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:

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.


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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.

image       image

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

Table 1

can diameter  (cm) 9.8
hose diameter (cm) 0.8
tape length (cm) 5
∆y (cm)  
h (cm)  
∆t (s)  
∆x (cm)  

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.

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

Data Analysis:

Fill in table 2.

Table 2

v1 = 5/∆t (cm/s)  
v2 = ∆x/(2∆y/980)1/2 (cm/s)  
A1v1 = (πd12/4)v1  (cm3/s)  
A2v2 = (πd22/4)v2  (cm3/s)  
R = (v22 - v12)/(2*980*h)  

Results:


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)


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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?

Record your explanations in your log.


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.

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

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.


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

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.

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

Record your results and conclusions in your log.


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.

Run Temperature (deg C) time (s)
1    
2    
3    
4    
5    

Record your results and conclusions in your log.


Convert your log into a session report, certify with you signature that you have actively participated, and hand it to your instructor.