In this studio session In this session, you will try to verify the equation of continuity for water flowing out of the bottom of an elevated can through a small-diameter hose.  You will measure the fraction of the ordered energy that is lost because of friction. You will also explore fluid flow using a simulation.  Then you will determine the viscosity of different brands of "Volumizing Shampoo" using Stokes' law.  You will use a fluid column as a viscometer and measure the rate of descent of a steel sphere, as it falls under the influence of gravity through the fluid, after the sphere has reached terminal velocity.

Equipment needed:

• can and hose
• water container
• meter stick
• Shampoo bottle with metal sphere and linear scale
• Magnet
• Webcam

Open a Microsoft Word document to keep a live journal of your experimental procedures and your results.  Include all deliverables, (data, graphs, analysis, outcome).  Write a 'mini-reflection' immediately after finishing each investigation, experiment or activity, while the logic is fresh in your mind.

The equation of continuity

Liquids are incompressible.  Their density ρ = mass/volume is constant.  When a liquid flows through a pipe, conservation of mass leads to the equation of continuity.
Consider the flow of a fluid through a pipe with varying cross sectional area A.
The volume V₁ of liquid flowing into the pipe equals the volumeV₂ flowing out of the pipe per unit time.
V₁/Δt = V₂/Δt,  A₁Δx₁/Δt = A₂Δx₂/Δt,  A₁v₁ = A₂v₂.

For the pipe we write the equation of continuity as A₁v₁ = A₂v₂, or Q = Av = constant.  Q is called the volume flow rate.

Experiment 1

Each table will work as a group on this experiment.

One end of a rubber hose is attached to a can with a spout in the bottom.  The can has an inside diameter of d₁ = 9.8 cm and the rubber hose has an inside diameter of d₂ = 0.8 cm.  The other end of the hose is taped to a horizontal rod a certain distance below the can.  While one member of your group plugs the hose with a finger, another member fills the can with water.  Then you will allow the water to drain from the can through the hose into a catch pan.  You will measure the speed with which the water level in the can drops, and the speed with which the water emerges from the hose.

       

Set up the apparatus on the floor, not on the table. 

• The horizontal position of the lower end of the hose should be a few cm over the edge of the catch pan.
• Measure the height of ∆y of the center of the hose above the bottom of the catch pan.
• Measure the vertical distance h from the center of the hose to the center of the 5 cm long black tape inside the can.
• Record your measurements of ∆y and h in table 1.

Table 1

Fill up the can with water to above the upper end of the black tape without allowing water to drain.  Distribute tasks to be performed after you start draining the water.

• One member of your group will notify the others when the water level in the can reaches the upper end and when it reaches the lower end of the black tape.
• Several members will measure the time interval ∆t between these two events.
• Other members will mark the spots where the water hits the bottom of the pan for those two events.  You will have to determine the average horizontal distance ∆x between the lower end of the hose and the spots where the water hits the pan.

Drain the water while making measurements and record your measurements of ∆x and ∆t in the table.

• Repeat the experiment to check for reproducibility.
   
• To find the speed v₁ with which the water level drops calculate v₁ = (5 cm)/∆t.
• To find the speed v₂ with which the water emerges from the hose, use the formulas for projectile motion.
  • For the vertical motion we have ∆y = ½gt².  We can solve for the time t it takes water to fall from the end of the hose to the bottom of the pan.
  • For the horizontal motion we have ∆x = v₂t.  Inserting the time from above we have v₂ = ∆x/(2∆y/g)^1/2.
• The kinetic energy of a volume V of water moving with speed v = ½ρVv² = ½mv², since ρV = m.
  The potential energy of the water changes as it moves.  While all the water moves, the change in potential energy is the same as that of a volume V, which has been moved from the top of the can to the exit of the hose.  The potential energy of the water in the rest of the can and hose is the same as the potential energy of the water that used to be there before the movement.
  The change in the potential energy of a mass ρV = m of water coming out of the hose is mgh.
  Here h is the distance from the center of the hose to the center of the 5 cm long black tape inside the can that you measured earlier.
  If no ordered energy is converted into disordered energy we expect from energy conservation that the change in kinetic energy is equal to the change in potential energy.
  ½mv₂² - ½mv₁² = mgh or ½(v₂² - v₁²) = gh.
  But some of the ordered energy will be converted into disordered energy, so we expect ½(v₂² - v₁²) < gh.
  Dividing the change in kinetic energy by the change in potential energy we find the fraction R =  (v₂² - v₁²)/(2gh) of the potential energy that is converted to kinetic energy.  If no ordered energy is lost, then R = 1.  We expect it to be less than one, because friction is always present.  But how much less?
• Fill in table 2.

Table 2

Experiment 1 Deliverables: (to be included in the your journal)

• Visuals:  Table 1 and table 2
• Analysis:
  • Did you verify the equation of continuity?  If not, explain what factors may be responsible for the difference in your two values of the volume flow rate?  Does the difference seem reasonable considering those factors?
  • What fraction of the potential energy of the water is converted into kinetic energy in this experiment?   What fraction is converted into thermal energy?  Does this result surprise you?  If yes, think about Poiseuille's law.

Open the simulation
http://phet.colorado.edu/en/simulations/fluid-pressure-and-flow
to investigate the flow of a liquid in a pipe.

The interface

• Try all the different tabs at the top of the simulation. 
• "Reset All" resets only the tab that you are presently using.
• You can use multiple tools to make measurements.
• Sensors are very sensitive, so you may expect some variations in the readings.

Flow tab

• The handles on the "Flow" tab let you change the shape/height of the water tube and end pipes.
• You can pause the simulation and then use increment time in discreet steps.

Water Tower tab

• Water is allowed to flow out of a hole at the bottom of a tall water tower. 

Exploration 1:

Click the Flow tab.

Part (a)

• Reset all.  Choose no friction.  For the fluid density choose water. 
  Keep the flow rate at 5000 liter/s = 5 m³/s.
• Case 1
  
  • Measure the diameter of the pipe and hose and the flow speed of the fluid.
  • Measure the pressure at the bottom of the pipe.
  • Record all values in table 1.
  • Click the red button to get a representation of the profile of the flow. 
     
• Case 2
  
  • Change the diameter of the pipe and hose by pulling all the lower handles down as far as possible.
  • Measure the diameter of the pipe and hose and the flow speed of the fluid.
  • Measure the pressure at the bottom of the pipe.
  • Record all values in table 1.
  • Click the red button to get a representation of the profile of the flow. 
• Case 3
  

  • Change the diameter of the pipe and hose again by pulling all the upper handles up as far as possible.

  • Measure the diameter of the pipe and hose and the flow speed of the fluid.
  • Measure the pressure at the bottom of the pipe.
  • Record all values in table 1.
  • Click the red button to get a representation of the profile of the flow. 
• Case 4
  • Keep the diameter of the pipe and hose the same as in case 3.
  • Change the fluid density first to gasoline and then to honey.
  • Measure the flow speed of the fluid.
  • Measure the pressure at the bottom of the pipe.
  • Record all values in table 1.
  • Click the red button to get a representation of the profile of the flow. 

Table 3

From your measurements, for all cases, calculate the volume flow rate in m³/s and record it in the table.

Part (b)

Reset all.

• Move the yellow handles in the middle of the pipe to their extreme up and down positions, while leaving everything else untouched. 
• Measure the pressure as close to the middle of the pipe as possible at two different locations in the pipe near the handles pointed to by arrows in the figure above. 
• At the same locations, measure the diameter of the pipe and the flow speed. 
• Record your measurements in table 2 and calculate the volume flow rate for both locations.

Table 4

Part (c)

Now reset all.  Turn on friction.  Adjust the flow rate so that the speed as close to the wall of the straight pipe as you can measure is 0.6 m/s. 

Exploration 1 Deliverables: (to be included in the your journal)

• Visuals:  Table 3 and table 4
• Analysis:
  • What information about the flow of a fluid in a pipe can you extract from table 3?
    Describe the the profile of the flow.  Is it the same for all cases?
  • What information about the flow of a fluid in a pipe can you extract from table 4?
  • For the viscous flow how does the speed of the liquid in the middle of the pipe compare to its speed close to the wall?  Describe the the profile of the flow.  Compare it to the profile without friction.

Viscosity

Viscosity is a measure of a fluids resistance to relative motion within the fluid.  Highly viscous fluids do not readily flow.  The viscosity of a fluid usually varies with temperature.  For a fluid flowing through a pipe in laminar flow, viscosity is one of the factors determining the volume flow rate.

Poiseuille's law: Q = π∆Pr⁴/(8ηL)

Volume flow rate = π*(pressure difference)*(pipe radius)⁴/[8*(pipe length)*viscosity)

Exercise

Blood is a viscous fluid circulating through the human body.  The circulatory system is a closed-loop system with two pumps.  One-way valves keep the flow unidirectional.  A sketch is shown below.  The unit of pressure in the sketch is mm Hg.  (1 atm = 760 mm Hg)

It is often important to know the viscosity of a fluid.  A viscosimeter is the instrument used to measure viscosity.  The study of the viscosity of substances is known as rheology.

Example:

In order to keep the pistons moving smoothly in the cylinders of the internal combustion engine in a car, a thin film of motor oil between the piston rings and the cylinder wall acts as a lubricant.  The oil must be able to keep the piston moving smoothly, when the engine first starts up and is still cold and when the engine reaches its high operating temperature.  One way of measuring an oil's ability to lubricate is to measure its viscosity.

In this session you will determine the viscosity of different brands of "Volumizing Shampoo" using Stokes' law.  You will use a fluid column as a viscometer and measure the rate of descent of a steel sphere, as it falls under the influence of gravity through the fluid, after the sphere has reached terminal velocity.

George Gabriel Stokes, an Irish-born mathematician, worked most of his professional life describing fluid properties.  Stokes' law gives the force required to move a sphere through a viscous fluid at a specific velocity, as long as the flow around the sphere is laminar and the Reynolds number is low (Reynolds number < 1).  Stokes' Law is written as

F = 6πηrv.

Here r is the radius of the sphere, v the speed and η the viscosity. 

Experiment 2

Measure the rate of descent of a steel sphere, as it falls under the influence of gravity through the shampoo.

• Start with the steel sphere held near the middle of the lid of the shampoo bottle by the magnet.  Make sure all the bubbles have risen and the fluid is quiet.
• Connect a webcam to one computer.
• Start the Windows 10 Camera App and select "Take Video".
• If necessary, click the "Change Camera" icon to select the USB video device.
• Under Settings, select the video quality of 720p, 4:3, 30 fps.
• Position the camera so it has a good view of the shampoo bottle and the scale.

  
 

• Remove the magnet and wait for the ball to drop.  Make sure it does not contact the wall of the bottle.
• When the ball has fallen ~2 cm start recording.
• Stop recording when the ball reaches the bottom of the bottle.  Your video clip is automatically saved in the Picture/Camera Roll directory.
• Close the Camera App and open the this webpage.
• Open your recorded video.  Step through it frame by frame.  The frame number and the time will be displayed.  When the sphere is in front of the scale at 12 cm, 11 cm, ..., 2 cm, record the frame number and time in an Excel spreadsheet.

  position (cm)	frame number	time (s)
  12
  11
  10
  9
  8
  7
  6
  5
  4
  3
  2

• Plot position (vertical axis) versus time (horizontal axis).  Verify that the plot resembles a straight line.  This verifies that the sphere moves with constant, terminal velocity.  Find the speed of the sphere by adding a linear trendline.  The magnitude of the slope of this trendline is the speed (positive number) of the sphere.

Experiment 2 Deliverables: (to be included in the your journal)

• Visuals:  Your data table and your plot of position versus time (with trendline).
• Analysis: 
  • When the ball is falling at a constant speed (terminal velocity), what must be true about the sum of the forces?  Draw a free-body diagram before looking at the provided one.

  The forces acting on the sphere are gravity, the buoyant force, and the viscous drag force given by Stokes' law.  A free body diagram is shown below.

  
  

  Since the sphere moves with constant velocity, the net force is zero. 

  F_drag + F_buoyant = mg

  6πηr_spherev + ρ_fluidV_sphereg = ρ_sphereV_sphereg

  6πηr_spherev + (4/3)πρ_fluid(r_sphere)³g = (4/3)πρ_sphere(r_sphere)³g

  η = 2(ρ_sphere - ρ_fluid)(r_sphere)²g/(9 v)

  The density of the "Volumizing Shampoo" is very close to that of water, ρ_fluid = 1.03 g/cm³.
  The density of the stainless steel ball is 7.866 g/cm³, and its diameter is 1/4 inch = 0.635 cm.

  • Calculate the viscosity η of the shampoo using your measured velocity in units of poise = g/(cm-s). 
    Use the densities in units of g/cm³, the speed in units of cm/s, the radius of the sphere in units of cm and g = 981 cm/s².
    (1 Pa-s = 1 kg/(m s) = 10 g/(cm-s) = 10 poise)
  • Calculate the Reynolds number  R = 2ρ_fluidr_spherev/η.  It is a dimensionless number.
    Check that the Reynolds number is less than 1, so that we are in the regime where Stokes' law is valid.
  • The table below lists typical viscosities of some viscous fluids at room temperature.  Does your value for the viscosity of the shampoo seem reasonable?  Why or why not?

  fluid	viscosity (Pa-s)
  honey	2 - 10
  molasses	5 - 10
  ketchup	50 - 100
  chocolate syrup	10 - 25

  • Based on your measured viscosity, if you tried to pour this shampoo through a funnel, would it flow faster or slower than chocolate syrup?  Why?

Convert your log into a lab report.

Name:
E-mail address:

Laboratory 8 Report

• Make sure you completed the entire lab and answered all parts.  Make sure you show your work and inserted and properly labeled relevant tables and plots in your journal.
• Add a summary reflection at the end of your report in a short essay format.  Was there a moment in this lab where your initial intuition about fluid behavior was wrong?  If yes, what evidence from the simulation changed your mind?"

Save your Word document (your name_lab8.docx), go to Canvas, Assignments, Lab 8, and submit your document.