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/ insects/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 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)

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 Surf Tens. Time 0 bowl Position 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) Surf Tens. Time 0 Surf Tens. Time 1 Surf Tens. Time 2 Position across Bowl 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 or change in y run change in x Slope: change in surface tension s2-s1 change in position x2-x1 Agnes Pockels can help you find s2 and s1!

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