Professor Rachel Levy
Mathematics Department
Harvey Mudd College
Goals: bring slope to life and
provide a window into current
applied mathematics research
Participants:
Faculty wanting to create outreach activities
Classes/workshops of 15 – 80 students
USA Science and Engineering Festival
What is Applied Mathematics?
There are many areas of Applied
Mathematics
I study the mathematics of Fluid Mechanics
using ideas from…
Mathematics
Chemistry
Biology
Physics
Engineering
Computer Science
Challenge questions:
What is a surfactant?
What is the difference between
Buoyancy
Surface tension
What is soap’s job?
How are SOAP and SLOPE related?
What is Soap?
Soap is a surfactant
http://commons.wikimedia.org/wiki/File:Surfactant.jpg
Fancy term for soap: surfactant
 Surface-active-agents lower surface tension
 Where there is more soap the surface tension is lower
 Surfactants are used in detergents to surround grease and
enable it to leave a surface and enter rinsing water.
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
Inflates the alveoli of lungs
Like blowing up balloons
Natural surfactants make
it easier to breathe by
lowering surface tension
http://hyperphysics.phy-astr.gsu.edu/Hbase/ptens2.html#alv
Your lungs need surfactant!
http://www.valuemd.com/usmle-step-1-forum/21404-alveoli-surfactant.html
Surfactants lower surface tension.
(soap’s job is to lower surface tension)
What is surface tension?
Surface tension is an attractive force
between molecules on the surface of a
fluid.
Wikipedia:WassermoleküleInTröpfchen.svg
Surfactants lower surface tension
by weakening the attraction
between surface molecules
A water strider is a bug that uses surface
tension to walk on water.
Water Strider
http://www.everythingabout.net/articles/biology/animals/arthropods/i
nsects/bugs/water_strider/
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,_Agnes@871234567.html
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? (Derivative?)
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
Three Experiments
1
Divide into teams of three.
2
Give each team member a number: 1, 2, 3.
3
Each team member will be in charge of one experiment.
Experiment 1
(Team member 1 conducts the experiment)
Supplies:
Clean hands (no soap, lotion)!
One paper plate
Cup of water
One large paperclip and one small paperclip
Piece of paper (optional)
Soap
Experiment 1
Float 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!
Experiment 2
(Team member 2 conducts the experiment)
Supplies:
Clean hands (no soap, lotion)!
One paper plate
Cup of water
Paper boat
Soap
Experiment 2
Float 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.
Time 1
Surf
Tens.
Position
across
Bowl
Time 2
Surf
Tens.
Position
across
Bowl
Position
across
Bowl
Time 0
Time 0 before soap is added: zero slope  no motion.
Time 1
Time 2
No slope (zero slope)  no more motion.
How does the sign of the slope
relate to the direction of the
boat motion?
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
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.
Experiment 3
(Team member 3 conducts the experiment)
Supplies:
Clean hands (no soap, lotion)!
One paper plate
Pepper
Soap
Experiment 3
(Team member 3 conducts the experiment)
Put some water on a plate
Sprinkle pepper on the water
Put a drop of soap in the middle
Graph 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 1
Time 2
The big idea:
To get the motion you saw in the experiments, there had
to be areas with different surface tension.
Slope:
rise
run
or
change in y
change in x
Slope:
change in surface tension
change in position
Agnes Pockels can help you find s2 and s1!
s2-s1
x2-x1
Challenge questions revisited:
What is a surfactant?
What is the difference between
Buoyancy
Surface tension
What is soap’s job?
How are SOAP and SLOPE related?
Extension Activities or
Homework or Quiz Material
Have students act out the experiments in 3D using
their location as position and their height as surface
tension
Count the number of drops of clean water that will stay
on a penny. Ask students to guess how the result will
be different when you put drops of soapy water on a
penny. Try it. Plot distribution of results.
My research:
 Thin liquid films and surfactants
 Surfactant moves the fluid
 Fluid moves the surfactant
 Changes in space and time
 Coupled partial differential equations
height of the film
surfactant concentration
Image from research group of
Prof. Sandra Troian
chemical engineering
Research with
Harvey Mudd College undergraduate
math majors:
Solve these 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?
Height equation
Surfactant concentration equation
The upside down triangle is a fancy sign for slope!
The other symbol in yellow stands for surface tension.
Professor Karen Daniels, NCSU
 HMC mathematics students conducted summer
research experiments in Physics Lab at NCSU
 Analytical and numerical solutions for thesis
Thank you very much!
 Prof. Rachel Levy
levy@hmc.edu