Transcript Chapter 9 - Mona Shores Blogs

Chapter 9
Fluid Mechanics
Chapter Objectives
Define fluid
 Density
 Buoyant force
 Buoyancy of floating objects
 Pressure
 Pascal's principle
 Pressure and depth
 Temperature
 Fluid flow continuity equation
 Bernoulli's principle
 Ideal gas law

What is a Fluid?
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So far we have studied the causes of motion
dealing with solids.
That leaves us with gases and liquids.
Liquids and gases are different phases, but
have common properties.
One common property of gases and liquids is
their ability to flow and alter their shape on
the process.
Materials that exhibit the property to flow are
called fluids.
Density

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It is a difficult concept to visualize the mass of a fluid because
its shape can change.
So a more useful measurement is the density of an object.
The density of an substance is the mass per unit volume of the
substance.
Because this uses mass, it is called the mass density.
If it uses weight, it is called weight density.
rho
ρ= m
v
SI units = Kg/m3
Densities of Common Substances
Substance
kg/m3
Hydrogen
0.0899
Helium
0.179
Steam (100 oC)
0.598
Air
1.29
Oxygen
1.43
Carbon Dioxide
1.98
Ethanol
8.06 x 102
Ice
9.17 x 102
Fresh water
1.00 x 103
Sea water
1.025 x 103
Iron
7.86 x 103
Mercury
13.6 x 103
Gold
19.3 x 103
Specific Gravity
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The specific gravity of a substance is a ratio of a
substance’s density to that of water.
It gives us an easier scale for comparison of whether
objects will float or not.
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Example: Lead has a specific gravity of 11.4.
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Which means that lead is 11.4 times more dense than water.
– Water has a density of 1.00 x 103 kg/m3, so lead has a density of
11.4 x 103 kg/m3.
– Or water has a density of 1 g/cm3, so lead has a density of 11.4
g/ 3.
cm

The object with the larger specific gravity will sink!
Buoyancy
The ability of a substance to float in a liquid is
based on the densities of the two substances.
 The less dense substance will move to the
top, or float.
 The force pushing on an object while in a
liquid or floating is called the buoyant force.
 The buoyant force acts opposite of gravity,
and that is why objects seem “lighter” in
water
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Archimedes’ Principle
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When an object is placed in water, the total
volume of water is raised the same volume as
the Portion of the Object that is submerged.
Archimedes Principle states the Buoyant Force
is equal to the weight of water displaced.

Use this formula if the object is totally submerged in the
fluid.
FB = Fg(displaced fluid) = mfg
Buoyant Force
Mass Fluid = Vf ρf
Buoyant Force on Floating Objects
For an object to float, the Buoyant Force
must be equal magnitude to the weight of the
object.
 The density of the object determines the
depth of the submersion.

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Use the following for an object that is floating on
top of the fluid.

The object is not totally submerged.
FB = Fg
(object)
= mog
Mass of Object
Unknown Densities/Objects
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When faced with the challenge of identifying an
unknown substance, we compare its density to that
of water.
To do this, we use the Buoyant Force of water to
determine the density of the unknown substance.
 This is done by comparing the weight of the object in air
and then again in water.

The difference between the two weights would show the
Buoyant Force.
FB = Fg
(in air)
-
Fg
(in water)
Fg
Use this when identifying
an unknown substance by
its density
(in air)
FB
object
=
fluid
Pressure
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F
P=
A
Pressure is a measure of how much force is
applied over a given area.
The SI Unit for pressure is the Pascal (Pa),
which is equal to 1 N/m2.
The air around us pushes with a pressure. This is
called atmospheric pressure, which is about 105
Pa.
That amount gives us another unit, the
atmosphere (atm).
105 Pa = 1 atm = 1 bar = 29.92 in Hg = 14.7 psi
Pascal’s Principle
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When you pump up a bicycle tire, it just
doesn’t grow sideways, but also in height.
Pascal’s Principle states that the
pressure applied to a fluid in a closed
container is transmitted equally to every
point of the fluid and to the walls of the
container.
Pressure in a closed
container
Pinc = F1 = F2
A1
A2
Pressure and Depth
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Water pressure increases with depth because the
water at a given depth has to support the weight of
the water above it.
Imagine an object suspended in a fluid. There is an
imaginary column that is the same cross-sectional
area of the object.
There is water trapped below pushing up on the
object. The water above is pushing down on the
object.
Since the water is suspended, the two pressures are
equal.
If one becomes larger, the object will sink or float.
Fluid Pressure Equation
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Pressure varies with the depth in a fluid.
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That is because there is a larger column of water
above the object pushing downward.
We must also account for atmospheric pressure
pushing down on top of the water.
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F
P=
A
mg ρVg ρAhg
=
=
=
= ρhg
A
A
A
And taking atmospheric pressure into account, we get the
following.
P = Po + ρhg
Temperature

Temperature is a measure of the average
kinetic energy of the particles in a substance.
 There are actually two SI units for
temperature, kelvin (K) and degrees
Celsius (oC).
 To convert, add 273 to the Celsius
measurement.
 Fahrenheit is the units for temperature in the
United States.
Ideal Gas Law
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The ideal gas law varies
slightly for physics
versus chemistry.
That is due to
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kB = 1.38 x 10-23 J/K

Boltzmann’s Constant
(kB).
PV = NkBT
N is number of particles
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Chemistry’s version
uses the ideal gas
constant (R).
R = 8.31 J/(mol *K)
PV = nRT
n is number of moles
P is pressure (N/m2)
V is volume (m3)
T is temperature (K)
3rd Version of Ideal Gas Law
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Assuming the amount of gas (N1= N2) remains
constant in a closed container, we can derive a 3rd
version of the ideal gas law.
This version will also help us to see the basis for the
3 gas laws from chemistry (Boyle’s Law, Charles’ Law,
and Gay-Lussac’s Law).
Notice that
PV
N=
kBT
And since N1=N2,
P1V1 P2V2
=
kBT1 kBT2
Leaving us with
P1V1 P2V2
=
T1
T2
Because kB is constant, it cancels itself out
Fluid Flow
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Fluid flows in one of two ways:
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Laminar flow is when every particle of fluid
follows the same smooth path.
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Turbulent flow is when there is irregular flow
due to objects in the path or sharp turns in the
flowing chamber.
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That path is said to be streamline.
Irregular motions of the fluid are called eddy
currents.
Since laminar flow is predictable and easy to
model, we will use its characteristics in this
book.
Continuity Equation
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Due to the conservation of mass, the amount
of fluid as it flows through a chamber is
consider to also be conserved.
But the mass of a gas is
hard to find and we do
know the density and
the space it takes up.
So m1= m2
ρV1= ρV2
ρA1Δx1= ρA2Δx2
ρA1v1Δt = ρA2v2Δt
It is hard to measure
displacement of a gas,
but we can measure the
time it takes to travel.
A1v1 = A2v2
But what happens when the
chamber changes size?
Density of the gas will be
constant and the time will
be constant, so…
Continuity Equation
Bernoulli’s Principle
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The pressure in a fluid decreases as the fluid’s
velocity increases.
This is the principle responsible for lift.
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As air flows over the top of the wing, the speed must
increase because it travels a longer distance.
Because the speed increased, the pressure then decreases.
Now there is more pressure on the bottom of the wing
pushing upward, creating lift!
Bernoulli’s Equation
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This equation relates pressure to energy in a moving
fluid.
Since energy is conserved, Bernoulli’s Equation is
set to be a constant.
For our use, we will then set Bernoulli’s Equation
equal to itself under initial and final conditions.
P1 + 1/2ρ1v12 + ρ1gh1 = P2 + 1/2ρ2v22 + ρ2gh2
Pressure
Density
Velocity
Height