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Chapter 9 Chapter 9 Fluid Mechanics Fluid Mechanics

Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

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Page 1: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Chapter 9Chapter 9

Fluid MechanicsFluid Mechanics

Page 2: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Chapter ObjectivesChapter Objectives Define fluidDefine fluid Density Density Buoyant force Buoyant force Buoyantly of floating objectsBuoyantly of floating objects Pressure Pressure Pascal's principlePascal's principle Pressure and depthPressure and depth Temperature Temperature Fluid flow continuity equationFluid flow continuity equation Bernoulli's principleBernoulli's principle Ideal gas lawIdeal gas law

Page 3: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

What is a Fluid?What is a Fluid?

So far we have studied the causes of So far we have studied the causes of motion dealing with motion dealing with solidssolids..

That leaves us with That leaves us with gasesgases and and liquidsliquids.. LiquidsLiquids and and gasesgases are different phases, are different phases,

but have common properties. but have common properties. One common property of One common property of gasesgases and and

liquidsliquids is their ability to flow and alter is their ability to flow and alter their shape on the process.their shape on the process.

Materials that exhibit the property to Materials that exhibit the property to flow are called flow are called fluidsfluids. .

Page 4: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

DensityDensity It is a difficult concept to visualize the mass of a fluid It is a difficult concept to visualize the mass of a fluid

because its shape can change. because its shape can change. So a more useful measurement is the density of an So a more useful measurement is the density of an

object.object. The The densitydensity of an substance is the of an substance is the massmass per unit per unit

volumevolume of the substance. of the substance. Because this uses mass, it is called the Because this uses mass, it is called the mass densitymass density.. If it uses weight, it is called If it uses weight, it is called weight densityweight density..

ρ =

SI units = Kg/m3rhomv

Page 5: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Densities of Common Densities of Common SubstancesSubstances

SubstanceSubstance kg/mkg/m33

HydrogenHydrogen 0.08990.0899

HeliumHelium 0.1790.179

Steam (100 Steam (100 ooC)C) 0.5980.598

AirAir 1.291.29

OxygenOxygen 1.431.43

Carbon DioxideCarbon Dioxide 1.981.98

EthanolEthanol 8.06 x 108.06 x 1022

IceIce 9.17 x 109.17 x 1022

Fresh waterFresh water 1.00 x 101.00 x 1033

Sea waterSea water 1.025 x 101.025 x 1033

IronIron 7.86 x 107.86 x 1033

MercuryMercury 13.6 x 1013.6 x 1033

GoldGold 19.3 x 1019.3 x 1033

Page 6: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Specific GravitySpecific Gravity

The The specific gravityspecific gravity of a substance is a of a substance is a ratioratio of of a substance’s a substance’s densitydensity to that of water. to that of water.

It gives us an easier scale for comparison of It gives us an easier scale for comparison of whether objects will float or not.whether objects will float or not. Example: Lead has a Example: Lead has a specific gravityspecific gravity of 11.4. of 11.4.

Which means that lead is 11.4 times more dense than Which means that lead is 11.4 times more dense than water.water.

– Water has a Water has a densitydensity of 1.00 x 10 of 1.00 x 103 kg3 kg/m/m33,, so lead has a so lead has a densitydensity of of

11.4 x 1011.4 x 103 kg3 kg//mm33..

– Or water has a Or water has a densitydensity of 1 of 1 gg//cmcm33, so lead has a , so lead has a densitydensity of 11.4 of 11.4 gg//cmcm33..

The object will the larger The object will the larger specific gravityspecific gravity will sink! will sink!

Page 7: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Buoyancy Buoyancy The ability of a substance to float in a liquid The ability of a substance to float in a liquid

is based of the densities of the two is based of the densities of the two substances.substances.

The less dense substance will move to the The less dense substance will move to the top, or float.top, or float.

The force pushing on an object while in a The force pushing on an object while in a liquid or floating is called the liquid or floating is called the buoyant buoyant forceforce..

The The buoyant forcebuoyant force acts opposite of gravity, acts opposite of gravity, and that is why objects seem “lighter” in and that is why objects seem “lighter” in waterwater

Page 8: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Archimedes’ PrincipleArchimedes’ Principle When an object is placed in water, the total When an object is placed in water, the total

volume of water is raised the same volume as the volume of water is raised the same volume as the Portion of the ObjectPortion of the Object that is submerged. that is submerged.

Archimedes PrincipleArchimedes Principle states the states the Buoyant ForceBuoyant Force is equal to the weight of water displaced.is equal to the weight of water displaced. Use this formula if the object is totally submerged in the fluid.Use this formula if the object is totally submerged in the fluid.

FB = Fg(displaced fluid) = mfg

Buoyant Force Mass Fluid = Vf ρf

Page 9: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Buoyant Force on Floating Buoyant Force on Floating ObjectsObjects

For an object to float, theFor an object to float, the Buoyant ForceBuoyant Force must be must be equal magnitudeequal magnitude to the weight of to the weight of the object.the object.

The The densitydensity of the object determines the of the object determines the depth of the submersion.depth of the submersion. Use the following for an object that is floating on Use the following for an object that is floating on

top of the fluid.top of the fluid. The object is not totally submerged.The object is not totally submerged.

FB = Fg (object) = mog

Mass of Object

Page 10: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Unknown Unknown Densities/ObjectsDensities/Objects

When faced with the challenge of identifying When faced with the challenge of identifying an unknown substance, we compare its an unknown substance, we compare its densitydensity to that of water. to that of water.

To do this, we use the To do this, we use the Buoyant ForceBuoyant Force of water to of water to determine the determine the densitydensity of the unknown substance.of the unknown substance.

This is done by comparing the weight of the object This is done by comparing the weight of the object in air and then again in water.in air and then again in water. The difference between the two weights would show The difference between the two weights would show

the the Buoyant ForceBuoyant Force..

FB = Fg (in air) - Fg (in water) =

Fg (in air)

FB

object

fluid

Use this when finding identifying an unknown substance by its density

Page 11: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

PressurePressure PressurePressure is a measure of how much force is a measure of how much force

is applied over a given area.is applied over a given area. The The SI UnitSI Unit for for pressurepressure is the is the Pascal Pascal ((PaPa)), ,

which is equal to 1 N/mwhich is equal to 1 N/m22.. The air around us pushes with a pressure. The air around us pushes with a pressure.

This is called This is called atmospheric pressureatmospheric pressure, which , which is about is about 101055 Pa Pa..

That amount gives us another unit, the That amount gives us another unit, the atmosphereatmosphere ((atmatm).).

105 Pa = 1 atm = 1 bar = 29.92 in Hg = 14.7 psi

P =

FA

Page 12: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Pascal’s PrinciplePascal’s Principle

When you pump up a bicycle tire, it just When you pump up a bicycle tire, it just doesn’t grow sideways, but also in doesn’t grow sideways, but also in height.height.

Pascal’s PrinciplePascal’s Principle states that the states that the pressure applied to a fluid in a closed pressure applied to a fluid in a closed container is transmitted equally to every container is transmitted equally to every point of the fluid and to the walls of the point of the fluid and to the walls of the container.container. Pinc = F1 = F2

A1 A2Pressure in a closed container

Page 13: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Pressure and DepthPressure and Depth Water pressure increases with depth because Water pressure increases with depth because

the water at a given depth has to support the the water at a given depth has to support the weight of the water above it.weight of the water above it.

Imagine an object suspended in a fluid. There Imagine an object suspended in a fluid. There is an imaginary column that is the same cross-is an imaginary column that is the same cross-sectional area of the object.sectional area of the object.

There is water trapped below pushing up on There is water trapped below pushing up on the object. The water above is pushing down the object. The water above is pushing down on the object.on the object.

Since the water is suspended, the two Since the water is suspended, the two pressures are equal.pressures are equal.

If one becomes larger, the object will sink or If one becomes larger, the object will sink or float.float.

Page 14: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Fluid Pressure EquationFluid Pressure Equation

PressurePressure varies with the depth in a fluid. varies with the depth in a fluid. That is because there is a larger column of That is because there is a larger column of

water above the object pushing downward.water above the object pushing downward. We must also account for We must also account for atmospheric atmospheric

pressurepressure pushing down on top of the water. pushing down on top of the water.

P =

FA

= == =ρAhg

ρVg

mg AAA

ρhgAnd taking atmospheric pressure into account, we get the following.

P =

ρhgPo +

Page 15: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

TemperatureTemperature

TemperatureTemperature is a measure of the is a measure of the average kinetic energy of the particles average kinetic energy of the particles in a substance.in a substance.

There are actually two There are actually two SI unitsSI units for for temperaturetemperature, , kelvinkelvin ( (KK) and ) and degrees degrees CelsiusCelsius ( (ooCC).).

To convert, add To convert, add 273273 to the to the CelsiusCelsius measurement.measurement.

Fahrenheit is the units for Fahrenheit is the units for temperaturetemperature in the United States.in the United States.

Page 16: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Ideal Gas LawIdeal Gas Law

The ideal gas law The ideal gas law varies slightly for varies slightly for physics versus physics versus chemistry.chemistry.

That is due to That is due to Boltzmann’s Boltzmann’s ConstantConstant ( (kkBB)). .

kkBB = 1.38 x 10 = 1.38 x 10-23-23 JJ//KK

Chemistry’s version Chemistry’s version uses the uses the ideal gas ideal gas constantconstant ( (RR).).

R = 8.31 R = 8.31 JJ//(mol *K)(mol *K)

PV = NkBT

PV = nRT

P is pressure (N/m2)V is volume (m3)T is temperature (K)N is number of particles

n is number of moles

Page 17: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

33rdrd Version of Ideal Gas Version of Ideal Gas LawLaw

Assuming the amount of gas Assuming the amount of gas (N(N11= N= N22)) remains remains constant in a closed container, we can derive a constant in a closed container, we can derive a 33rdrd version of the ideal gas law. version of the ideal gas law.

This version will also help us to see the basis This version will also help us to see the basis for the 3 gas laws from chemistry for the 3 gas laws from chemistry (Boyle’s (Boyle’s Law, Charles’ Law, and Gay-Lussac’s Law)Law, Charles’ Law, and Gay-Lussac’s Law)..

And since N1=N2,

N=kBTPV

Notice that

kBT1

P1V

1 kBT2

P2V2=

Because kB is constant, it cancels itself out

Leaving us with

T1

P1V

1 T2

P2V2=

Page 18: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Fluid FlowFluid Flow

Fluid flows in one of two ways:Fluid flows in one of two ways: Laminar flowLaminar flow is when every particle of is when every particle of

fluid follows the same smooth path.fluid follows the same smooth path. That path is said to be That path is said to be streamlinestreamline..

Turbulent flowTurbulent flow is when there is irregular is when there is irregular flow due to objects in the path or sharp flow due to objects in the path or sharp turns in the flowing chamber.turns in the flowing chamber.

Irregular motions of the fluid are called Irregular motions of the fluid are called eddy currentseddy currents..

Since Since laminar flowlaminar flow is predictable and is predictable and easy to model, we will use its easy to model, we will use its characteristics in this book.characteristics in this book.

Page 19: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Continuity EquationContinuity Equation

Due to the conservation of mass, the Due to the conservation of mass, the amount of fluid as it flows through a amount of fluid as it flows through a chamber is consider to also be conserved.chamber is consider to also be conserved.

So m1= m2But the mass of a gas is hard to find and we do know the density and the space it takes up.

ρV1= ρV2

ρA1Δx1= ρA2Δx2

But what happens when the chamber changes size?ρA1v1Δt =

ρA2v2ΔtIt is hard to measure displacement of a gas, but we can measure the time it takes to travel.

Density of the gas will be constant and the time will be constant, so…A1v1 =

A2v2

Continuity Equation

Page 20: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Bernoulli’s PrincipleBernoulli’s Principle

The pressure in a fluid decreases as the fluid’s The pressure in a fluid decreases as the fluid’s velocity increases.velocity increases.

This is the principle responsible for This is the principle responsible for liftlift.. As air flows over the top of the wing, the speed must As air flows over the top of the wing, the speed must

increase because it travels a longer distance.increase because it travels a longer distance. Because the speed increased, the pressure then Because the speed increased, the pressure then

decreases.decreases. Now there is more pressure on the bottom of the Now there is more pressure on the bottom of the

wing pushing upward, creating lift!wing pushing upward, creating lift!

Page 21: Chapter 9 Fluid Mechanics. Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and

Bernoulli’s EquationBernoulli’s Equation

This equation relates pressure to energy in a This equation relates pressure to energy in a moving fluid.moving fluid.

Since energy is conserved, Since energy is conserved, Bernoulli’s Bernoulli’s EquationEquation is set to be a constant. is set to be a constant.

For our use, we will then set For our use, we will then set Bernoulli’s EquationBernoulli’s Equation equal to itself under initial and final conditions.equal to itself under initial and final conditions.

P1 + 1/2ρ1v12 + ρ1gh1 = P2 + 1/2ρ2v2

2 + ρ2gh2

Pressure

DensityVelocity

Height