown in the figure below with the dimensions labeled.  If the plunger of the syringe is pushed with a speed of $10 \, mm/s$, at what speed does the fluid inside the syringe leave the tip of the needle?

This is an image of a syringe with a gauge needle. The needle has an inner diameter of zero point three hundred and five millimeters. The syringe has a diameter of four point seven six two five millimeters.

3.  The diameter of a typical garden hose is $0.625 \, inches$ and contains water that flows at a speed of $1 \, m/s$.  What diameter nozzle must you connect to the end of the hose to get the water to shoot out with a speed of $10 \, m/s$ ?

 

CLICK HERE for solutions 

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

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1.  Consider the figure below.  The pressure drops $5 \, kPa$ from location 1 to location 2.  The speed of the water at location 1 is known to be $3.87 \, m/s$.  What is the speed at location 2?

This is an image of a tube where the first part of the tube has a greater cross sectional area than the second part. Both the cross sectional areas has the center at the same elevation.

2.  Consider the figure below.  The pressure drops $5 \, kPa$ from location 1 to location 2.  The speed of the water at location 1 is equal to the speed at location 2.  What is the change in height between these two locations?



This is an image of a tube where the first part of the tube has a greater cross sectional area than the second part. The center of the second part of the tube is higher than the center of the first part of the tube. This is indicated by y two minus y one and a vector arrow pointing up.

3.  Consider the figure below.  The pressure drops $5 \, kPa$ from location 1 to location 2.  The diameter of the pipe at location 1 is $3 \, cm$ and at location 2 is $2 \, cm$.  What is the speed of the water at each location?

This is an image of a tube where the first part of the tube has a greater cross sectional area than the second part. Both the cross sectional areas has the center at the same elevation.

 

 

CLICK HERE for solutions.

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

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Practice Problems

BoxSand practice problems

Conceptual problems

BoxSand's multiple select problems

BoxSand's quantitative problems

Recommended example practice problems 


For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.

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