Physics 106 Lecture 13
Fluid Mechanics
SJ 7th Ed.: Chap 14.1 to 14.5
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What is a fluid?
Pressure
Pressure varies with depth
Pascal’s principle
Methods for measuring pressure
Buoyant forces
Archimedes principle
y
assumptions
p
Fluid dynamics
An ideal fluid
Continuity Equation
Bernoulli’s Equation
What is a fluid?
Force
Solids
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definite volume and shape
Resist shear stress
Shear stress
Force
Fluids: substances that can “flow”, that is, liquid and gas
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no definite shape, cannot resist shear stress
Liquids: definite volume
ƒ often almost incompressible under pressure (from all sides)
ƒ Molecules move while weakly interacting with each other
Gases: no definite volume
ƒ molecules move independently of each other
ƒ comparatively easy to compress: density depends on
temperature and pressure
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Study on fluids
Fluid Statics - fluids at rest (mechanical equilibrium)
Fluid Dynamics – fluid flow (continuity, energy conservation)
Mass and Density
• Density is mass per unit volume at a point:
ρ≡
Δm
ΔV
or
ρ≡
m
V
• scalar
• units are kg/m3, gm/cm3..
• ρwater= 1000 kg/m3= 1.0 gm/cm3
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Volume and density vary with temperature - slightly in liquids
Incompressible liquid: density is constant
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The average molecular spacing in gases is much greater than in
liquids.
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Forces/Stresses in Fluids
• Fluids do not allow shearing stresses or tensile stresses
(pressures)
Tension
Shear
Compression
• The only stress that can be exerted on an object
submerged in a static fluid is one that tends to compress
the object from all sides
• The force exerted by a static fluid on an object is always
perpendicular to the surfaces of the object
Area vector
Imagine an area, either on the surface of or inside a fluid
For this area, define an area vector:
G
A = Anˆ
G
A = Anˆ
Magnitude, A: area
Direction: perpendicular to the area
n̂ : unit vector perpendicular to the area
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Pressure
• The pressure P on a “small” area ΔA is the ratio of the
magnitude of the net force to the area
P=
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ΔF
or P = F / A
ΔA
G
G
ΔF = PΔA = PΔAn̂
n̂
PΔA n
Pressure is a scalar while force is a vector
The direction of the force producing a pressure is perpendicular
to some area of interest
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h
At a point in a fluid (in mechanical
equilibrium) the pressure is the same in
any direction
Pressure units:
ƒ 1 Pascal (Pa) = 1 Newton/m2 (SI)
ƒ 1 PSI (Pound/sq. in) = 6894 Pa.
ƒ 1 milli-bar = 100 Pa.
A sensor to measure absolute pressure
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The spring is calibrated by a known force
Vacuum is inside
The force due to fluid pressure presses on the top of the piston and
compresses the spring until the spring force and F are equal.
The force the fluid exerts on the piston is then measured
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The pressure inside a commercial airliner is maintained at 1
ATM (=10^5 N/m^2). What is the net force exerted on a 1 m x 2
m cabin door by the air inside and outside airliner if the
outside pressure (at 10 km height) is 0.3 ATM?
Pressure versus depth
in a incompressible fluid in static equilibrium
Fluid is in static equilibrium
The net force on the shaded volume = 0
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Incompressible liquid - constant density ρ
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Horizontal surface areas = A
Forces on the shaded region:
– Weight of shaded fluid: Mg
– Downward force on top:
– Upward force on bottom:
∑ Fy = 0 =
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F1 =P1A
F2 = P2A
y=0
y1
F1
P1
h
F2
y2
P2
Mg
P2 A − P1 A − Mg
In terms of density, the mass of the
shaded fluid is:
The extra pressure at
extra depth h is:
M = ρΔV = ρAh
ΔP = P2 − P1 = ρgh
∴ P2 A ≡ P1 A + ρghA
h ≡ y1 − y 2
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Pressure relative to the surface of a liquid
air
Example: The pressure at depth h is:
P0
Ph = P0 + ρgh
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h
P0 is the local atmospheric (or ambient)
pressure
Ph is the absolute pressure at depth h
The difference is called the gauge pressure
Ph
liquid
All points at the same depth are at the
same pressure; otherwise, the fluid could
not be in equilibrium
The pressure at depth h does not depend
on the shape of the container holding the
fluid
Preceding
P
di equations
ti
also
l
hold approximately for
gases such as air if the
density does not vary
much across h
P0
Ph
Atmospheric pressure and units conversions
• P0 is the atmospheric pressure if the liquid is open to the
atmosphere.
• Atmospheric pressure varies locally due to altitude,
temperature, motion of air masses, other factors.
• Sea level atmospheric pressure P0 = 1.00 atm
= 1.01325 x 105 Pa = 101.325 kPa = 1013.25 mb (millibars)
= 29.9213” Hg = 760.00 mmHg ~ 760.00 Torr
= 14.696 psi (pounds per square inch)
Pascal
(Pa)
1 Pa
1 bar
bar (bar)
2
≡ 1 N/m
100,000
−5
10
≡
106 dyn/cm2
atmosphere
(atm)
−6
9 8692×10
9.8692×10
0.98692
torr
(Torr)
pound-force per
square inch (psi)
−3
7 5006×10
7.5006×10
750.06
−6
145 04×10
145.04×10
14.5037744
1 atm
101,325
1.01325
≡ 1 atm
760
14.696
1 torr
133.322
1.3332×10−3
1.3158×10−3
≡ 1 Torr;
≈ 1 mmHg
19.337×10−3
1 psi
6.894×103
68.948×10−3
68.046×10−3
51.715
≡ 1 lbf/in2
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Pressure Measurement: Barometer
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near-vacuum
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Invented by Torricelli (1608-47)
Measures atmospheric pressure P0 as it varies
with the weather
The closed end is nearly a vacuum (P = 0)
One standard atm = 1.013 x 105 Pa.
P0 = ρ Hggh
Mercury (Hg)
How high is the Mercury column?
P
1.013 × 10 5 Pa
h = 0 =
= 0.760 m
ρ Hgg (13.6 × 10 3 kg / m3 )(9.80 m/s 2 )
One 1 atm = 760 mm of Hg
= 29.92 inches of Hg
How high would a water column be?
h =
P0
ρ water g
=
1.013 × 10 5 Pa
(1.0 × 103 kg / m3 )(9.80 m/s 2 )
= 10.34 m
Height limit for a suction pump
iClicker Q
Superman with “infinitely”
powerful lungs tries to drink a
soda in an open cup on the
ground from the top of a 20 m
high building through a straw.
straw
Will he succeed?
A)Yes
B)No
C)Not enough information
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Pressure Measurements: Manometer
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Measures the pressure of the gas in the
closed vessel
One end of the U-shaped tube is open to
the atmosphere at pressure P0
The other end is connected to the pressure
PA to be measured
Points A and B are at the same elevation.
The pressure PA supports a liquid column
of height h
The Pressure of the gas is:
PA = PB = P0 + ρgh
Absolute vs. Gauge Pressure
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The gauge pressure is pressure measured relative to atmospheric pressure,
PA – P0= ρgh
– What you measure in your tires
Gauge pressure can be positive or negative.
Find the pressure in atmospheres at the base of
Dworshak Dam if the water in the reservoir is 200
meters deep. (10^5 N/m^2 = 1 ATM.)
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