## Studio Session 9

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 also measure the fraction of the ordered energy that is lost because of friction.  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 containers, one with overflow spout
• meter stick
• Shampoo bottle with metal sphere and linear scale
• Magnet
• Webcam

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.  This log will form the basis of your lab report.  Address the points highlighted in blue.  Answer all questions.

### 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

can diameter  (cm) 9.8 0.8 5

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.

Data Analysis:

• 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

Results:

• 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.

### 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)

During heavy exercise, the blood's volume flow rate is 5-10 times higher than when the body is at rest.  Discuss different possible ways that a body can accomplish this?

• Is increasing blood pressure 5 - 10 times higher a viable option?  What percentage increase in blood pressure is reasonable?  Explain!
• Is decreasing the length of your blood vessels a viable option?  Explain!
• The arterioles (small arteries) are surrounded by circular muscles.

In order to increase the blood flow rate by a factor of 5, what percentage increase in the radius of a blood vessel is needed?  (This is called vasodilatation.)
• Arteries in the human body can be constricted when plaque builds up on the inside walls.  How does this affect the blood flow rate through this artery?  Is it possible for the body to keep the flow rate constant?  Explain!

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

Three students will work as a group on this experiment.  All groups will compare their results and predictions.

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 Movie Maker.
• Choose "Webcam video".
• 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 capturing.
• Stop capturing when the ball reaches the bottom of the bottle.  Save your movie.
• Review your movie on the timeline.  You can pause it at any frame.  Pause the video when the sphere is in front of the scale at 12 cm, 11 cm, ..., 2 cm, and enter these times into an Excel spreadsheet.
position (cm) 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.

Data Analysis:

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.

• Fdrag + Fbuoyant = mg
• 6πηrspherev + ρfluidVsphereg = ρsphereVsphereg
• 6πηrspherev + (4/3)πρfluid(rsphere)3g = (4/3)πρsphere(rsphere)3g
• η = 2(ρsphere - ρfluid)(rsphere)2g/(9 v)

The density of the "Volumizing Shampoo" is very close to that of water, ρfluid = 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 = 2ρfluidrspherev/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.
• Compare your value for η with the value obtained by the other groups.  Do the values agree?
• The table below lists typical viscosities of some viscous fluids at room temperature.  Does your value for the viscosity of the shampoo seem reasonable?  Discuss.
fluid viscosity (Pa-s)
honey 2 - 10
molasses 5 - 10
ketchup 50 - 100
chocolate syrup 10 - 25
• Predict the terminal velocity of a sphere made of the same material but with diameter of 3/8 inch in the same fluid.  One group should measure it.

Experiment 3

Capillary viscometers make use of Poiseuille's law to measure the relative viscosity of liquids or solutions.  They consist of a fine capillary tube in which a liquid is placed and measurements made of the time for a fixed volume of liquid to flow through the tube.  Poiseuille's law (Q = π∆Pr4/(8ηL)) tells us that the volume flow rate is inverse proportional to the viscosity, so the time it takes a fluid to move through a fixed length of the tube is proportional to viscosity over the density.

t = constant*η/ρ

If the density ρ is constant, the time is directly proportional to the viscosity η.

Each group of three students will perform a virtual experiment, measuring the percentage change in the viscosity of glycerol with temperature.

• Open the virtual experiment.
Depending on the browser you are using you may have to click "Get Flash", "Always allow", and also display a scalable version of this experiment.
• Choose a temperature.
• Perform the experiment and record the temperature and the time it takes for the glycerol to move through the length of the tube between the markers.
Run Temperature (deg C) time (s)
1
2
3
4
5
• Plot the time (viscosity, arbitrary units) versus temperature.
• Does the viscosity change linearly with temperature?
• What is the percentage change in the viscosity, when the temperature changes from 28 oC to 62 oC?  Compare your result with that from other groups.