Work by pressure
A1
A2
v1
v2
As an element of fluid moves during a short interval dt, the ends move distances ds1 and ds2.
Work by pressure during its motion:
ρ1
ρ2
ds2
ds1
dV
dV
If the fluid is incompressible, the volume should remain constant:
Kinetic and gravitational potential energy
Change in kinetic energy:
A1
A2
v1
v2
ρ1
ρ2
ds2
ds1
Change in potential energy:y: height of each element relative to some initial level (eg: floor)
Bernoulli’s equation
Putting everything together:
otherd K U dW
2 21 1 1 2 2 2
1 12 2
p v g y p v g y 21constant
2p v g y
NB: Bernoulli’s equation is only valid for incompressible, non-viscous fluids with a steady laminar flow!
Static vs flowing fluid
Cylindrical container full of water.
Pressure at point A (hA below surface):
A atm Ap p gh
Or gauge pressure:
hA
x A
hA
A x
Now we drill a small hole at depth hA.
Point A is now open to the atmosphere!
A atmp p
Container with hole
Assume the radius of the container is R = 15 cm, the radius of the hole is r = 1 cm and hA = 10 cm. How fast does water come out of the hole?
R = 15 cm
hA = 10 cm
yA
yBA x
B x
Bernoulli at points A and B (on the surface):2 2
A A A B B B
1 12 2
p v g y p v g y
A B atm B A Awhere and p p p y y h
Continuity at points A and B:
A A B BA v Av
2 2A B 2 Av v gh (Eqn 1)
2 2A Br v R v (Eqn 2)
2 2A B 2 Av v gh
2 2A Br v R v
2
B A
rv v
R
4
2A1 2 A
rv gh
R
For once, let’s plug in some numbers before the end:
4 41 cm
0.00002015 cm
rR
4
2A1 ~1 2 A
rv gh
R
Therefore,
This is equivalent to taking vB ~ 0 (the container surface moves very slowly because the hole is small ―compared to the container’s base)
2A 2 2 9.8 m/ s 0.10 m 1.4 m/ sAv gh
DEMO: Container with holes
h
●A ●B
flow
Measuring fluid speed: the Venturi meter
A horizontal pipe of radius RA carrying water has a narrow throat of radius
RB. Two small vertical pipes at points A and B show a difference in level of h. What is the speed of water in the pipe?
2 2A A B B A A B B
Continuity:
A v Av R v R v
A B
Statics:
p p gh
Venturi effect:High speed, low pressureLow speed, high pressure
2 equations for vA, vB
2 2B A A B
1
2v v p p
2 2A A B B
2 2A A B B
1 12 2
p v p v
R v R v
4
2AA A B
B
11
2
Rv p p
R
A Band p p gh
2
AB A
B
Rv v
R
A 4
A
B
2
1
ghv
R
R
DEMO: Tube with changing diameter
Partially illegal Bernoulli
Gases are NOT incompressible
Bernoulli’s equation cannot be used
It can be used if the speed of the gas is not too large (compared to the speed of sound in that gas).
But…
i.e., if the changes in density are small along the streamline
Example: Why do planes fly?
High speed, low pressure
Low speed, high pressure
Net force up (“Lift”)
bottom
2 2top topbottomLif t area of wing area of wing
2p p v v
DEMO: Paper sucked by
blower.
DEMO: Beach ball
trapped in air.
ACT: Blowing across a U-tube
A U-tube is partially filled with water. A person blows across the top of one arm. The water in that arm:
A. Rises slightly
B. Drops slightly
C. It depends on how hard is the blowing.
The air pressure is lower where the air is moving fast.
This is how atomizers work!
Aerodynamic grip
Tight space under the car ➝ fast moving air ➝ low pressure
Race cars use the same effect in opposite direction to increase their grip to the road (important to increase maximum static friction to be able to take curves fast)
Lower pressure
Higher pressure
Net force down
Tornadoes and hurricanes
Strong winds ➝ Low pressures
vin = 0
vout = 250 mph (112 m/s)
p
in−p
out=12v
out
2 =12
1.2 kg/ m3( ) 112 m/ s( )
2
=7500 Pa
Upward force on a 10 m x 10 m roof: 2 57500 Pa 10 m 7.5 10 NF
Weight of a 10 m x 10 m roof (0.1 m thick and using density of water –wood is lighter than water but all metal parts are denser):
4 2 510 kg 10 m/ s 10 Nmg
The roof is pushed off by the air inside !
The suicide door
The high speed wind will also push objects when the wind hits a surface perpendicularly!
Air pressure decreases due to air moving along a surface.
Modern car doors are never hinged on the rear side anymore.
If you open this door while the car is moving fast, the pressure difference between the inside and the outside will push the door wide open in a violent movement.
In modern cars, the air hits the open door and closes it again.
Delahaye Type 135
Curveballs
Speed of air layer close to ball is reduced (relative to ball)
Boomerangs are based on the same principle (Magnus effect)
Speed of air layer close to ball is increased (relative to ball)
Beyond Bernoulli
In the presence of viscosity, pressure may decrease without an increase in speed.
Example: Punctured hose (with steady flow).
Speed must remain constant along hose due to continuity equation.
Ideal fluid (no viscosity) Real fluid (with viscosity)
Friction accounts for the decrease in pressure.
Lower jet.
The syphonThe syphon
The trick to empty a clogged sink:
A x
x B
h
Thin hose → vA ~ 0
B 2v gh
PA = PB = Patm
ACT: Wooden brick
When a uniform wooden brick (1 m x 1 m x 2 m) is placed horizontally on water, it is partially submerged and the height of the brick above the water surface is 0.5 m. If the brick was placed vertically, the height of brick above the water would be:
A. 0.5 m
B. 1.0 m
C. 1.5 m.
0.5 m
The displaced volume in both cases needs to be the same: half of the volume of the brick.
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