Soap and Slope Mathematical adventures in fluid dynamics

# Soap and Slope Mathematical adventures in fluid dynamics

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## Soap and Slope Mathematical adventures in fluid dynamics

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1. 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

2. These slides were developed for people to use, modify and share. Please cite this source if you use this material. Thank you!http://creativecommons.org/licenses/by-nc-sa/3.0/

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

4. What is Soap?

5. Soap is a surfactant http://commons.wikimedia.org/wiki/File:Surfactant.jpg The head is hydrophilic (loves water) and the tail is hydrophobic (hates water).

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

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

8. Your First Breath 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 http://hyperphysics.phy-astr.gsu.edu/Hbase/ptens2.html#alv

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

10. Surfactants lower surface tension.(Answer to challenge question:Soap’s job is to lower surface tension.)What is surface tension?

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

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

13. 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/

14. 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

15. 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.

16. What is Slope?

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

18. What is the slope of this line?

19. What is the slope of this line?

20. What is the slope of this line?

21. What is the slope of this line?

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

23. 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.

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

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

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

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

28. Experiments • Divide into teams of three. • Give each team member a number: 1, 2, 3. • Possible roles: experimenter, recorder, reporter

29. Paperclip 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

30. 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?

31. What does it mean for something to float? Sink? Hint: there are two possible answers

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

33. 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?

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

35. 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.

36. 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

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

38. Surface Tension!

39. Boat Experiment Supplies: Clean hands (no soap, lotion)! One paper plate Cup of water Paper boat Soap

40. 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?

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

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

43. 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

44. 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)

45. 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

46. 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

47. 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

48. Time 0 Time 0 before soap is added: zero slope  no motion.