Applied Mathematics: Derivatives and Fluid Dynamics
Soap and Slope Mathematical adventures in fluid dynamics
Supplementary Materials for Teachers and Workshop Presenters
Dr. Rachel Levy
Mathematics Department
Harvey Mudd College
Creative Commons License CC BY-NC-SA 3.0 by Rachel Levy
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Challenge questions:What is a surfactant?What is the difference betweenBuoyancySurface tensionWhat is soaps job?How are SOAP and SLOPE related?
What is Soap?
http://commons.wikimedia.org/wiki/File:Surfactant.jpg
Soap is a surfactant
The head is hydrophilic (loves water) and the tail is hydrophobic (hates water).
Fancy term for soap: surfactant
Surface-active-agents lower surface tension
With soap the surface tension of water is lower because the bonds between soap and water on the surface are weaker than the bonds on the surface of water with no soap.
Surfactants are used in detergents to surround grease in balls called micelles. The hydrophobic tails attach to the bits of dirt and grease. The hydrophilic heads stay on the outer surface of the micelle and help it move into the water and off clothes and dishes.
Surfactants can attack dirt
Carlota Oliveira Rangel-Yagui1, Adalberto Pessoa Junior, Leoberto Costa Tavares,
J Pharm Pharmaceut Sci (www.cspscanada.org) 8(2):147-163, 2005
Your First Breath
http://hyperphysics.phy-astr.gsu.edu/Hbase/ptens2.html#alv
Air inflates the alveoli of lungs
Process is like blowing up balloons
Natural surfactants make
it easier to breathe by lowering surface tension in the alveoli
Your lungs need surfactant!
http://www.valuemd.com/usmle-step-1-forum/21404-alveoli-surfactant.html
Surfactants lower surface tension.(Answer to challenge question:Soaps job is to lower surface tension.)What is surface tension?
Wikipedia:WassermolekleInTrpfchen.svg
Surface tension is an attractive force between molecules on the surface of a fluid.
Surfactants lower surface tension by weakening the attraction between surface molecules
http://www.everythingabout.net/articles/biology/animals/arthropods/insects/bugs/water_strider/
Water Strider
A water strider is a bug that uses surface tension to walk on water.
Agnes Pockels (1862 1935) was one of the first people to carefully study surface tension.
Lord Rayleigh to Nature magazine (1891):
I shall be obliged if you can find space for the accompanying translation of an interesting letter which I have received from a German lady, who with very homely appliances has arrived at valuable results respecting the behaviour of contaminated water surfaces.
http://cwp.library.ucla.edu/Phase2 Pockels,[email protected]
Agnes Pockels: Nature Magazine
I will describe a simple method, which I have employed for several years, for increasing or diminishing the surface of a liquid in any proportion, by which its purity may be altered at pleasure.
A rectangular tin trough, 70 cm. long, 5 cm. wide, 2 cm. high, is filled with water to the brim, and a strip of tin about 1 1/2 cm. laid across it perpendicular to its length, so that the underside of the strip is in contact with the surface of the water, and divides it into two halves.
By shifting this partition to the right or the left, the surface on either side can be lengthened or shortened in any proportion, and the amount of the displacement may be read off on a scale held along the front of the trough.
What is Slope?
Definitions of slope
Slope = rise
run
Slope:change in y change in x
Slope: y2-y1 x2-x1
What is the slope of this line?
What is the slope of this line?
What is the slope of this line?
What is the slope of this line?
What is the slope of this curve?
What is the slope of this curve?
Consider tangent lines along the
curve -- at each point you can
measure a slope using the slope
of the tangent line.
Where is the slope of this curve positive? negative? zero?
Color gradient graph
Let x = position (left to right)
y= intensity (darkness) of the blue
y
x
Color gradient graph
Let x = position
y= intensity of the blue
y
x
Definition of slope
Slope: change in one quantity change in another quantity
blue intensity
position
Definition of slope
Slope: change in intensity of blue change in position
blue intensity
position
Experiments
Divide into teams of three.
Give each team member a number: 1, 2, 3.
Possible roles: experimenter, recorder, reporter
Paperclip Experiment Supplies:Clean hands (no soap, lotion)!One paper plateCup of water One large paperclip and one small paperclipPiece of paper (optional)Soap
Paperclip ExperimentFloat a paperclip (or two) on the surface of the water. If this is tough, float the paperclip on a scrap of paper, then sink the paper, allowing the paperclip to remain on the surface.Put a drop of detergent near it. What happens? Why?
What does it mean for something to float? Sink?
Hint: there are two possible answers
What does it mean for an object to float?
Float could refer to buoyancy
Float could refer to surface tension
Buoyancy
Objects less dense than the water will rise to the surface.
But metal ships (more dense than water) float! Why?
When do metal ships sink?
Sinking
Gravity pulls the mass of the boat down.
The mass of the boat is black.
Buoyancy
Buoyancy pushes the boat up.
Archimedes (~250BC): Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.
Displaced water is green.
Buoyancy
If the boat springs a leak and takes on water,
how much water can it hold before it sinks?
Why do you feel light when you are floating in the water?
Can you explain why it is easier to float
in salt water than fresh water?
Hints: weight = mass* gravitational constant
mass = density *volume
In your experiment, did the paper clip float because of (a) buoyancy or (b) surface tension?
Surface Tension!
Boat Experiment Supplies:Clean hands (no soap, lotion)!One paper plateCup of water Paper boatSoap
Boat ExperimentFloat a paper boat on one side of the bowl. Put a drop of detergent behind it(between the boat and the edge of the bowl). What happens? Why?
Plotting Surface Tension
What is happening to the surface tension of the water in the boat experiment?
Plotting Surface Tension
Graph x = position y = surface tension
(you can also draw your boat!)
Plotting Surface Tension
Graph x = position y = surface tension
(you can also draw your boat!)
Time 0: Before you put the soap in the water
Plotting Surface Tension
Graph x = position y = surface tension
(you can also draw your boat!)
Time 0: Before you put the soap in the water
Time 1: The second after you put the soap in (before the boat has moved much)
Plotting Surface Tension
Graph x = position y = surface tension
(you can also draw your boat!)
Time 0: Before you put the soap in the water
Time 1: The second after you put the soap in (before the boat has moved much)
Time 2: After the boat has stopped moving
Graph x = position y = surface tension and pic of boat
Time 0: Before you put the soap in the water
Time 1: The second after you put the soap in (before the boat has moved much)
Time 2: After the boat has stopped moving
Time 0
Surf
Tens.
Position
bowl
View of bowl from top
Graph surface tension along this line
Graph x = position y = surface tension
Time 0: Before you put the soap in the water
Time 1: The second after you put the soap in (before the boat has moved much)
Time 2: After the boat has stopped moving(reminder: soap lowers surface tension)
Time 0
Surf
Tens.
Position
across
Bowl
Time 1
Surf
Tens.
Position
across
Bowl
Time 2
Surf
Tens.
Position
across
Bowl
Time 0
Time 0 before soap is added: zero slope no motion.
Time 1
No slope (zero slope) no more motion.
Time 2
Time 0 before soap: zero slope no motion.
How does the sign of the slope relate to the direction of the boat motion?
Time 1 after soap: positive slope motion to right.
How does the sign of the slope relate to the direction of the boat motion?
Time 2 after soap: zero slope no motion
How does the sign of the slope relate to the direction of the boat motion?
Surface tension can be high (time 0) or low (time 2), but if there is no change, the surface tension does not cause the fluid on the surface (and the boat) to move.
When there is a change in surface tension (time 1) across the bowl, there is surface motion.
Pepper ExperimentSupplies:Clean hands (no soap, lotion)!One paper platePepper or Tea leavesSoap
Pepper ExperimentPut some water on a plateSprinkle pepper on the waterPut a drop of soap in the middleGraph your results at time 0: before you added the soap time 1: right after you added the time 2: longer after you added the soap
Time 0
Time 2
Time 1
The big idea:
To get the motion you saw in the experiments, there had to be areas with different surface tension.
Change in amount of soap change in surface tension
motion!
Slope: riseorchange in y run change in x
Slope: change in surface tensions2-s1 change in positionx2-x1
Agnes Pockels can help you find s2 and s1!
Challenge questions revisited:What is a surfactant?What is the difference betweenBuoyancySurface tensionWhat is soaps job?How are SOAP and SLOPE related?
Dr. Levys research
Thin liquid films and surfactants
Surfactant moves the fluid
Fluid moves the surfactant
Changes in space and time
Two partial differential equationsone equation for height of the filmone equation for surfactant concentration
Levys research with Harvey Mudd College undergraduate math majors:
Students solve these partial differential equations modeling a thin liquid film
and surfactant using computer programs
How does the film height and surfactant concentration evolve
in space and time?
How do solutions of this model compare to experiments?
The upside down triangle is a fancy sign for slope!
The other symbol in yellow is a function that describes
how surfactant affects surface tension.
Height equation
Surfactant concentration equation
Collaborator Dr. Karen Daniels, Physics, North Carolina State University
In the left picture, you are looking down at an experiment. The red line is from a laser that indicates the glycerine film height. The yellow areas show the variation in the concentration of surfactant on the thin film of glycerine (slope!). The right picture is an enlarged horizontal slice of the image on the left. We compare this experimental data to mathematical solutions of the two equations.
Thank you very much!
Dr. Rachel Levy [email protected]
Lecture 3_notes Page 1
Lecture 3_notes Page 1
Lecture 3_notes Page 1
Lecture 3_notes Page 1
Lecture 3_notes Page 1
surface tensionof the water
position across diameter of bowl
clean water
soapy water
BOWL
slope = 0,no boat motion
surface tensionof the water
position across diameter of bowl
clean water
soapy water
BOWL
SOAP
slope > 0moves boat
to right
surface tensionof the water
position across diameter of bowl
clean water
soapy water
BOWL
slope = 0,no boat motion
Lecture 3_notes Page 1
