Upload
vantruc
View
218
Download
2
Embed Size (px)
Citation preview
A couple of house rules
Be on time
Switch off mobile phones
Put away laptops
Being present = Participating actively
Applied Natural SciencesLeo Pel
e‐mail: [email protected]
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
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
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
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
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
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
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
• 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
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 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
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
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
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