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Fluid Mechanics

Fluid Mechanics. What’s a fluid?. Is it a state of matter? Well, what are the states of matter?. A riddle. Which weighs more? A pound of feathers or a pound of bricks?. The riddle. Why do people answer incorrectly? Same mass, different volume Let’s make a new quantity to differentiate:

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Fluid Mechanics

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  1. Fluid Mechanics

  2. What’s a fluid? • Is it a state of matter? • Well, what are the states of matter?

  3. A riddle • Which weighs more? A pound of feathers or a pound of bricks?

  4. The riddle • Why do people answer incorrectly? • Same mass, different volume • Let’s make a new quantity to differentiate: • Anybody know it? • It’s a great name for an auto-repair garage • Density!

  5. Density • density = mass/volume •  = m/V • Units: • kg/m3 • Vector or scalar? • scalar

  6. Density • Is density always constant for a material? • no. V is variable • What about by states of matter? • solids: • little change possible • liquids: • little change possible • gases • huge changes • 1000x less dense than solids & liquids (average)

  7. Difficult vocabulary time • Specific gravity (of a substance) • ratio of the density of a substance to the density of water at 4ºC (=1000kg/m3) • Ex: Substance q has a specific gravity of 6.3. What’s its density?

  8. Example • Substance q has a specific gravity of 6.3. What’s its density? 6.3 = q/H20 6.3H20 = q q = 6.3(1000kg/m3) = 6.3E3 kg/m3 • If I have 63cm3 of q, how much mass do I have? .4kg

  9. Practice • An oil has a density of 550 kg/m3 • Find its specific gravity. • Find the volume of oil with a mass of 200kg .55 .36 m3

  10. Another question • I walk across the snow and sink to my knees. I put on snowshoes and don’t. Why? • FBD & TFE? • Ftot = Fn – Fg • Clearly, there is a difference that a TFE cannot explain.

  11. How force is applied makes a difference • We already know this! • torque: location of force could cause rotation • impulse: more time applied, more Δ • spreading out force over time • Now, let’s spread out force over space • Pressure = force/area • P = F/A

  12. Pressure • P = F/A • Units: • N/m2 = kgms-2/m2 = kg/ms2 • Pascal, Pa • English units • lb/in2 (psi) or atm (atmospheric pressure) • Atmospheric pressure at sea level = 1atm = 14.7lb/in2 = 1.01E5 Pa

  13. Practice • A water bed with massless fibers is a square 2m on each side and 30cm deep. How much does it weigh? • How much pressure does it exert on the floor? 12000N 3000 Pa

  14. Warmup • Which is deadlier: getting punched or getting poked? Assume equal forces behind both strikes. • What are some problems that could occur?

  15. Hydrostatics: fluids at rest • Fluid is at rest (with respect to me) • All portions are “at rest” – static equlibrium • All points at same depth are at the same pressure • What would happen if this isn’t the case? • What does pressure do to fluid?

  16. Time to draw! • container of water at equilibrium • add a small mass • fill mass with fluid & make walls massless • ta-da! • And now, FBD time in the Rockies.

  17. Analogy – what’s putting pressure on her body?

  18. Practice • You dive into your pool to pick up a quarter lying on the floor, 2m below the surface. How much pressure is exerted on that quarter? • P = P0 + ρgh • P = 1.01*105Pa + (1000kgm-3)(10ms-2)(2m) • P = 1.21*105 Pa • How much pressure is exerted by the water? 2*104 Pa – gauge pressure

  19. Practice • Salt water floods an oil tank to 5m. On top is oil 8m deep. Find pressure at the bottom. ρoil = .7g/cm3ρH20 = 1025kg/m3 8m Pboundary = 1.57*105 Pa 5m Pbottom = 2.07*105 Pa

  20. Pascal’s Principle • If pressure in fluid only depends on depth & surface pressure, what happens if surface pressure increases? • Pascal’s principle: A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the containing walls

  21. Let’s visit Jiffy Lube… They put Dent City out of business • How do they lift cars? • What a nice drawing of a lift, Mr. Galster! • hydraulics • what quantity stays the same?

  22. Practice • A car lift at a service station uses compressed air to exert force on a small piston with circular cross section r1=5cm. This pressure is transmitted through an incompressible fluid to another circular piston r2 = 15cm. • What force must be exerted on the small piston to lift a car weighing 13300 N? • What air pressure is needed to produce a force of that magnitude? • Show the work done by each piston is the same. 1480N 1.88*105 Pa

  23. Practice • A hydraulic jack applies a force for 1000N onto a circular plate of r=3cm. The force is transmitted through an incompressible fluid to a circular piston. The jack lifts a go-kart of m=900kg. Find the diameter of the piston. d = .18m

  24. Measuring Pressure: Tools • evacuated cylinder • phet example • barometer • manometer • sphygmomanometer

  25. Floating and sinking • You through a steel beam into the ocean. Does it float or sink? • So how can an ocean liner float when its made of tons of steel beams? • Why is it easier to lift a brick in a pool than on land?

  26. Floating and sinking • Recall our FBD of object in water • Different depths means different pressures. • Is there more pressure at the top or bottom? • bottom – by the amount of the displaced fluid • Buoyant force

  27. Floating and sinking • In equilibrium: Fbuoy = Mg • Replace the fluid with an object and this equation still holds • just mass acted on is changed • If Mball > Mfluiddisplaced, sink • If Mball < Mfluiddisplaced, float

  28. Archimedes Principle • A submerged or partially submerged object in a fluid is buoyed upwards by a force w/ magnitude equal to the weight of the fluid displaced by the object • Force is due to the surrounding fluid, not the object

  29. Buoyancy • FB = Mg • M = ρfluidVdisplaced • FB = ρfluidVdisplacedg

  30. Practice • A wooden raft ρ=600 kg/m3 has surface area 5.7m2 and volume .6m3. When it’s placed in fresh water, find the depth of the raft that is submerged.

  31. Practice • A crown of “gold” weighs 7.84N in air. However, when it’s immersed in water, it weighs 6.86N. Is the crown made of pure gold? ρAu = 19.3 kg/m3

  32. Practice • A 1000kg cube of Aluminum is placed in a tank. Water is added until the block is half-submerged. What’s the normal force? ρAl = 2.7*103kg/m3 • Now, Hg is poured in until Fn = 0. How deep is the Hg? ρHg = 13.6*103kg/m3 8148N .116m

  33. On to fluid flow! • Why does a shower curtain “ride in?” • What do airplanes and curveballs have in common? • How does topspin make you a good tennis player?

  34. Representing motion • Lines = path taken by fluid @ a point • never cross • Density of lines proportional to velocity

  35. Ideal fluids • Nonviscous: no internal resistance • Incompressible: constant density • steady: velocity, density, pressure constant over time at a given location • no turbulence: a disc will translate, not rotate • Laminar (streamline) flow • no turbulence: streams jumble and entangle

  36. Pipe filled with ideal fluid • In some time t, the same volume of water flows through areas 1 &2. • B/C of larger radius, x1 <x2 • V = Ax so Ax1 = Ax2 • Flow rate: Ax/t = Av • Av1 = Av2 • Eq of continuity http://www.4physics.com/phy_demo/fluid-flow.gif

  37. Practice • You’re washing your car with a garden hose when your friend walks by. You try to squirt them but they cleverly back up so the water falls just short. What do you do and why?

  38. Practice • A hose is 2.50cm in diameter. You notice it takes exactly 1.0 min to fill a 30L bucket. (1L = 1000cm3) Then, you attach a nozzle with A=.5cm2 and shoot horizontally, holding it 1.0m above ground. How far can you shoot? 4.47m

  39. Practice • Each second, 5525m3 of water flows over the 670m wide Niagra falls. If the water is 2m deep, how fast is it moving? • A proposed hydroelectric dam will collect all the fallen water into 3 identical pipes. If the max speed the pipes can handle is 30m/s, find the smallest possible pipe radius. 4m/s 2.55m

  40. Practice • A pipes radius doubles. By what factor will the velocity of the fluid in the pipe change? • Going from one part of a pipe to another, the velocity of fluid doubled. What can you say about the radius of the pipe? 4x slower √2x smaller in the fast part

  41. Back to the pipe… • We see from continuity that large area = slow speed • But how are the pressures affected? • And what happens if we change heights?

  42. Bernoulli’s prinicple • Daniel Bernoulli, 1738 • the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume is constant for all points in a streamline • essentially, conservation of energy for fluids

  43. Practice • A barrel full of water is open to the atmosphere. All of the sudden, .5m below the water line, a tiny leak forms. With what velocity does water squirt out? .5m v=3.2m/s

  44. Bernoulli’s principle • Corollary: high velocity = low pressure; low velocity = high pressure • Think about it: • where is velocity greatest? • why? • heights are equal • so P1 + (1/2)v12 = P2 + (1/2)v22 • One must be big, the other, small

  45. Bernoulli’s principle applications • Venturi tube – explain it!

  46. Bernoulli’s principle applications • Airplanes! • Surely you can draw this one.

  47. Airplane practice • A plane has wings with a total area of 4m2. The air velocity over the top of the wing is 245m/s while underneath it is 222m/s. Assuming lift is straight up, find the maximum plane mass that can be supported. air = 1.29kg/m3 • hint: find lifting force and use a TFE

  48. More applications • Sports • curveballs • topspin • ballooning • bend it like Beckham • Health • arteriosclerosis & vascular flutter • aneurysms (oh no!)

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