Lecture 4Bernouilli’s equation.

ACT: Aluminum and lead

Two blocks of aluminum and lead with identical sizes are suspended from the ceiling with strings of different lengths and placed inside a bucket of water as shown. In which case is the buoyant force greater?

A. Al

B. Pb

C. It’s the same for both

Al

The displaced volume (= volume of the block) is the same in both cases.

Depth or object density do not play any role. Pb

ceiling

The different weight is compensated with a different tension in the strings.

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.

1 1 1 2 2 2 1 2dW pAds p A ds p p dV Work by pressure during his motion:

ρ1

ρ2

ds2

ds1

1 1 2 2dV Ads A ds

dV

dV

If the fluid is incompressible, the volume should remain constant:

Kinetic and gravitational potential energy

2 22 1

1 12 2

dK dV v dV v Change in kinetic energy:

A1

A2

v1

v2

ρ1

ρ2

ds2

ds1

Change in potential energy:

2 1dU g dV y y

y: height of each element relative to some initial level (eg: floor)

Bernouilli’s equation

Putting everything together:

2 22 1 2 1 1 2

1 12 2

dV v dV v g dV y y p p dV

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: Bernouilli’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:gaugeA A atm Ap p p gh

hA

x A

hA

A x

Now we drill a small hole of radius 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

Bernouilli 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 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 B

Bernouilli:

1 1

2 2p v p v y y

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 BernouilliPartially illegal Bernouilli

Gases are NOT incompressible

Bernouilli’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…

Ie, if the changes in density are small along the streamline

Example: Why do planes fly?Example: Why do planes fly?

High speed, low pressure

Low speed, high pressure

Net force up (“Lift”)

bottom

2 2top top bottom

1 1Bernouilli: is negligible

2 2p v p v g h

bottom

2 2top topbottomLif t area of wing area of wing

2p p v v

DEMO: Paper and spool.

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

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 at A is lower where the air is moving fast.

This is how atomizers work!

Tornadoes and hurricanes

Strong winds Low pressures

vin = 0

vout = 250 mph (112 m/s)

22 3out in out

1 11.2 kg/ m 112 m/ s 7500 Pa

2 2p p v

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.

Curveballs

Lower speed (relative to ball)

Higher speed (relative to ball)

Boomerangs are based on the same principle.

Beyond BernouilliBeyond Bernouilli

In general, when there is no friction (no viscosity), the Ventury effect applies (it’s just a consequence of the work-kinetic energy theorem).

In the presence of viscosity, lower speed does not necessarily mean higher pressure.

Example: Punctured hose (with steady flow):

Ideal fluid (no viscosity)

Real fluid (with viscosity)

Here friction accounts for the decrease in pressure. (Speed must remain constant due to continuity equation).