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Soap and Slope: Derivatives and Fluid Dynamics

Soap and Slope: Derivatives and Fluid Dynamics. Professor Rachel Levy Mathematics Department Harvey Mudd College.

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Soap and Slope: Derivatives and Fluid Dynamics

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  1. Soap and Slope:Derivatives and Fluid Dynamics Professor Rachel Levy Mathematics Department Harvey Mudd College

  2. Goals: bring slope to life and provide a window into current applied mathematics researchParticipants:Faculty wanting to create outreach activitiesClasses/workshops of 15 – 80 studentsUSA Science and Engineering Festival

  3. What is Applied Mathematics?

  4. There are manyareasof Applied Mathematics I study the mathematics of Fluid Mechanics using ideas from… Mathematics Chemistry Biology Physics Engineering Computer Science

  5. Challenge questions:What is a surfactant?What is the difference between Buoyancy Surface tensionWhat is soap’s job?How are SOAP and SLOPE related?

  6. What is Soap?

  7. Soap is a surfactant http://commons.wikimedia.org/wiki/File:Surfactant.jpg

  8. Fancy term for soap: surfactant • Surface-active-agentslower 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.

  9. Surfactants can attack dirt Carlota Oliveira Rangel-Yagui1, Adalberto Pessoa Junior, Leoberto Costa Tavares, J PharmPharmaceutSci (www.cspscanada.org) 8(2):147-163, 2005

  10. 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

  11. Your lungs need surfactant! http://www.valuemd.com/usmle-step-1-forum/21404-alveoli-surfactant.html

  12. Surfactants lower surface tension.(soap’s job is to lower surface tension)What is surface tension?

  13. Surface tension is an attractive force between molecules on the surface of a fluid. Wikipedia:WassermoleküleInTröpfchen.svg

  14. Surfactants lower surface tension by weakening the attraction between surface molecules

  15. 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/

  16. 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

  17. 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.

  18. What is Slope? (Derivative?)

  19. Definitions of slope Slope = rise run Slope: change in y change in x Slope: y2-y1 x2-x1

  20. What is the slope of this line?

  21. What is the slope of this line?

  22. What is the slope of this line?

  23. What is the slope of this line?

  24. What is the “slope”of this curve?

  25. 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.

  26. Where is the slope of this curve… positive? negative? zero?

  27. Color gradient graph Let x = position (left to right) y= intensity (darkness) of the blue y x

  28. Color gradient graph Let x = position y= intensity of the blue y x

  29. Definition of slope Slope: change in one quantity change in another quantity blue intensity position

  30. Definition of slope Slope: change in intensity of blue change in position blue intensity position

  31. Three Experiments • Divide into teams of three. • Give each team member a number: 1, 2, 3. • Each team member will be in charge of one experiment.

  32. 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

  33. Experiment 1Float 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?

  34. What does it mean for something to float? Sink? Hint: there are twopossible answers

  35. What does it mean for an object to “float”? Float could refer to buoyancy… Float could refer to surface tension…

  36. 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?

  37. Sinking Gravity pulls the mass of the boat down. The mass of the boat is black.

  38. 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.

  39. 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

  40. In your experiment, didthe paper clip “float” because of (a) buoyancy or (b) surface tension?

  41. Surface Tension!

  42. Experiment 2 (Team member 2 conducts the experiment) Supplies: Clean hands (no soap, lotion)! One paper plate Cup of water Paper boat Soap

  43. Experiment 2Float 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?

  44. Plotting Surface Tension • What is happening to the surface tension of the water in the boat experiment?

  45. Plotting Surface Tension Graph x = position y = surface tension (you can also draw your boat!)

  46. 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

  47. 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)

  48. 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

  49. 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. View of bowl from top Graph surface tension along this line Position bowl

  50. 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 Time 1 Time 2 Surf Tens. Surf Tens. Surf Tens. Position across Bowl Position across Bowl Position across Bowl

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