# Fluid Dynamics

## Inquiry Lab Kit for AP® Physics 2

### Materials Included In Kit

Bottles, plastic, 1 L, 6
Bottle caps, 6
Rulers, metric 6
Tubing connector, ½" i.d., 1
Tubing connector, " i.d., 1

Butane safety lighter or Bunsen burner
Calipers (optional)
Cork borer, 5 mm i.d. or soldering iron
Sink or water catch basin
Tubing, rubber/latex

### Prelab Preparation

Teacher Demonstration

1. Using a 3" x 5" index card, make 1 cm tabs on the 3" end to make a “table” (see Figures 3 and 4).
{14008_Preparation_Figure_3}
{14008_Preparation_Figure_4}
2. This and a straw will be needed for the teacher demonstration before Prelab Questions are answered. Set the card on a flat surface and ask students to answer Prelab Question 1. Then, using a straw, blow a constant fast stream of air underneath the surface of the card. The card will bend down. Students can now complete the remaining Prelab Questions.
Prelab Preparation
To prepare 1-L bottles with small holes:
1. Find a vertical seam on the bottle. Measure 5 cm from the bottom of the bottle and use a marker to place a dot at that location.
2. Using a butane safety lighter, heat the end of a 5 mm i.d. cork borer by holding the end in the flame of the lighter for 20 seconds.
3. Use the cork borer to pierce the bottle at the location of the dot. Care should be taken that the hole is made perpendicular to the bottle so the stream emerges in a horizontal direction. Note: The holes can be cut into 6 bottles using the method above (or using the round end of a soldering iron) in under 15 minutes.

### Safety Precautions

Use caution when working with hot objects. Quickly wipe up any spills to prevent injury. Wear safety glasses. Wash hands thoroughly with soap and water before leaving the laboratory. Follow all laboratory safety guidelines.

### Disposal

All materials may be stored and saved for future use.

### Lab Hints

• This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Also, all student-designed procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions, in this case, are required to be completed after a teacher demonstration.
• In the Introductory Activity, the use of Teflon tape can improve the fit between the tubing and the faucet and reduce the amount of leaks.
• When preparing the 1-L bottles, take care to create round holes with smooth edges to reduce the effects of turbulent flow. Ensure that the cork borer is hot enough so that a clean hole is made, if the end of the cork borer does not immediately go through the bottle, heat it another 20 seconds.

### Teacher Tips

• Blowing air over a sheet of paper is a quick demonstration that students could do to further familiarize themselves with Bernoulli’s principle.
• In the Guided-Inquiry Activity, students may choose to directly measure fluid depth vs. time. This data can still be used to come to the same conclusion about the relationship between flow rate and fluid depth but requires further analysis than the fluid depth vs. fluid velocity method.
• If students opt to study fluid height in the container as a function of time, it is important to guide them away from the confusion that velocity of the water level as it drops in height is the same as the velocity of the water as it exits the hole.
• Giving different groups bottles with different sized holes would add another element of analysis to the investigation.

### Further Extensions

Opportunities for Inquiry
Investigate how changing the density of the fluid affects the results. A salt water mixture may be used for a higher density fluid and an alcohol/water mixture for a lower density fluid.

Alignment to the Curriculum Framework AP® Physics 2

Enduring Understandings and Essential Knowledge
Materials have macroscopic properties that result from the arrangement and interactions of the atoms and molecules that make up the material. (1E)
1E1: Matter has a property called density.

The energy of a system is conserved. (5B)
5B10: Bernoulli’s equation describes the conservation of energy in fluid flow.

Classically, the mass of a system is conserved. (5F)
5F1: The continuity equation describes conservation of mass flow rate in fluids. Examples should include volume rate of flow and mass flow rate.

Learning Objectives
5B10.1: The student is able to use Bernoulli’s equation to make calculations related to a moving fluid.
5B10.2: The student is able to use Bernoulli’s equation and/or the relationship between force and pressure to make calculations related to a moving fluid.
5B10.3: The student is able to use Bernoulli’s equation and the continuity equation to make calculations related to a moving fluid.
5B10.4: The student is able to construct an explanation of Bernoulli’s equation in terms of the conservation of energy.
5F1.1: The student is able to make calculations of quantities related to flow of a fluid, using mass conservation principles (the continuity equation).

Science Practices
1.2 The student can describe representations and models of natural or man-made phenomena and systems in the domain.
1.4 The student can use representations and models to analyze situations or solve problems qualitatively and quantitatively.
2.1 The student can justify the selection of a mathematical routine to solve problems.
2.2 The student can apply mathematical routines to quantities that describe natural phenomena
3.1 The student can pose scientific questions.
4.1 The student can justify the selection of the kind of data needed to answer a scientific question.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
4.3 The student can collect data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns and relationships.
5.2 The student can refine observations and measurements based on data analysis.
6.1 The student can justify claims with evidence.
6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.
7.2 The student can connect concepts in and across domain(s) to generalize or extrapolate in and/or across enduring understanding and/or big ideas.

### Science & Engineering Practices

Developing and using models
Planning and carrying out investigations
Analyzing and interpreting data
Obtaining, evaluation, and communicating information
Using mathematics and computational thinking
Engaging in argument from evidence

### Disciplinary Core Ideas

HS-PS3.B: Conservation of Energy and Energy Transfer

### Crosscutting Concepts

Patterns
Systems and system models
Scale, proportion, and quantity
Cause and effect
Stability and change
Energy and matter

### Performance Expectations

HS-PS3-1. Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.

1. Predict will happen when your teacher blows air underneath the index card “table.”

2. Describe what happened when the teacher blew air underneath the index card. What are the forces acting on the card that account for your observations? How does the pressure below the card compare to the pressure above the card?

The card bent down. The forces acting on the card are gravity, air pressure up, and air pressure down. The air pressure below the card is lower than the pressure above the card. The higher pressure air moves towards the lower pressure air, causing the card to bend down.

3. What is the difference between flow rate of a fluid and the speed of the fluid? Hint: Consider the units of measurement of each quantity.

Flow rate is the movement of a unit of volume of a fluid per unit of time. Speed of fluid is a unit of distance pre unit of time. A flow rate could be measured in m3/s whereas a fluid speed is measured in m/s.

4. A boy holding a water gun shoots at his friend standing 5 meters away but misses to the right. At the time of firing, the gun nozzle is 120 cm from the ground. At what speed does the water exit the nozzle?

h = 1.2 m
g = 9.8m/s2
h = ½gt2
t = sqrt(2h/g)
t = sqrt(2 x (1.2)/(9.8))
t = 0.49 seconds
Speed = x/t
x = 5 m
t = 0.49 s
Speed = 5/0.49
Speed = 10.1 m/s

5. An Olympic swimmer dives to the bottom of a 3 meter deep pool. How much pressure does the diver experience at the bottom of the pool?

Pressure = ρgh = (1000 kg/m3)(9.8 m/s2)(3 m) = 29.4 kPa

### Sample Data

Introductory Activity

• From the data, calculate the speed of the water exiting each tubing connector with the projectile motion equations.

Answers will vary. To calculate the speed of water, first measure the distance, h, from the nozzle to the floor. Calculate the time it takes for the water to fall to floor with t = √(2h/g) where g is the acceleration due to gravity. Using the t calculated, measure the speed of the exiting water with x/t with x being the horizontal distance the water traveled before impacting with the floor.

• Are the flow rate values for the different connectors expected to be the same? Calculate and compare the flow rates when using the smaller connector and the larger connector.

Yes, the flow rates are expected to be the same due to the conservation of mass. To calculate the flow rate, first measure the inner diameter of the respective connector. Then divide by 2 to get the radius and use πr2 to calculate area. Flow rate is area multiplied by speed of fluid.

{14008_Data_Table_1}
• If the flow rates differ, explain why that may occur.

Answers will vary. The flow rates are very close in value but differ slightly. A reason for this difference could be due to the faucet not being open to exactly the same location for each connector tested. Therefore, the flow rates differed slightly.

Guided-Inquiry Activity
1. Obtain a 1-L bottle that has a hole pierced near the bottom.
2. Choose the max height of water on the bottle.
3. With the materials available, devise a procedure for collecting data that can be used to demonstrate the relationship between fluid depth above the exit hole and the rate at which fluid flows from the container.

One method students could use is to measure fluid depth vs. velocity. Using a vertical piece of masking tape, create a marked scale on the outside of the bottle. Set the bottle on the end of an elevated surface and record the horizontal distance traveled by the stream from the exit hole to the container on the floor for different fluid depths of water. One could use masking tape to measure spots on the floor that correspond with tick marks on the vertical scale on the bottle. The exit velocity can then be calculated with the projectile motion equation and the data gathered is that of fluid depth and exit velocity.

Analyze the Results
• Present a graph(s) that represents the data collected.
{14008_Data_Figure_5}
• From the created graph(s), derive a mathematical relationship between fluid depth above the exit hole and the rate at which water leaves the bottle.

The equation produced from the graph of fluid height vs velocity squared gives us an equation of h = 0.0513v2. When solving for v: v = √(h /0.051) which allows for the conclusion that the rate at which waters flows from the container is proportional to the square root of the water depth above the exit hole.

• What are some possible sources of error when collecting data points?

Possible experimental error may arise from imprecise measurements due expected systematic error (e.g., delayed reaction times, non-uniform tick mark distances). Another source of error leading to losses in kinetic energy of the fluid may arise from slight turbulence at the exit point due to the perforated hole not being smooth enough.

Review Questions for AP® Physics 2

1. Consider the same experimental setup as in the Guided-Inquiry Activity.
1. Using Bernoulli’s equation, derive a mathematical relationship between fluid height in the container and the speed with which the water exits the container.

P1 + ½ρv12 + ρgh1 = P2 + ½ρv22 + ρgh2; here P1 and P2 are atmospheric pressure and can be cancelled out. Further, v1 is zero and the second term on the left side can be cancelled out. Likewise, h2 is zero (at the position of the hole) and the third term on the right side can be cancelled out. We are left with ρgh1 = ½ρv22. The densities cancel out. When solving for v2, we get v2 = √2gh1

2. With the relationship derived in Question 1a, do you expect the experimental values to be higher than or lower than the theoretical values? Calculate what the theoretical values of fluid speed should be for each point of fluid depth measured.

Student answers will vary. Experimental values are expected to be lower than theoretical values due to losses in energy from friction and turbulence. Theoretical values of fluid speed can be calculated using the expression derived in 1a. Use the height for each respective fluid depth data point to calculate the respective theoretical fluid speed.

3. Give an explanation for any differences in value between the theoretical and exprimental fluid speeds.

The theoretical values are higher than the experimental data because in reality you lose energy due to friction and turbulence effects, which results in lower expected speeds.

4. How does the flow rate of water dropping in height throughout the bottle compare to the flow rate of water exiting the hole?

The flow rates are the same due to conservation of mass as portrayed in the continuity equation.

2. In your own words, explain how the systems analyzed in the Introductory Activity and the Guided-Inquiry Activity follow known conservation laws.

The investigation conducted in the Introductory Activity was a demonstration of conservation of energy, and more specifically, conservation of mass. This is because flow rates are constant due to conservation of mass flow, which is observed in the activity. The experiment in the Guided-Inquiry Activity follows the law of conservation of energy because potential energy due to gravity is completely converted to kinetic energy at the exit hole. Bernoulli’s equation is a statement of conservation of energy in fluids, and when used to calculate expected values, it confirmed that energy was conserved.

3. A volume of 2 m3 of water is flowing in a level, horizontal pipe with a flow rate of 1 m3/s. The water flows from a pipe section with a cross sectional area of 0.2 m2 to a pipe section with a 0.1 m2 cross sectional area (see Figure 2). Before entering the constricted segment of pipe, the water is at a pressure 20 kPa above atmospheric pressure. What is the average acceleration that the volume of water undergoes when flowing from the unconstricted pipe segment to the constricted pipe segment?
P1 = 121,325 pascals. Volume of water = 2 m3. A1 = 0.2 m2. A2 = 0.1m2. Flow rate Q = 1 m3/s.
Acceleration is found from P1A1 – P2A2 = F1 – F2 = Fnet = ma. a is acceleration. Mass of volume is density of water times volume = 1000 kg/m3 x 2 m3 = 2000 kg.
P2 is found using Bernoulli’s equation: P2 = P1 + ½ρ(v12 – v22). (Due to the level fluid flow the terms containing fluid height are eliminated). The speeds v1 and v2 can be calculated using the continuity equation where Q = A1v1 = A2v2, which is 1 m3/s = (0.2 m2)v1 = (0.1 m2)v2. v1 = 5 m/s and v2 = 10 m/s. When using these values in Bernoulli’s equation, P2 is found to be 83,825 pascals. Fnet can then be calculated to be 15,882.5 N. Equating Fnet to ma and solving for a gives an acceleration of 7.94 m/s2.
4. Far into the future, Mars has been successfully colonized. A farmer on Mars has a personal water tower for which the water level is always kept constant. This water tower is used to feed the farmer’s irrigation system. Water from the water tower flows down through a single tube and is held under pressure behind an irrigation system valve (the water is no longer moving at this point). In order to irrigate the field, the water is held at 100 kPa when the valve is closed.
1. How tall must the water tower be in order to irrigate the field? The water pressure in the tower is 1 kPa and the acceleration due to gravity on Mars is 3.8 m/s2.

Using Bernoulli’s equation: 1,000 pascals + 0 + ρ(3.8 m/s2)h = 100,000 pascals + 0 +0.
h = 26.05 meters.

2. Once the valve is opened, how fast does the water flow? The atmospheric pressure on the Martian surface is about 600 pascals.

100,000 pascals = 600 pascals + ½ρv2 + 0.
v = 14.1 m /s.

### References

AP® Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014.

# Fluid Dynamics

## Inquiry Lab Kit for AP® Physics 2

### Introduction

Understanding the motion of fluids is important to build a greater understanding of engineering fields such as aerodynamics, hydrodynamics and traffic engineering. Fluid dynamics is used to calculate the forces that act on a flying aircraft, predict weather patterns, and even determine the rate of flow of blood in the body. By exploring the laws of conservation of mass and energy, we can discover the laws that govern fluid dynamics.

### Concepts

• Conservation of energy
• Continuity equation
• Bernoulli’s law
• Pressure

### Background

The concept of pressure is central to the study of fluid dynamics. A fluid exerts pressure in all directions and particularly in a direction perpendicular to any wall with which it makes contact. French physicist, Blaise Pascal (1623–1662), contributed greatly to the principles of hydraulic fluids. When experimenting with hydraulic pistons, he determined that an increase in the pressure of a static, enclosed fluid at one place in the fluid caused a uniform increase in pressure throughout the fluid. Pascal’s second law states that the amount of pressure exerted at any given point depends upon the height of the liquid above that particular point. Pressure is expressed as a force over an area (Equation 1).

{14008_Background_Equation_1}
When considering fluid flow, it is important to note that liquids are regarded to be almost incompressible. Gases can be compressed due to the large spaces between molecules. The molecules of a liquid are much closer together and cannot be compressed in the same sense. Liquid molecules instead slide past each other. Now consider water flowing in a pipe as shown in Figure 1.
{14008_Background_Figure_1}
Since liquids are almost incompressible, the amount of water flowing through cross section A1 is the same as the amount of water flowing through cross section A2. This means that the flow rate through cross section A1 is the same as the flow rate through cross section A2. Flow rate is defined as the volume of fluid that flows through a cross section of a pipe in the time interval t2 – t1 (Equation 2).
{14008_Background_Equation_2}
where

Q = flow rate,
V = volume of fluid,
Δt = time interval (t2 – t1)

In the narrower section of the pipe, the speed of the water will increase in order to keep the flow rate constant. This is shown in the continuity equation:

{14008_Background_Equation_3}
where

v1 = average speed of fluid flowing through cross section of area A1
v2 = average speed of fluid flowing through cross section of area A2

Swiss physicist Danile Bernoulli (1700–1782) conducted experiments focusing on the conservation of energy using liquids. He observed that in a segment of pipe with a large diameter, water flowed slowly whereas the same stream of water would flow more quickly through a segment of smaller diameter (as in Figure 1). It was clear that some force had to be acting on the water for it to increase speed. The force for the acceleration could be accounted for by considering pressure differences. If the pressure in the larger diameter segment is greater than the pressure in the smaller diameter segment, then there is a net force from the left (at high pressure) toward the constricted pipe segment on the right (low pressure). Therefore, the liquid accelerates as it enters the constricted segment of the pipe. Bernoulli’s principle states that the pressure a fluid exerts on a surface decreases as the speed with which the fluid moves across the surface increases. This principle is applied to not only liquids but to all fluids and has as many applications with regard to airflow as it does to liquid flow. To quantify this principle, three assumptions must be made: the fluid is incompressible, the fluid moves without friction, and the flow is streamline (without turbulence). Bernoulli’s equation is expressed as:
{14008_Background_Equation_4}
where

P = pressure, Pa (Pascals)
ρ = fluid density, kg/m3
v = fluid speed, m/s
g = acceleration due to gravity, m/s2
h = depth of the fluid, m

This equation describes the conservation of energy as applied to fluids; it relates the pressures, speeds and elevations of two points along a single streamline in a fluid. The conservation of energy statement is most apparent when all terms are multiplied by volume. The density of a substance multiplied by its volume equates to mass. The respective terms then become PV (work), ½mv2 (kinetic energy) and mgh (potential energy).

### Experiment Overview

The purpose of this advanced inquiry investigation is to gain an understanding of fluid motion by applying the laws of conservation of energy and mass. An introductory class demonstration of conservation of mass flow is performed by changing the ending cross-sectional area of tubing connected to a faucet. This allows for a class discussion focused on the continuity equation. The guided-inquiry activity explores the relationship between fluid depth in a container and the rate at which the fluid flows from the container.

### Materials

Bottle, plastic, 1-L
Bottle cap
Bucket to catch water
Calipers (optional)
Ruler, metric
Sink or water catch basin
Tubing connector, ½" i.d.
Tubing connector, " i.d.
Tubing, rubber/latex

### Prelab Questions

1. Predict will happen when your teacher blows air underneath the index card “table.”
2. Observe the teacher demonstration.
3. Describe what happened when the teacher blew air underneath the card. What are the forces acting on the card that account for your observations? How does the pressure below the card compare to the pressure above the card?
4. What is the difference between flow rate of a fluid and the speed of the fluid? Hint: Consider the units of measurement of each quantity.
5. A boy holding a water gun shoots at his friend standing 5 meters away but misses to the right. At the time of firing, the gun nozzle is 120 cm from the ground. At what speed does the water exit the nozzle?
6. An Olympic swimmer dives to the bottom of a 3-meter deep pool. How much pressure does the diver experience at the bottom of the pool?

### Safety Precautions

Wear safety glasses. Quickly wipe up any spills to prevent injury. Wash hands thoroughly with soap and water before leaving the laboratory. Follow all laboratory safety guidelines.

### Procedure

Introductory Activity

1. Connect the rubber tubing to a sink faucet.
2. Adjust the tubing so that the end where water will exit is at table height (or higher) and fixed parallel to the floor (see Figure 2).
{14008_Procedure_Figure_2}
3. Place towels along the floor where water might splash.
4. Place a bucket in front the of tubing to collect the exiting stream of water.
5. Insert the smaller tubing connector into the end of the tubing.
6. Measure the inner diameter of the tubing connector. You may use calipers if available.
7. Turn on the faucet so a stream of water falls into the bucket (see Figure 3). Mark how far the faucet handle is turned. Note: In order to reduce the amount of water spilled, have someone hold the bucket up to the exit nozzle. Then, turn on the faucet gradually while the person holding the bucket walks away with water continuously falling into the bucket until the bucket is able to be placed on the floor.
{14008_Procedure_Figure_3}
8. Measure the distance, h, from where the water exits to the floor.
9. Measure the horizontal distance, x, that the stream of water travels until it makes contact with the bottom of the bucket.
10. Turn off the faucet. Note: In order to reduce the amount of water spilled, have someone pick up the bucket and follow the stream of water back toward the exit nozzle while the faucet is gradually closed.
11. Repeat steps 5–10 with the larger tubing connector. Be sure to open the faucet handle to the same point used in step 7. Adjust the position of the bucket to collect the water.
12. Wipe up any water off the floor.
Analyze the Results
• From the data, calculate the speed of the water exiting each tubing connector with the projectile motion equations.
• Are the flow rate values for the different connectors expected to be the same? Calculate and compare the flow rates when using the smaller connector and the larger connector.
• If the flow rates differ, explain why that may occur.
Guided-Inquiry Design and Procedure
1. Obtain a 1-L bottle that has a hole pierced near the bottom
2. If the 1-L bottle is filled to the top with water, a stream will pour from the pierced hole. What happens to fluid pressure at the exit hole as fluid depth decreases?
3. Choose the initial max height of water in the bottle.
4. Devise a method for determining fluid as water exits the container.
5. Formulate a mathematical technique for determining fluid velocity as it exits the container. Hint: Consider how the projectile motion equations were used to calculate fluid velocity in the Introductory Activity.
6. With the materials available, devise a procedure for collecting data that can be used to demonstrate the relationship between fluid depth above the exit hole and the rate at which fluid flows from the container. Be sure to gather sufficient values of fluid depth and respective fluid exit velocity in order to provide scientifically accurate data.
Analyze the Results
• Present a graph(s) that represents the data collected.
• From the created graph(s), derive a mathematical relationship between fluid depth above the exit hole and the rate at which water leaves the bottle.
• What are some possible sources of error when collecting data points?

### Student Worksheet PDF

14008_Student1.pdf

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