# introduction to fluid mechanics

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24 Οκτ 2013 (πριν από 4 χρόνια και 7 μήνες)

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R
EVIEW

ρ

= m/V

ρ
: density (kg/m
3
)

m: mass (kg)

V: volume (m
3
)

Units::

kg/m
3

You should remember how to do density calculations
from chemistry! (and from last year)

Draw a FBD of an object in water

What happens if F
b

is larger than W?

W =
ρ
gV

F
b

=
ρ
gV

FBD
S

OF

THE

FOLLOWING

OBJECTS

(
FLOATING
)

What causes buoyant forces?

P =
ρ
gh P = F/A

S
OME

POINTS
:

Buoyant force on a floating object is equal to the
object’s weight

Buoyant force equals weight of fluid displaced

Buoyant force depends ONLY on density of fluid
and volume of displaced fluid

T
HURSDAY
, D
ECEMBER

3

Homework due tomorrow!

F
LUID

P
RESSURE

Depends on depth P =
ρ
gh

This type of pressure is often called
gauge
pressure.. Why?

F
LUID

P
RESSURE

Depends on depth P =
ρ
gh

This type of pressure is often called
gauge
pressure.. Why?

A pressure gauge measures an “unpressurized”
tank as 0 N/m
2
.

F
LUID

P
RESSURE

Depends on depth P =
ρ
gh

This type of pressure is often called
gauge
pressure.. Why?

Absolute pressure
:

P =
ρ
gh + P
0

G
REAT

VIDEO

While swimming near the
bottom of a pool, you let
out a small bubble of air.
As the bubble rises toward
the surface, what happens
to its diameter?

1) bubble diameter decreases

2) bubble diameter stays the same

3) bubble diameter increases

Question 15.4

Bubbling Up

While swimming near the
bottom of a pool, you let
out a small bubble of air.
As the bubble rises toward
the surface, what happens
to its diameter?

1) bubble diameter decreases

2) bubble diameter stays the same

3) bubble diameter increases

As the bubble rises, its depth decreases, so the water pressure
surrounding the bubble also decreases. This allows the air in the
bubble to expand (due to the decreased pressure outside) and so
the bubble diameter will increase.

Question 15.4

Bubbling Up

In a mercury barometer at atmospheric
pressure, the height of the column of
mercury in a glass tube is 760 mm. If
another mercury barometer is used that has
a tube of
larger diameter
, how high will the
column of mercury be in this case?

1) greater than 760 mm

2) less than 760 mm

3) equal to 760 mm

Question 15.9

The Straw III

In a mercury barometer at atmospheric
pressure, the height of the column of
mercury in a glass tube is 760 mm. If
another mercury barometer is used that has
a tube of
larger diameter
, how high will the
column of mercury be in this case?

1) greater than 760 mm

2) less than 760 mm

3) equal to 760 mm

While the weight of the liquid in the tube has increased (volume =
height
×

area) due to the larger area of the tube, the net upward force
on the mercury (force = pressure
×

area) has also increased by the
same amount! Thus, as long as the pressure is the same, the height of
the mercury will be the same.

Question 15.9

The Straw III

Q
UESTION

15.11

T
WO

B
RICKS

1

2

1) greater

2) the same

3) smaller

Imagine holding two identical
bricks in place underwater. Brick
1 is just beneath the surface of the
water, and brick 2 is held about 2
feet down. The force needed to
hold brick 2 in place is:

The force needed to hold the brick in
place underwater is
W

F
B
.

According
to Archimedes’ Principle,
F
B

is equal to
the weight of the fluid displaced.
Because each brick displaces the same
amount of fluid, then
F
B

is the same in
both cases.

Q
UESTION

15.11

T
WO

B
RICKS

1

2

1) greater

2) the same

3) smaller

Imagine holding two identical
bricks in place underwater. Brick
1 is just beneath the surface of the
water, and brick 2 is held about 2
feet down. The force needed to
hold brick 2 in place is:

Q
UESTION

15.14

O
N

G
OLDEN

P
OND

1) rises

2) drops

3) remains the same

4) depends on the size
of the steel

A boat carrying a large chunk of
steel is floating on a lake. The
chunk is then thrown overboard
and sinks. What happens to the
water level in the lake (with
respect to the shore)?

Initially the chunk of steel “floats” by
sitting in the boat. The buoyant force
is equal to the

weight

of the steel, and
this will

require a lot of displaced water

to equal the weight of the steel. When
thrown overboard, the steel sinks and

only displaces its
volume

in water
.
This is not so much water

certainly
less than before

and so the water
level in the lake will drop.

Q
UESTION

15.14

O
N

G
OLDEN

P
OND

1) rises

2) drops

3) remains the same

4) depends on the size
of the steel

A boat carrying a large chunk of
steel is floating on a lake. The
chunk is then thrown overboard
and sinks. What happens to the
water level in the lake (with
respect to the shore)?

Q
UESTION

15.17

A
RCHIMEDES

III

An object floats in
water

with
¾

of its volume
submerged. When more
water

is poured on top of
the water, the object will:

1) move up slightly

2) stay at the same place

3) move down slightly

4) sink to the bottom

5) float to the top

density of the object is of the density of
water
,
so it floats in water

(i.e., the buoyant force is greater than its
weight). When covered by more water, it must therefore float to the
top.

Q
UESTION

15.17

A
RCHIMEDES

III

1) move up slightly

2) stay at the same place

3) move down slightly

4) sink to the bottom

5) float to the top

An object floats in water with
¾

of its volume submerged. When
more water is poured on top of
the water, the object will:

Q
UESTION

15.18

A
RCHIMEDES

IV

1) move up slightly

2) stay at the same place

3) move down slightly

4) sink to the bottom

5) float to the top

An object floats in
water

with

of its volume submerged.
When
oil

is poured on top of
the water, the object will:

With the oil on top of the water, there is an

on
the object equal to the weight of the displaced oil. The effect of this
extra force is to

move the object upward

slightly, although it is not
enough to make the object float up to the top.

Q
UESTION

15.18

A
RCHIMEDES

IV

1) move up slightly

2) stay at the same place

3) move down slightly

4) sink to the bottom

5) float to the top

An object floats in
water

with
of its volume submerged.
When
oil

is poured on top of
the water, the object will:

Three containers are filled with water to
the same height and have the same surface
area at the base, but the total weight of
water is different for each. Which
container has the greatest total force acting
on its base? (1, 2, 3, or same)

Q
UESTION

15.21

W
OOD

IN

W
ATER

I

same for both

Two beakers are filled to the brim with water. A wooden
block is placed in the beaker 2 so it floats. (Some of the
water will overflow the beaker). Both beakers are then

larger weight
?

The block in 2
displaces an amount of
water equal to its weight
, because it is
floating. That means that the
weight
of the overflowed water is equal to the
weight of the block
, and so the
beaker
in 2 has the same weight as that in 1
.

Q
UESTION

15.21

W
OOD

IN

W
ATER

I

Two beakers are filled to the brim with water. A wooden
block is placed in the beaker 2 so it floats. (Some of the
water will overflow the beaker). Both beakers are then

larger weight
?

same for both

P
RACTICE

P
ROBLEM

A diver descends from a salvage ship to the ocean
floor at a depth of 35 m below the surface. Ocean
water has a density of 1025 kg/m
3
.

Calculate the gauge pressure on the diver on the
ocean floor.

Calculate the absolute pressure on the diver on the
ocean floor

The diver finds a rectangular aluminum plate having
dimensions 1m x 2m x .03m. A hoisting cable is lowered
from the ship and the diver connects it to the plate. The
density of aluminum is 2700 kg/m
3
. Ignore the effects of
viscosity.

Calculate the tension in the cable if it lifts the plate
upward at a slow, constant velocity

N
OW

DEALING

WITH

MOVING

FLUIDS

Law of conservation of mass

If the fluid is incompressive, volume is also
conserved

(Spraying water out of a hose)

Water coming out of a faucet (what happens to the
area toward the bottom?)

A
1
v
1

= A
2
v
2

W
HAT

THIS

MEANS
….

If area decreases, velocity has to increase for the
same mass to flow through

F
LOW

C
ONTINUITY

Volume flow = A
1
v
1

= A
2
v
2

A
1
, A
2
: cross sectional areas at points 1 and 2

v
1
, v
2
: speed of fluid flow at points 1 and 2

Q
UESTION

15.23

F
LUID

F
LOW

(1) one
-
quarter

(2) one
-
half

(3) the same

(4) double

(5) four times

Water flows through a
1
-
cm diameter

pipe
connected to a
½
-
cm diameter

pipe.
Compared to the speed of the water in the
1
-
cm pipe
, the speed in the
½
-
cm pipe

is:

The area of the small pipe is
less
, so we know that the water will flow
faster

there. Because

A

r
2
, when the

one
-
half
,
the
area is reduced by one
-
quarter
,

so the

speed must increase by four
times

to keep the flow rate

(
A

v
)

constant.

Q
UESTION

15.23

F
LUID

F
LOW

(1) one
-
quarter

(2) one
-
half

(3) the same

(4) double

(5) four times

Water flows through a
1
-
cm diameter

pipe
connected to a
½
-
cm diameter

pipe.
Compared to the speed of the water in the
1
-
cm pipe
, the speed in the
½
-
cm pipe

is:

v
1

v
2

D
EMONSTRATION

Fold a piece of paper into ¼. Place it like a tent
on your desk, and blow air underneath it.

Place to cans next to each other. What happens
when the air between them starts flowing?

Use Bernoulli’s equation to explain your findings

B
ERNOULLI

S

E
QUATION

P +
ρ

g h + ½

ρ
v
2

= Constant

(conservation of energy)

F
LUID

F
LOW

THROUGH

P
IPES

Where in this pipe does the fluid flow the fastest?

Where in this pipe is the pressure the highest?
How do you know this?

A
FEW

RULES
:

Any time a fluid is exposed to the atmosphere,

P = P
atm

or 0 ( absolute or gauge pressure)

At the top of a large tank, we assume v=0 (and
P=0)

G
OOD

QUESTION
!

Why does P =
ρ
gh?

Always compare two points!!!

P +
ρ
gh + .5

ρ
v
2

= P +
ρ
gh + .5

ρ
v
2

P
RACTICE

P
ROBLEM

A large container is filled with a liquid of density
1100 kg/m
3
. A hole of area 2.5x10
-
6

m
2

is opened in
the side of the container a distance h below the
liquid surface, which allows a stream of liquid to
flow through the hole and into a beaker placed to
the right of the container. At the same time, liquid
is also added to the container so that h remains
constant. The amount of liquid collected in the
beaker in 2.0 minutes is 7.2x10
-
4

m
3
.

Calculate the volume flow rate of liquid from the hole.

Calculate the speed of the liquid as it exits from the hole

Calculate the height, h, of the liquid above the hole to
cause the speed determined in part (b).

Q
UESTION

15.24

B
LOOD

P
RESSURE

I

1) increases

2) decreases

3) stays the same

4) drops to zero

A blood platelet drifts along with the
flow of blood through an artery that
is partially blocked. As the platelet
moves from the wide region into the
narrow region, the blood pressure:

The

speed increases in the narrow
part
, according to the continuity
equation. Because the

speed is

higher
, the

pressure is lower
, from
Bernoulli’s principle.

Q
UESTION

15.24

B
LOOD

P
RESSURE

I

1) increases

2) decreases

3) stays the same

4) drops to zero

A blood platelet drifts along with the
flow of blood through an artery that
is partially blocked. As the platelet
moves from the wide region into the
narrow region, the blood pressure:

speed is higher here

(so pressure is lower)

A person’s blood pressure is
generally measured on the
arm, at approximately the
same level as the heart. How
would the results differ if the

1) blood pressure would be lower

2) blood pressure would not change

3) blood pressure would be higher

Q
UESTION

15.25

B
LOOD

P
RESSURE

II

A person’s blood pressure is
generally measured on the
arm, at approximately the
same level as the heart. How
would the results differ if the

1) blood pressure would be lower

2) blood pressure would not change

3) blood pressure would be higher

Assuming that the flow speed of the blood does not change, then
Bernoulli’s equation indicates that at a lower height, the pressure
will be greater.

Q
UESTION

15.25

B
LOOD

P
RESSURE

II

How is the smoke
drawn up a chimney
affected when there is
a wind blowing
outside?

1) smoke rises more rapidly in the chimney

2) smoke is unaffected by the wind blowing

3) smoke rises more slowly in the chimney

4) smoke is forced back down the chimney

Q
UESTION

15.26

T
HE

C
HIMNEY

How is the smoke
drawn up a chimney
affected when there is
a wind blowing
outside?

1) smoke rises more rapidly in the chimney

2) smoke is unaffected by the wind blowing

3) smoke rises more slowly in the chimney

4) smoke is forced back down the chimney

Due to the speed of the wind at the top of the chimney, there is a
relatively lower pressure up there as compared to the bottom.
Thus, the smoke is actually drawn up the chimney more rapidly,
due to this pressure difference.

Q
UESTION

15.26

T
HE

C
HIMNEY