Chapter 8: Fluid Mechanics

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29 Οκτ 2013 (πριν από 4 χρόνια και 12 μέρες)

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Chapter 8:

Fluid Mechanics

Learning Goal


To define a fluid.


To distinguish a gas from a liquid

States of Matter


Solids


definite volume, definite shape



Liquids


definite volume, indefinite shape



Gases


indefinite volume, indefinite shape



(Also plasma and Bose
-
Einstein condensates
but we don’t need to worry about those.)

What state of matter is glass?

1.
Solid

2.
Liquid

3.
Gas

What state of matter is honey?

1.
Solid

2.
Liquid

3.
Gas

The Nature of Fluids

Fluids:


Liquids and Gases
comprise the
category of what we call
fluids.


Fluids exhibit certain characteristics
that solids do not


they flow when
subjected to shear stress


PROPERTIES OF STATIC FLUIDS

Learning Goal


To use density to describe a fluid.


To apply buoyant force to explain why some
objects float or sink in a fluid.

Static Fluid Properties


Density (

) = mass / volume


Viscosity = internal resistance to flow


Note: Atmospheric pressure and
temperature influence a fluid’s density
and viscosity

Density

The density of an object is
represented by:

Density = mass / volume


While this formula is familiar to us,
we will use it in subsequent
derivations.

Specific Gravity


In order to have a constant comparison, we
use
specific gravity
instead of density
sometimes.


Since water has a density of 1 g/mL or 1 x
10
3

kg/m
3
, we eliminate the units and call the
number specific gravity.


Ex. For iron which has a density of 7.86 g/mL,
the specific gravity is 7.86 (or 7.86 as dense
as water).

Which is more dense, a pound
of feathers or a pound of
bricks?

1.
A pound of bricks

2.
A pound of feathers

3.
They are the same

Common Density
Misconceptions


Let’s expel some common misconceptions
about density.



Refer to your worksheet for the following
Turning Point questions about whether the
object will float or sink.

A. (Refer to worksheet)

1.
Sink

2.
Float

B. (Refer to worksheet)

1.
Sink

2.
Float

C. (Refer to worksheet)

1.
Sink

2.
Float

D. (Refer to worksheet)

1.
Sink

2.
Float

E. (Refer to worksheet)

1.
Sink

2.
Float

F. (Refer to worksheet)

1.
Sink

2.
Float

G. (Refer to worksheet)

1.
Sink

2.
Float

H. (Refer to worksheet)

1.
Sink

2.
Float

I. (Refer to worksheet)

1.
Sink

2.
Float

J. (Refer to worksheet)

1.
Sink

2.
Float

Buoyancy


The upward force present when an object
floats in a fluid, or feels lighter, is the
buoyant
force


on the object.



The weight of an object immersed in a fluid is
the
apparent weight

of the object (versus the
actual weight).


Apparent weight = F
G

-

F
B

Buoyant Force


F
B

= F
g

(displaced fluid) = m
f
g



Magnitude of


= weight of


of buoyant force


fluid displaced

Apparent Weight


The
apparent weight

of an object is the net
weight between the force of gravity and the
buoyant force.


F
net

= F
B



F
g



The apparent weight of an object in a
fluid, F
B



F
g

, could also be called
what?

1.
Net Force

2.
Tensional Force

3.
Buoyant Force

4.
Actual Weight

If an object is sinking to the bottom of
a glass of water, the buoyant force
must be?

1.
Equal to the Net
Force

2.
Less than
Fg

3.
More than
Fg

4.
Equal to
Fg

If an object is sinking to the bottom of
a glass of water, the buoyant force
must be?

1.
Equal to the Net
Force

2.
Less than
Fg

3.
More than
Fg

4.
Equal to
Fg

What must be true for the buoyant force
to be greater than gravitational force?

1.
Object is floating
continuously
upward

2.
Object is floating
at the top of the
fluid

3.
Object is sinking

Floating Objects


By Newton’s third law, if an object is floating,
and there is a force downward due to
acceleration of gravity, there must be an
equal buoyant force upward to bring about
equilibrium



F
b

= F
w

= m
o
g

Archimedes’ Principle


Displaced volume

of a fluid is the increase in
volume of a fluid due to the insertion of an
object.



Archimedes’ Principle

states that any object
completely or partially submerged in a fluid
experiences an upward buoyant force equal
to the
weight of the fluid displaced
.

If a rock is completely submerged in
a fluid, what must be true?

1.
The volume of the
displaced fluid = the
volume of the rock

2.
The weight of the rock =
weight of the fluid that
was displaced.

3.
Both 1 and 2

4.
None of the above

If a raft is floating and is partially
submerged in a fluid, what must be
true?

1.
The volume of the
displaced fluid = the
volume of the raft

2.
The weight of the raft =
weight of the fluid that
was displaced.

3.
Both 1 and 2

4.
None of the above

Archimedes Principle example


A bargain hunter purchases a “gold” crown at
a garage sale. After she gets home, she
hangs the crown from a scale and finds its
weight to be 7.84 N. She then weighs the
crown while it is immersed in water, and the
scale reads 6.86N. Is the crown made of
pure gold?

Pressure in Fluids


Pressure occurs within fluids due to the
constant motion of their molecules.



Common Pressure Units


For example,
standard atmospheric
pressure
is:


14.7 psi (pounds per square inch)


1.01 x 10
5
Pa (Pascal) = N/m
2


760 mmHg (millimeters mercury)


1
atm

(atmosphere)

Pressure cont.


Pressure is a measure of force per
given area.



P = F / A


Karate Chop demo

Bed of Nails


Pascal’s Principle



Pascal’s Principle






Because force is inversely proportional to
area, one can vary the cross
-
sectional area to
provide more force.



Eg
. Hydraulic brakes, car jacks, clogging of
arteries

In order to use a lesser force to accomplish a
difficult task, you should apply the force on the
hydraulic cylinder with

1.
Smaller radius

2.
Larger radius

3.
Doesn’t matter

Ex. 2


A car weighing 12000 N sits on a hydraulic
press piston with an area of 0.90 m
2
.
Compressed air exerts a force on a second
piston, which has an area of 0.20m
2
. How
large must this force be to support the car?

Pressure as a function of depth

Which hole will have the water
shoot out the furthest?

1.
Top hole

2.
Middle Hole

3.
Bottom Hole

4.
All will be equal

Absolute and Gauge Pressure


Absolute pressure = Atmospheric + Gauge


Pressure
Pressure




Atmospheric pressure is the pressure due to
the gases in the atmosphere (always present)


Gauge pressure is the pressure due to a fluid
(not counting atmospheric pressure)


Absolute pressure is the total pressure


Ex. 3


Calculate the absolute pressure at an ocean
depth of 1,000m. Assume that the density of
water is 1,025 kg/m
3

and that


P
o
= 1.01 x 10
5
Pa.


What is the gauge pressure as well?

Laminar versus Turbulent Flow

Laminar flow:


Low velocity relative to fluid medium


Streamline path

Turbulent flow:


High velocity relative to fluid medium


Irregular Flow (Eddy currents)

15
-
6

Ideal Fluids


Laminar flow


Nonviscous


Incompressible


Constant density and pressure



All these characteristics must be true for
these equations to hold true. (Hence, the
name for the ideal gas laws.)

Continuity Equation


Based on Law of Conservation of Mass


what comes in has
gotta

come out

Bernoulli’s Equation


Results from conservation of energy.



Taken into account are kinetic energy,
potential energy, and also
pressure

because
we are dealing with fluids

Bernoulli’s Principle


Bernoulli’s Principle

states that the flow
speed (Av) in a constriction must be greater
than the flow speed before or after it.




Also, swiftly moving fluids exert less pressure
than do slowly moving fluids.


Eg. Tornadoes and blown off roofs

Bernoulli’s principle



Pressure in a fluid varies inversely
with the velocity