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Applied Natural SciencesLeo Pel

e‐mail: 3nab0@tue.nl

http://tiny.cc/3NAB0

Het basisvak Toegepaste Natuurwetenschappen

http://www.phys.tue.nl/nfcmr/natuur/collegenatuur.html

Copyright © 2012 Pearson Education Inc.

PowerPoint® Lectures forUniversity Physics, Thirteenth Edition

– Hugh D. Young and Roger A. Freedman

Lectures by Wayne Anderson

Chapter 12

Fluid MechanicsNewton’s Laws in Fluid Language!

LEARNING GOALS

• The meaning of the density of a material and the average density of

a body.

• What is meant by the pressure in a fluid, and how it is measured.

• How to calculate the buoyant force that a fluid exerts on a body

immersed in it.

• The significance of laminar versus turbulent fluid flow, and how the

speed of flow in a tube depends on the tube size.

• How to use Bernoulli’s equation to relate pressure and flow speed at

different points in certain types of flow.

4

Introduction

States of Matter

Solid

Has a definite volume and shape

Liquid

Has a definite volume but not a definite shape

Gas – unconfined

Has neither a definite volume nor shape

These definitions are somewhat artificial and both liquids and gases are fluids

5

Plays the role for fluids that mass plays for solid objects

Density

The density of a material is its mass per unit volume:

= m/V (kg/m3 = 10-3 g/cm3)

ρ (lower case Greek rho, NOT p!)

6

Densities of common substances

x 1000 x 10

x 100000000000000

7

Note: ρ = (M/V)

• Mass of body, density ρ, volume V is

M = ρV

• Weight of body, density ρ, volume V is

w=M g= ρVg

Quiz

8

Find the mass and weight of the air at 20°C in a living room with a 4.0x 5.0 M floor and a ceiling 3.0 m high?

V = 4 x 5 x 3 = 60 m3

mair = ρair V = 1.2 x 60 = 72 kg

Wair = mair x 9.81= 700 N

Example

9

The sphere on the right has twice the mass and twice the radius of the sphere on the left.

Compared to the sphere on the left, the larger sphere on the right has

1. twice the density.

2. the same density.

3. 1/2 the density.

4. 1/4 the density.

5. 1/8 the density.

mass mradius R

mass 2mradius

2R

Quiz

V= 4/3 π R3

10

Consider a cross sectional area A oriented horizontally inside

a fluid. The force on it due to fluid above it is F.

FPA

Pressure

Definition: Pressure = Force/Area

F is perpendicular to ASI units: N/m2

1 N/m2 = 1 Pa (Pascal)

11

Consider a solid object submerged in a STATIC

fluid as in the figure.

The pressure P of the fluid at the level to

which the object has been submerged is the

ratio of the force (due to the fluid surrounding it

in all directions) to the areaFPA

If 1. & 2. weren’t true, the fluid would be in motion, violating the statement that it is static!

At a particular point, P has the following properties: 1. It is same in all directions. 2. It is to any surface of the object.

12

Pressure is a scalar and force is a vector.

The direction of the force producing a pressure is perpendicular to the area of interest.

Unit of pressure is pascal (Pa)

1 Pa= 1 N/m2

The atmosphere exerts a pressure on the surface of the Earth and all objects at the surface

Atmospheric pressure is generally taken to be1.00 atm = 1.013 x 105 Pa = Po

13

When the pressure is distributed over many nails, each individual nail exerts too small an amount to pop the balloon. When a less dense population of nails is used, each individual nail exerts more pressure and the balloon pops 14

16

Top: F=(p+dp)A

Bottom: F=pA

Weight: dw = ρgV = ρgA dy

Fy = 0

pA-(p+dp)A- ρg A dy=0

17

Take depth as:

We can rewrite this as

18

19

Pressure deep sea

20

Force on a dam pressure

21

Force on a dam pressure

Calculate pressure:

P= ρgh= ρg(H-y)

Force small part wall:

dF=PdA= ρ g(H-y)w dy

So total force on dam is:

F= ∫dF =∫PdA= ∫ ρ g(H-y)w dy

= 1/2 ρ g wH2

22

1623-1662French mathematician, physicist, inventor, writer and Christian philosopher

Blaise Pascal

Because the pressure in a fluid depends on depth and on the value of P0, any increase in pressure at the surface must be transmitted to every other point in the fluid.

Pascal’s law:Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

Pascal’s law

23

Pascal’s law

24

21 PP2

2

1

1

AF

AF

11

22 F

AAF

Pascal’s law

25

22

11 F

AAF

Since the volumes are equal,A1 x1 = A2 x2

Combining the equations,

This is a consequence of Conservation of Energy

Pascal’s law and work

222221

22

2

1111 xmgWxFx

AA

FAA

xFW

26

For exampleLift cylinder : 25 cm diameterSmall cylinder: 1.25 cm in diameter

Lift ratio: 400

To lift a 6000 newton car,

Only 6000 N/400 = 15 N on the fluid in the small cylinder needed

However to lift the car 10 cm, you would have to move the oil 400 x 10cm = 40 meters

Car lift

27

Car break

28

Bottom of bottle

29

Tennis ball on the eye

30

Pressure Measurements:  BarometerInvented by Torricelli (1608-1647)

A long closed tube is filled with mercury

and inverted in a dish of mercury.

The closed end is nearly a vacuum.

Measures atmospheric pressure as

Po = ρHg g h

One 1 atm = 0.760 m (of Hg)

(Water h 10 m)

Pressure gauge

31

Finding absolute and gauge pressure

Pressure from the fluid and pressure from the air above it are determined separately and may or may not be combined.

p + pgy1= patm + ρgy2

p-patm=ρg(y2-y2)= ρgh

MOST ARE RELATIVE

?

32

Pressure gausses

The spring is calibrated by a known force

The force due to the fluid presses on the top of the piston and compresses the spring

The force the fluid exerts on the piston is then measured

33

There are many clever ways to measure pressure

34

Archimedesc. 287 – 212 BC

Perhaps the greatest scientist of antiquity

Greek mathematician, physicist and engineer

Computed ratio of circle’s circumference to

diameter

Calculated volumes and surface areas of

various shapes

Discovered nature of buoyant force

Inventor: Catapults, levers, screws, etc.

Buoyancy and Archimedes Principle

36

The pressure at the bottom of the cube is greater than 

the pressure at the top of the cube

• The pressure at the top of the cube causes a downward 

force of Ptop A.

• The pressure at the bottom of the cube causes an 

upward force of Pbot A.

B = (Pbot – Ptop) A = (ρfluid g h) A

B = ρfluid  g Vdisp

Vdisp = A h= volume of the fluid displaced.B = M g

Mg is the weight of the fluid displaced.

37

The upward buoyant force is B = fluid g Vobject

The downward gravitational force is Fg = Mg = obj g Vobj

The net force is B ‐ Fg = (fluid – obj) g Vobj

38

Density of ball

Buoyancy and nails 

Measuring the density of a liquid

41

A block of ice (density 920 kg/m3) and a block of iron (density 7800 kg/m3) are both submerged in a fluid. Both blocks have the same volume. Which block experiences the greater buoyant force?

1. The block of ice

2. The block of iron

3. Both experience the same buoyant force.

4. The answer depends on the density of the fluid.

Quiz

42

If I squeeze the bottle what will happen?

2. The diver will sink

1. The diver will stay at same position

3. The diver will go to the top

Quiz

43

44

• Squeezing on the top of the sealed plastic container decreases the volume and therefore increases air pressure above the water.• By Pascal's principle, that pressure is transmitted to all parts of the container. This increases the pressure inside the small glass vial.• The increased pressure decreases the volume of air at the top of the vial, and in so doing, decreases the amount of water displaced by the vial. This decreases the buoyant force on it enough to cause it to sink.

45

Crown of gold?  Archimedes

Mass from weight in air = 7.84N/g

F = B + T2 – Fg = 0

B = Fg – T2=1 N

(Weight in air – apparent “weight” in water) 

Archimedes’s principle says B = watgV

crown =mc/Vc= mcrown in airwatg/B

= 7.84 103 kg/m3

7.84N 6.84N

Gold=11.3 103 kg/m3

46

Archimedes’s Principle, Iceberg Example

What fraction of the iceberg is below water?

The iceberg is only partially submerged;

so Vdisp / Vice  =   ice / seawater applies

89% of the ice is below the water’s surface.

47

A block of ice (density 920 kg/m3) and a block of iron (density 7800 kg/m3) are both submerged in a fluid. Both blocks have the same volume. Which block experiences the greater buoyant force?

2. the block of iron

1. the block of ice

3. both experience the same buoyant force

Antwoord: 3.

Quiz

4. the answer depends on the density of the fluid

48

Water tank: I put in a diet pepsi and a regular pepsi. Which one will float?

2. Diet pepsi

1. Regular pepsi

Quiz

49

Sugar is a lot more dense than nutrisweet. Thus, a regular soda is more dense than a diet soda. When placed in water, the regular soda, being more dense than water, will sink. The diet, being less dense than water, will float. 50

Two weight are hanging on a balance. If this is pumped vacuum, what will happen?

2. Sphere will rise

1. Nothing

Quiz

3. Sphere will sink

51

When in the atmosphere, the globe experiences a buoyancy force upward exerted by air. When the air is removed, the buoyancy force is also removed and it is clear that the globe weighs more than the cylinder. 52

Surface tension

Emperor penguin huddle, Antarctica© Doug Allan/Naturepl.com

http://www.arkive.org/education/ 53

Zero gravity

http://spaceflightsystems.grc.nasa.gov/WaterBalloon/

54

Floating paperclip

55

FxxF

AW

lengthforce

Surface tension

Surface tension (10-3 Nm-1)alcohol 23benzene 29glycerol 62kwik 500milk 45water 73

influence surfactants (soap)(often dynes/cm dyne=10-5 N)

56

mass

Weight: Fw=m 10

Estimate ~ 10 mm of feet in contact water

Surface tension Fs= 2x0.073x0.01

mass,max~ 0.15 gram=150 mgr (~10 mgr)

Water strider

57

42: above 38: feet slightly lower

35: feet lower 33: feet broken through surface,head & body still dry

31: feet & body even lower 30: feet & body under water

Polution

58

Fluid flow : types of flow 

Laminair Turbulent

(Mathermatics difficult)

59

In general there can be transition from one into the otherabove critical speed

60

Fluid flow II

The incompressibility of fluids allows 

calculations to be made even as pipes 

change.

2211 vAvA

61

How large is volume flow through A ?

Mass‐flux

Mass conservation

uitminm qq ,,

skginvAqq Vm /

sminvAqV /3

Continuity equation

222111 vAvA

)( 212211 alsvAvA

62

How many capillaries?

Aorta 1.2 cm 40 cm/s

Capillary 4 10-4cm, Vcap=5 x10-4 m/s

2211 vAvA

22

12 vrNvr capaorta

N=7 109

63

Bernoulli’s equation

dVppdsApdsApdW )( 21222111

)(21 2

122 vvdVdK

)( 12 yydVgdU

• Bernoulli’s equation allows the user to consider all variables that might be changing in an ideal fluid.

)(21)( 2

12221 vvghpp

dKdUdW

64

Rearranging and expressing in terms of density:

This is Bernoulli’s Equation as applied to an ideal fluid and is 

often expressed as

as the speed increases, the pressure decreases.

• When the fluid is at rest, this becomes P1 – P2 = g h which is consistent 

with the pressure variation with depth we found earlier .

• The general behavior of pressure with speed is true even for gases.

65

22221

211 2

121 gyvpgyvp

cgyvp 12

21

Summary liquids  (Newtons laws)

pressure

buoyancy

mass

Bernoulli

zgpzp 0)(

VgF

VgF

netto

opw

1

1122 AvAv

constant

221 vgzp

2v1v

11 , Ap 22 , Ap

0z

z

0p

1

66

Water pressure in a home

Inlet 2 cm 4 105 Pa (4 atm) and 1.5 m/sSecond floor at 5 m,1 cmFlow, pressure?

Continuity

(areas)V2= 6 m/s

2211 vAvA

Bernouilli

P2 =P1 ‐½ v22‐v12 ) – ρg(y2‐y1)

= 3.3 105 Pa (3.3 atm)

67

Top water A2>>A1 V2=0 m/s

P1 + ½ v12 + ρgy1 = P2 + ½ v22 + ρgy2

h=y2-y1 and P1=Po , P2=P

ghPPv o 2)(21

If the tank is open to the atmosphere, then P –Po=0. In this case the speed of the liquid leaving a hole a distance h below the surface is equal to that acquired by an object falling freely through a vertical distance h.

This phenomenon is known as Torricelli’s law.

68

Demo Torricelli

The Venturi meter

222

211 2

121 vpvp

1

21

2

2

12121 A

Avghpp

No height difference of flow

2211 vAvA

Continuity

1

22

2

1

1

AA

ghv

70

Air flowing through a narrow pipe created a lower pressure than air flowing through a wider pipe.

71

An incompressible fluid flows through a pipe of varying radius (shown in cross-section). Compared to the fluid at point P, the fluid at point Q has

2. Greater pressure and the same volume flow rate

1. Greater pressure and greater volume flow rate

3. The same pressure and greater volume flow rate

Antwoord: 4

Quiz

4. Lower pressure and the same volume flow rate

radius 2Rradius R

P Q

5. None of the above

72

Applications of Fluid Dynamics – Airplane WingStreamline flow around a moving airplane wing.

Lift is the upward force on the wing from the air.Drag is the resistance.

The curvature of the wing surfaces causes the pressure above the wing to be lower than that below the wing due to the Bernoulli effect.

The lift depends on the speed of the airplane, the area of the wing, its curvature, and the angle between the wing and the horizontal.

airplane wing

73

74

A ping pong ball can be "floated" on a stream of air. The air rushing around the ball creates a pressure low enough to lift and support the ball. Even when the ball is not exactly over the air source! As long as the low pressure spot is under the center of mass of the ball, it will stay "afloat". 75

When a ping pong ball is placed in a funnel with the air blowing out, the ball won't fall out of the funnel. The rushing air creates an area of low pressure that holds the ball in place.

76

Ships Bernoulli

• Bank effect• Squat effect

77

Magnus Effect

This pressure differential creates a lift force directed from the 

high pressure region to the low pressure region (curve ball).

78

A moving, spinning ball will curve due to the addition or subtraction of pressure effects. The side that is spinning in the direction of the motion will have a higher pressure than the side spinning away, thus the ball will curve to the low pressure side.

79

Summary

80

Summary

81