1. Chapter 8: Fluid Mechanics

2. Learning Goal
To define a fluid.
To distinguish a gas from a liquid

3. States of Matter Solids – definite volume, definite shape
Liquids – definite volume, indefinite shape
Gases – indefinite volume, indefinite shape
(Also plasma and Bose-Einstein condensates but we don’t need to worry about those.)

4. What state of matter is glass?
Solid
Liquid
Gas

5. What state of matter is honey?
Solid
Liquid
Gas

6. The Nature of Fluids Fluids:
Liquids and Gases comprise the category of what we call fluids.
Fluids exhibit certain characteristics that solids do not – they flow when subjected to shear stress
Although in general conversation the term fluid is often used interchangeably with the term liquid, from a mechanical perspective, a fluid is any substance that tends to flow or continuously deform when acted on by a shear force.
Fluid: substance that flows when subjected to a shear stress.

7. Properties of static fluids

8. Learning Goal To use density to describe a fluid.
To apply buoyant force to explain why some objects float or sink in a fluid.

9. Static Fluid Properties
Density () = mass / volume
Viscosity = internal resistance to flow
Note: Atmospheric pressure and temperature influence a fluid’s density and viscosity
Other factors that influence the magnitude of the forces a fluid generates are the fluid’s density, specific weight, and viscosity.
Density
Specific weight
The denser and heavier the fluid medium surrounding a body, the greater the magnitude of the forces the fluid exerts on the body.
The property of fluid viscosity involves the internal resistance to a fluid to flow
Increased fluid viscosity results in increased forces exerted on bodies exposed to the fluid.

10. Density The density of an object is represented by:
Density = mass / volume
While this formula is familiar to us, we will use it in subsequent derivations.

11. Specific Gravity
In order to have a constant comparison, we use specific gravity instead of density sometimes.
Since water has a density of 1 g/mL or 1 x 103 kg/m3, we eliminate the units and call the number specific gravity.
Ex. For iron which has a density of 7.86 g/mL, the specific gravity is 7.86 (or 7.86 as dense as water).

12. Which is more dense, a pound of feathers or a pound of bricks?
They are the same

13. Common Density Misconceptions
Let’s expel some common misconceptions about density.
Refer to your worksheet for the following Turning Point questions about whether the object will float or sink.

14. A. (Refer to worksheet)
Sink
Float

15. B. (Refer to worksheet)
Sink
Float

16. C. (Refer to worksheet)
Sink
Float

17. D. (Refer to worksheet)
Sink
Float

18. E. (Refer to worksheet)
Sink
Float

19. F. (Refer to worksheet)
Sink
Float

20. G. (Refer to worksheet)
Sink
Float

21. H. (Refer to worksheet)
Sink
Float

22. I. (Refer to worksheet)
Sink
Float

23. J. (Refer to worksheet)
Sink
Float

24. Buoyancy
The upward force present when an object floats in a fluid, or feels lighter, is the buoyant force on the object.
The weight of an object immersed in a fluid is the apparent weight of the object (versus the actual weight).
Apparent weight = FG - FB (when sinking)

25. Floating Objects Only if floating
If, and only if, an object is floating on the surface:
The buoyant force exerted by the fluid that is displaced is equal in magnitude to the weight of the floating object
This is because when an object is floating, it is not moving up or down
therefore the net force is zero and the buoyant force must equal the weight
Float something, and talk about the equilibrium state that floating is. Things will bobble up and down until they are displacing just the right amount of water to balance the weight of the object.
Only if floating

26. Archimedes’ Principle
Any object completely or partially submerged in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the object
Buoyant force of displaced air
Weight of the hot air balloon
The hot air balloon rises because of the large volume of air that it displaces

27. Apparent Weight
The apparent weight of an object is the net weight between the force of gravity and the buoyant force.
Apparent Weight= Fnet = FG – FB

28. The Red line

29. A boat has a mass of 8450kg. What is the minimum volume of water it will need to displace in order to float on the surface of pure water without sinking? This is something you will have to think about with your cardboard boats!
Volume Displaced

30. If an object is sinking to the bottom of a glass of water, the buoyant force must be?
Equal to the Net Force
Less than Fg
More than Fg
Equal to Fg

31. What must be true for the buoyant force to be greater than gravitational force?
Object is floating continuously upward
Object is floating at the top of the fluid
Object is sinking

32. If a rock is completely submerged in a fluid, what must be true?
The volume of the displaced fluid = the volume of the rock
The weight of the rock = weight of the fluid that was displaced.
Both 1 and 2
None of the above

33. The apparent weight of an object in a fluid, FB – Fg , could also be called what?
Net Force
Tensional Force
Buoyant Force
Actual Weight

34. If a raft is floating and is partially submerged in a fluid, what must be true?
The volume of the displaced fluid = the volume of the raft
The weight of the raft = weight of the fluid that was displaced.
Both 1 and 2
None of the above

35. Archimedes Principle example
A bargain hunter purchases a “gold” crown at a garage sale. After she gets home, she hangs the crown from a scale and finds its weight to be 7.84 N. She then weighs the crown while it is immersed in water, and the scale reads 6.86N. Is the crown made of pure gold?

36. Pressure in Fluids
In solids, pressure is defined as the amount of force per unit area.
P = F/A
Pressure occurs within fluids due to the constant motion of their molecules but it is more difficult to determine the area.

37. Common Pressure Units For example, standard atmospheric pressure is:
14.7 psi (pounds per square inch)
1.01 x 105 Pa (Pascal) = N/m2
760 mmHg (millimeters mercury)
1 atm (atmosphere)

38. Pressure as a function of depth

39. Which hole will have the water shoot out the furthest?
Top hole
Middle Hole
Bottom Hole
All will be equal

40. Absolute and Gauge Pressure
Absolute pressure = Atmospheric + Gauge
Pressure Pressure
Atmospheric pressure is the pressure due to the gases in the atmosphere (always present)
Gauge pressure is the pressure due to a fluid (not counting atmospheric pressure)
Absolute pressure is the total pressure

41. Ex. 3
Calculate the absolute pressure at an ocean depth of 1,000m. Assume that the density of water is 1,025 kg/m3 and that
Po= 1.01 x 105Pa.
What is the gauge pressure as well?

42. Pascal’s Principle

43. Pascal’s Principle
Because force is directly proportional to area, one can vary the cross-sectional area to provide more force.
Eg. Hydraulic brakes, car jacks, clogging of arteries

44. Smaller radius Larger radius Doesn’t matter
In order to use a lesser force to accomplish a difficult task, you should apply the force on the hydraulic cylinder with
Smaller radius
Larger radius
Doesn’t matter

45. Ex. 2
A car weighing N sits on a hydraulic press piston with an area of 0.90 m2. Compressed air exerts a force on a second piston, which has an area of 0.20m2. How large must this force be to support the car?

46. Laminar versus Turbulent Flow
Laminar flow:
Low velocity relative to fluid medium
Streamline path
Turbulent flow:
High velocity relative to fluid medium
Irregular Flow (Eddy currents)
When an object such as a human hand or a canoe paddle moves through the water, there is little apparent disturbance of the immediately surrounding water if the relative velocity of the object with respect to the water is low.
However, if the relative velocity of motion through the water is sufficiently high, waves and eddies appear
When an object moves with sufficiently low velocity relative to an fluid medium, the flow of the adjacent fluid is termed laminar flow.r
Laminar flow: flow characterized by smooth, parallel layers of fluid
When an object moves with sufficiently high velocity relative to a surrounding fluid, the layers of fluid near the surface of the object mix, and the flow is termed turbulent.
Turbulent flow:flow characterized by mixing of adjacent fluid layers

48. 15-6

50. Ideal Fluids Laminar flow Nonviscous Incompressible
Constant density and pressure
All these characteristics must be true for these equations to hold true. (Hence, the name for the ideal gas laws.)

51. Fluids in Motion Steady, Laminar Flow (Ideal Fluid):
-Every fluid particle passing trough the same point in the stream has the same velocity.

52. Flow Rate
Flow rate stays constant (at constant pressure in a closed system)
Flow Rate = Av = V/t
A1v1 = A2v2
A = cross-sectional area (m2)
v = speed (m/s)
V = volume (m3/s)
t = time (s)

53. How are cross-sectional area and velocity of fluids proportional?
Inversely
Directly
No relationship

54. Continuity Equation
Based on Law of Conservation of Mass – what comes in has gotta come out

55. What will happen to the yellow foam ball?
It will stay in the funnel
It will shoot out
It will explode into yellow chunks

56. What will happen to the pop cans when air is blown between them?
They will come together and collide.
They will move apart from each other
It will remain motionless.
Pop will fly out from the openings.

57. How are pressure and velocity of fluids proportional?
Inversely
Directly
No relationship

58. Bernoulli’s Equation P1 + ρgh1 + ½ ρv12 = P2 + ρgh2 + ½ ρv22
Helpful notes:
P = Patm if either side is open.
Set bottom height (h2 ) = 0
If there is a large volume up top, (v1 ) = 0

59. Bernoulli’s Equation P + ρgh + ½ ρv2 = constant
Results from conservation of energy.
P = Pressure energy resulting from internal forces within the fluid
ρgh = similar to gravitational potential energy
½ ρv2 = similar to kinetic energy

60. Bernoulli’s Principle
Bernoulli’s Principle states that the flow speed (Av) in a constriction must be greater than the flow speed before or after it.
Also, swiftly moving fluids exert less pressure than do slowly moving fluids.
Eg. Tornadoes and blown off roofs

61. Bernoulli’s principle
Pressure in a fluid varies inversely with the velocity