# What is a fluid?

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29 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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PROGRAM OF “PHYSICS”

Lecturer
:

Dr. DO Xuan Hoi

Room 413

E
-
mail :
dxhoi@hcmiu.edu.vn

PHYSICS 2

(FLUID MECHANICS AND THERMAL PHYSICS)

02 credits (30 periods)

Chapter 1 Fluid Mechanics

Chapter 2 Heat, Temperature and the Zero
th

Law of Thermodynamics

Chapter 3 Heat, Work and the First Law of

Thermodynamics

Chapter 4 The Kinetic Theory of Gases

Chapter 5 Entropy and the Second Law of

Thermodynamics

References :

Halliday D., Resnick R. and Walker, J. (2005),
Fundamentals of Physics, Extended seventh edition.
John Willey and Sons, Inc.

Alonso M. and Finn E.J. (1992). Physics, Addison
-
Wesley
Publishing Company

Hecht, E. (2000). Physics. Calculus, Second Edition.
Brooks/Cole.

Faughn/Serway (2006), Serway’s College Physics,
Brooks/Cole.

Roger Muncaster (1994), A
-
Level Physics, Stanley
Thornes.

http://ocw.mit.edu/OcwWeb/Physics/index.htm

http://www.opensourcephysics.org/index.html

http://hyperphysics.phy
-
astr.gsu.edu/hbase/HFrame.html

http://www.practicalphysics.org/go/Default.ht
ml

http://www.msm.cam.ac.uk/

http://www.iop.org/index.html

.

.

.

Chapter 1 Fluid Mechanics

1. Variation of Pressure with Depth

2. Fluid Dynamics

3. Bernoulli’s Equation

Question

What is a fluid?

1. A liquid

2. A gas

3. Anything that flows

4. Anything that can be made to
change shape.

States of matter: Phase Transitions

ICE

WATER

STEAM

heat

heat

These are three states of matter

(plasma is another one)

States of Matter

Solid

Liquid

Gas

Plasma

States of Matter

Solid

Liquid

Gas

Plasma

Has definite volume

Has definite shape

Molecules are held in specific
location by electrical forces and

Can be modeled as springs
connecting molecules

Solid

Liquid

Gas

Plasma

Crystalline solid

Atoms have an ordered structure

Example is salt (red spheres are
Na
+

ions, blue spheres represent Cl
-

ions)

Amorphous Solid

Atoms are arranged randomly

Examples include glass

States of Matter

States of Matter

Solid

Liquid

Gas

Plasma

Has a definite volume

No definite shape

Exist at a higher temperature than solids

The molecules “wander” through the liquid in a
random fashion

The intermolecular forces are not strong
enough to keep the molecules in a fixed
position

Random motion

States of Matter

Solid

Liquid

Gas

Plasma

Has no definite volume

Has no definite shape

Molecules are in constant random motion

The molecules exert only weak forces on each
other

Average distance between molecules is large
compared to the size of the molecules

States of Matter

Solid

Liquid

Gas

Plasma

Matter heated to a very high temperature

Many of the electrons are freed from the nucleus

Result is a collection of free, electrically charged ions

Plasmas exist inside stars or experimental reactors or
fluorescent light bulbs!

http://fusedweb.pppl.gov/CPEP/Chart_Pages/4.CreatingConditions.html

Is there a concept that helps to distinguish between
those states of matter?

Density

The density of a substance of uniform composition is defined as its
mass per unit volume
:

some examples:

Object is denser

Density is greater

The densities of most liquids and solids vary
slightly
with changes
in temperature and pressure

Densities of gases vary
greatly

with changes in temperature and
pressure (and generally 1000 smaller)

Units

SI

kg/m
3

CGS

g/cm
3
(
1 g/cm
3
=1000
kg/m
3

)

Pressure

Pressure of fluid is the
ratio of the force exerted
by a fluid on a submerged
object to area

Units

SI

Pascal (Pa=N/m
2
)

Example:

100 N over 1 m
2

is P=(100 N)/(1 m
2
)=100 N/m
2
=100 Pa.

1.1 Pressure and Depth

If a fluid is at rest in a container,
all portions of the fluid must be in
static equilibrium

All points at the same depth must
be at the same pressure
(otherwise, the fluid would not be
in equilibrium)

Three external forces act on the
region of a cross
-
sectional area A

External forces: atmospheric, weight, normal

1. Variation of Pressure with Depth

Test 1

You are measuring the pressure at the depth of 10 cm
in three different containers. Rank the values of
pressure from the greatest to the smallest:

1. 1
-
2
-
3

2. 2
-
1
-
3

3. 3
-
2
-
1

4. It’s the same in all three

10 cm

1

2

3

Pressure and Depth equation

P
o

is normal atmospheric
pressure

1.013 x 10
5

Pa = 14.7
lb/in
2

The pressure does not
depend upon the shape of
the container

Other units of pressure:

76.0 cm of mercury

One atmosphere 1 atm =

1.013 x 10
5

Pa

14.7 lb/in
2

Example 1:

Find pressure at 100 m below
ocean surface.

1.2
Absolute Pressure and Gauge Pressure

The excess pressure above atmospheric pressure is
usually called
gauge pressure (

gh
)
, and the total
pressure is called
absolute pressure
.

A storage tank 12.0 m deep is filled with water. The top of

the tank is open to the air. What is the absolute pressure at

the bottom of the tank? The gauge pressure?

The absolute pressure :

The gauge pressure :

PROBLEM 1

SOLUTION

The U
-
tube in Fig. 1 contains two liquids in static

equilibrium: Water of density p
w

= 998 kg/m
3

is in the right

arm, and oil of unknown density p
x

is in the left.

Measurement gives
l

= 135 mm and
d

= 12.3 mm.

What is the density of the oil?

In the right arm:

In the left arm:

PROBLEM 2

SOLUTION

1.3 Pascal’s Principle

A change in pressure applied to
an enclosed fluid is transmitted
undiminished to every point of
the fluid and to the walls of the
container.

The hydraulic press is an
important application of
Pascal’s Principle

Also used in hydraulic brakes,
forklifts, car lifts, etc.

Since A
2
> A
1
, then F
2
> F
1
!!!

1.4 Measuring Pressure

The spring is calibrated by a
known force

The force the fluid exerts on
the piston is then measured

One end of the U
-
shaped tube
is open to the atmosphere

The other end is connected to
the pressure to be measured

Pressure at B is P
o
+
ρgh

A long closed tube is
filled with mercury and
inverted in a dish of
mercury

Measures atmospheric
pressure as
ρgh

Question

Suppose that you placed an extended
object in the water. How does the
pressure at the top of this object
relate to the pressure at the
bottom?

1. It’s the same.

2. The pressure is greater at the top.

3. The pressure is greater at the
bottom.

4. Whatever…

1.5 Buoyant Force

This force is called the buoyant force.

What is the magnitude of that force?

P
1
A

P
2
A

= mg

Buoyant Force

The magnitude of the buoyant force always equals
the
weight of the displaced fluid

The buoyant force is the same for a totally
submerged object of any size, shape, or density

The buoyant force is exerted by the fluid

Whether an object sinks or floats depends on the
relationship between the buoyant force and the
weight

Archimedes' Principle

Any object completely or partially submerged in a
fluid is buoyed up by a force whose magnitude is
equal to the
weight of the fluid displaced by
the object
.

This force is
buoyant force.

Physical cause:

pressure difference between the top and

the bottom of the object

Archimedes’ Principle:

Totally Submerged Object

The upward buoyant force is
B =
ρ
fluid
gV
obj

The downward gravitational force is
w = mg = ρ
obj
g V
obj

The net force is
B

w = (ρ
fluid
-

ρ
obj
) g V
obj

Depending on the direction
of the net force, the object
will either float up or sink!

The object is
less dense

than the fluid
ρ
fluid
< ρ
obj

The object experiences a
net
upward

force

The net force is
B
-

w=(ρ
fluid
-

ρ
obj
) g V
obj

The object is
more dense

than the fluid
ρ
fluid
> ρ
obj

The net force is downward,
so the object accelerates
downward

Test 2

Two identical glasses are filled to the same level
with water. One of the two glasses has ice cubes
floating in it.Which weighs more?

1. The glass without ice cubes.

2. The glass with ice cubes.

3. The two weigh the same.

NOTE :

Ice cubes displace exactly their own
weight in water.

An iceberg floating in seawater, as shown in figure, is extremely

dangerous because much of the ice is below the surface. This

hidden ice can damage a ship that is still a considerable distance

from the visible ice. What fraction of the iceberg lies below the

water level ? The densities of seawater and of iceberg are

W

= 1030 kg/m
3

and

I

= 917 kg/m
3

Weight of the whole iceberg :

Buoyant force :

(V
W

: volume of the displaced water = volume of the ice beneath the water)

The fraction of ice beneath the water’s surface:

PROBLEM 3

SOLUTION

Chapter 8 Fluid Mechanics

1. Variation of Pressure with Depth

2. Fluid Dynamics

2.1 Fluids in Motion: Streamline Flow

Streamline flow (also called
laminar flow)

every particle that passes a
particular point moves exactly
along the smooth path
followed by particles that
passed the point earlier

Streamline is the path

different streamlines cannot
cross each other

the streamline at any point
coincides with the direction of
fluid velocity at that point

Laminar flow around an
automobile in a test
wind tunnel.

2.1 Fluids in Motion: Turbulent Flow

The flow becomes irregular

exceeds a certain velocity

any condition that causes abrupt changes in
velocity

Eddy currents are a characteristic of turbulent flow

Hot gases from a

particles. The smoke first moves in

laminar flow at the bottom and

then in turbulent flow above

Fluid Flow: Viscosity

Viscosity is the degree of internal friction in the
fluid

The internal friction is associated with the
resistance between two adjacent layers of the fluid
moving relative to each other

2.2 Characteristics of an Ideal Fluid

The fluid is nonviscous

There is no internal friction between adjacent layers

The fluid is incompressible

Its density is constant

Its velocity, density and pressure do not change in time

The fluid moves without turbulence

No eddy currents are present

2.3 Equation of Continuity

The product of the cross
-
sectional area of a pipe
and the fluid speed is a
constant

Speed is high where
the pipe is narrow and
speed is low where
the pipe has a large
diameter

Av is called the
volume
flow rate

The mass is
conserved :

Equation of Continuity :

(a) The speed of the oil:

The
mass

flow rate:

(b)

PROBLEM 4

SOLUTION

As part of a lubricating system for heavy machinery, oil of density

850 kg/m
3

is pumped through a cylindrical pipe of diameter 8.0 cm

at a rate of 9.5 liters per second. The oil is incompressible.

(a)
What is the speed of the oil? What is the mass flow rate?

(b)
If the pipe diameter is reduced to 4.0 cm, what are the new

values of the speed and volume flow rate?

Oil incompressible: volume flow rate has the same value:

3. Bernoulli’s Equation

Magnitude of the force

exerted by the fluid in

section 1:
P
1
A
1

( V: volume of section 1)

The work done by this

force

W
1

=
F
1
x
1

=
P
1
A
1
x
1

=
P
1
V

The work done by by the fluid in section 2:

W
2

=
-

F
2
x
2

=
-

P
2
A
2
x
1

=
-

P
2
V

(W
2

< 0 : the fluid force opposes the displacement)

The net work done by two forces:
W =

(
P
1

-

P
2
)
V

Theorem of the variation of kinetic energy :

Bernoulli’s equation applied to an ideal fluid :

Bernoulli’s Equation

Relates pressure to fluid speed and elevation

Bernoulli’s equation is a consequence of Conservation
of Energy applied to an ideal fluid

Assumes the fluid is incompressible and nonviscous,
and flows in a nonturbulent, steady
-
state manner

States that the sum of the pressure, kinetic energy
per unit volume, and the potential energy per unit
volume has the same value at all points along a
streamline

Measure the speed of the fluid flow: Venturi Meter

Shows fluid flowing through a
horizontal constricted pipe

Speed changes as diameter
changes

Swiftly moving fluids exert less
pressure than do slowly moving
fluids

Application of Bernoulli’s Equation

How to measure the speed
v
2

?

EXAMPLE

Measure the speed of the fluid flow: Venturi Meter

Application of Bernoulli’s Equation

Equation of Continuity :

Rate of flow :
the volume of fluid which passes
through a given surface per unit time (m
3
/s)

4. Poiseuille’s law

Poiseuille's equation :

R

P
1

P
2

v

L

:
viscosity of the fluid

PROBLEM 5

A horizontal pipe of 25
-
cm
2

cross
-
section carries water
at a velocity of 3.0 m/s. The pipe feeds into a smaller
pipe with cross section of only 15 cm
2
.

W
=10
3
kg/m
3

(a)

What is the velocity of water in the smaller pipe ?

(b)

Determine the pressure change that occurs from
the larger
-
diameter pipe to the smaller pipe.

SOLUTION

A
1

v
1

A
2

v
2

(a)

(b)

PROBLEM 6

A large pipe with a cross
-
sectional area of 1.00 m
2

descends 5.00 m and narrows to 0.500 m
2
, where it
terminates in a valve. If the pressure at point 2 is
atmospheric pressure, and the valve is opened wide
and water allowed to flow freely, find the speed of the
water leaving the pipe.

SOLUTION

h

v
2

v
1

2

P
2
=P
0

P
1
=P
0

SOLUTION

h

v
2

v
1

2

P
2
=P
0

P
1
=P
0

PROBLEM 7

There is a leak in a water tank. The hole is very small
compared to the tank’s cross
-
sectional area.

(a)

If the top of the tank is open to the atmosphere,
determine the speed at which the water leaves the
hole when the water level is 0.500 above the hole.

SOLUTION

(a)

y
2

y
1

h

A
1

v
1

P
0

P
2

=P
0

A
2

PROBLEM 7

There is a leak in a water tank. The hole is very small
compared to the tank’s cross
-
sectional area.

(b)

Where does the stream hit the ground if the hole is
3.00 m above the ground ?

SOLUTION

(b)

y
2

y
1

h

A
1

v
1

P
0

P
2

=P
0

A
2

x

y

PROBLEM 8

An airplane has wing, each wing area 4.00 m
2
,
designed so that air flows over the top of the wing at
245 m/s and under the wing at 222 m/s. Find the
mass of the airplane such that the lift on the plane will
support its weight, assuming the force from the
pressure difference across the wings is directed
straight upwards.

SOLUTION

The lift on the plane supports the plane’s weight :

PROBLEM 9

BLOOD PRESSURE WITH DEPTH:

Human blood has a density of approximately

1.05 x 10
3

kg/m
3
.

(a)

Use this information to estimate the difference in
blood pressure between the brain and the feet in a
person who is approximately 1.6 m tall.

SOLUTION

The difference in pressure is given by:

(a)

PROBLEM 9

BLOOD PRESSURE WITH DEPTH:

Human blood has a density of approximately

1.05 x 10
3

kg/m
3
.

(b)

Estimate the volume flow rate of blood from the
head to the feet of this person. Assume an effective