Fluid MechanicsChapter 9

Defining a FluidDefining a Fluid• A fluid is a nonsolid state of matter in which

the atoms or molecules are free to move past each other, as in a gas or a liquid.

• Both liquids and gases are considered fluids

because they can flow and change shape.

9-1: Fluids and Buoyant Force

• Liquids have a definite volume• Gasses do not

• Liquids do not have a definite shape

Density and Buoyant Force

• In general, density is a measure of how much there is of a quantity in a given amount of space.– Quantity can be anything; # of trees or

people, or amount of mass or energy

Mass Density

• The concentration of matter of an object is

called the mass density.

(Amount of mass per unit volume of a substance)

• Represented with the Greek symbol, ρ (rho)

Mass Density = mass/volume

ρ = m/v

S.I. Unit Kg / m3

Common fluids Common fluids and solidsand solids

• Solids and liquids tend to be almost incompressible.

i.e., change very

little with changes

in pressure.

Buoyant Force

• Buoyant Force – a force that acts upward Buoyant Force – a force that acts upward on an object submerged in a liquid or on an object submerged in a liquid or floating on a liquid’s surface.floating on a liquid’s surface.

• This force can keep objects afloatThis force can keep objects afloat

• Acts in direction OPPOSITE of gravityActs in direction OPPOSITE of gravity

Buoyant Force

Archimedes Principle

• Law of Archimedes:

The buoyant forcebuoyant force is equalequal to the weightweight of the replaced liquid or gas.

Buoyant Force and Archimedes’Buoyant Force and Archimedes’PrinciplePrinciple

• The Brick, when added will cause the water to be displaced and fill the smaller container.

What will the volume be inside the smaller

Container?

The same volume as the brick!

Archimedes’ principle describes the magnitudeof a buoyant force.

Archimedes’ principle:Archimedes’ principle: Any object completely or Any object completely orpartially submerged in a fluid experiences an partially submerged in a fluid experiences an upward buoyant force equal in magnitude to the upward buoyant force equal in magnitude to the weight of the fluid displaced by the objectweight of the fluid displaced by the object..

FB FB = = Fg (displaced fluid) Fg (displaced fluid) = = mfgmfgmagnitude of buoyant force = weight of fluid displacedmagnitude of buoyant force = weight of fluid displaced

Buoyant Force and Buoyant Force and Archimedes’ PrincipleArchimedes’ Principle

Buoyant Force

The raft and cargo

are floating

because their

weight and

buoyant force are

balanced.

Buoyant Force• Now imagine a small holeis put in the raft.

• The raft and cargo sinkbecause their density isgreater than the density ofthe water.

• As the volume of the raftdecreases, the volume ofthe water displaced by theraft and cargo also decreases, as does the magnitude of the Buoyant force.

Buoyant ForceBuoyant Force• For a floating objectfloating object, the buoyant force equals the object’s

weight.

• The apparent weightapparent weight of a submerged object depends on the densitydensity of the object.

• For an object with density ρo submerged in a fluid of density ρf, the buoyant force FB obeys the following ratio:

Fg (object) = ρo

FB ρf

Example

A bargain hunter purchasesa “gold” crown at a fleamarket. After she getshome, she hangs the crownfrom a scale and finds itsweight to be 7.84 N. Shethen weighs the crown whileit is immersed in water, andthe scale reads 6.86 N. Isthe crown made of puregold? Explain.

SolutionSolution

Choose your equations:

Fg – FB = apparent weightFg = ρo

FB ρf

Rearrange your equations:

FB = Fg – (apparent weight) Ρo = Fg (ρf)

FB

Plug and ChugPlug and Chug

FB = 7.84 N – 6.86 N = 0.98 N

Ρo = 7.84 N (1.00 10 kg/m) = 8.0 x 103 kg/m3

0.98 N

• From the table in your book, the density of gold is 19.3 x 103 kg/m3.

• Because 8.0 x 103 kg/m3 < 19.3 x 103 kg/m3, the crown cannotcannot be pure gold.

HOMEWORK

• Page 324– Practice 9A # 1 and 3– Section Review # 1-3

QOTD: Wed 1/13/2010QOTD: Wed 1/13/2010Get a calculator from the red bin and complete Get a calculator from the red bin and complete

the following:the following:

1: Calculate the actual weight, buoyant force, and apparent weight of a 5.0 X 10-5 m3 iron ball floating at rest in mercury. (density of mercury is 13.6 x 103 kg/m3)

2: How much of the ball’s volume is immersed in mercury?

QOTD: Wed 1/13/2010QOTD: Wed 1/13/2010

• Calculate the actual weight, buoyant force, and apparent weight of a 5.0 X 10-5 m3 iron ball floating at rest in mercury. (density of mercury is 13.6 x 103 kg/m3)

3.86N, 3.86 N, 0 N3.86N, 3.86 N, 0 N

• How much of the ball’s volume is immersed in mercury?

2.89 x 102.89 x 10-5-5 m m33

Fluid Pressure and TemperatureFluid Pressure and Temperature

• Objectives:– Calculate pressure exerted by a fluid– Calculate how the pressure varies with depth

in a fluid– Describe fluids in terms of temperature

PRESSUREPRESSURE

• THE MAGNITUDE OF THE FORCE ON A SURFACE PER UNIT AREA

• PRESSURE = FORCE/AREA

P = F

A

PRESSUREPRESSURE

SI unit is the pascalpascal (Pa)1 pascalpascal is equal to 1 N/m2

The pressure of the atmosphere at sea level is about 105 Pa

* this amount of air pressure under normal conditions is the basis

for another unit, the atmosphere (atm)

The absolute air pressure inside a typical automobile tire is about 300,000 Pa, or about 3 atm.

Applied Pressure

• Transmitted equallyequally throughout a fluid

• In general, if the pressure in a fluid is increased at any point in a container, the pressure increases at all points inside the container by exactly the same amount.

• This is noted by PASCAL• Which leads us to PASCAL’S PRINCIPLE

PASCAL’S PRINCIPLEPASCAL’S PRINCIPLE• 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.

Pascal’s PrincipalPascal’s Principal• A hydraulic lift uses this principle.

• A small force, F1, applied to a small piston of area A1 causes a pressure increase in a fluid, such as oil.

• This increase in pressure, Pinc, is transmitted to a larger piston of area A2 and the fluid exerts a force, F2, on this piston.

See figure 9-7 in textSee figure 9-7 in text

• Applying Pascal’s principle and the definition of pressure gives us the following equation :

Pinc = F1 = F2

A1 A2

F2 = F1 A2 A1

Rearranging the equation to solve for F2:

From this equation, we see that F2 is greater than F1 by a factor equal to the ratio of the areas of the 2 pistons.

Pascal’s Principle

Because the pressure is the same on both sides of

the enclosed fluid in a hydraulic lift, a small forceon the smaller piston (left) produces a much

largerforce on the larger piston (right).

Pressure

Pressure varies with depth in a fluid

• Water pressure increases with depth because the water at a given depth must support the weight of the water above it.

Fluid pressure as a function of depth

• By using the symbol ρo for the atmospheric pressure at the surface, we can express the total pressure, or absolute pressure, at a given depth in a fluid of uniform density, ρ, as follows:

ρ ρ = = ρρoo + + ρρghgh

Absolute pressureAbsolute pressure = atmospheric pressure + (density x free fall accel x depth)

Temperature

• The temperature in a gas can be understood as The temperature in a gas can be understood as what is happening to the atomic scale.what is happening to the atomic scale.

• The higher the temperature of a gas, the fasterThe higher the temperature of a gas, the faster the particles move.the particles move.

• The faster the particles move, the higher the rate The faster the particles move, the higher the rate of collisions against the walls of the container.of collisions against the walls of the container.

• More momentum is transferred to container More momentum is transferred to container walls in given time interval, thus an increase in walls in given time interval, thus an increase in pressure results.pressure results.

SI units for Temperature:SI units for Temperature: Kelvins and degrees

Celsius

To convert from Celsius to To convert from Celsius to Kelvin, add 273.Kelvin, add 273.

Ex: Room temp is about 20° Celsius Ex: Room temp is about 20° Celsius or 293 k.or 293 k.

HOMEWORKHOMEWORKPg 327 # 1-2

Pg 330 # 1

Pg 331 # 1, 3-4

QOTD Friday: 1/15/2010

FLUIDS IN MOTION• FLUID FLOW:

– When a fluid is in motion, the flow can be characterized in one of two ways. The flow is said to be laminar if every particle that passes a particular point moves along the same smooth path traveled by the particles that passed that point earlier.

– This path is called streamline. – Different streamlines CANNOT cross each other.

Fluid Flow Cont.

Flow of fluid becomes irregular, or turbulentturbulent, above a certain velocity or under conditions that can cause abrupt changes in velocity.– obstacles or sharp turns

• Irregular motion of the fluid – eddy currents– Turbulent flow

• Ex: water in wake of ship or in air currents of severe thunderstorm

• Ideal Fluid Ideal Fluid : fluid that has no internal friction or viscosity and is incompressible

• ViscosityViscosity: amount of internal friction within a fluid.– Can occur when one layer of fluid slides past another.

Principles of Fluid Flow

• The mass flowing into the pipe MUST equal the mass flowing out of the pipe in the same time interval. Thus, m1 = m2

• Remember…m = pv• Volume of a cylinder: V = A (delta X)

So, p1V1 = p2V2

p1A1 (Delta X 1) = p2A2 (Delta X 2)

p1V1 = p2V2

p1A1 (Delta X 1) = p2A2 (Delta X 2)

*The length of the cylinder is also the distance the fluid travels, which is equal to the speed of flow multiplied by the time interval (delta X = v delta t)

• So,

p1A1 v1 (Delta t) = p2A2 v2 (Delta t)

** for an ideal fluid, both the time interval and the density are the same on each side of the equation, so they cancel each other out.

p1A1 v1 (Delta t) = p2A2 v2 (Delta t)

Continuity Equation

• After canceling each other out, we are left with the CONTINUITY EQUATION:

A1v1 = A2v2

Simply state: area x speed in region 1 = area x speed in region

2

• Flow rate is CONSTANT throughout a pipe

• Pressure in a fluid is related to the speed of flow.

– BERNOULLI’S PRINCIPAL:The pressure in a fluid

decreases as the fluid’s velocity increases

Bernoulli’s Equation cont.

• Explains the conservation of energy in fluids. Expressed as:

P + P + ½½ ρρvv22 + + ρρgh = constantgh = constant

Pressure +KE per unit volume + PEg per unit volume =

constant along a given streamline

Homework

• Pg 337: Practice #1

• Section Review # 1

QOTD Tuesday 1/19/2010

• What is the continuity equation?

• Using the equation, if a pipe narrows from a cross section of 2.0 m2 to 0.30 m2 and the speed through the wider area is 8.0 m/s, what is the speed of the water flowing through the narrow part?

9-4: Properties of Gases

• Volume, Pressure, and Temperature are the three variables that completely describe the macroscopic state of an ideal gas.

IDEAL GAS LAW

• PV = NkBT

Pressure x volume = # gas particles x Boltzmann’s constant x temperature

KB = 1.38 x 10-23 J/K

Temperature MUST be in Kelvins!!!!

IDEAL GAS LAW

• PV = nRT

Pressure x volume = # moles of gas x molar gas constant x temperature

n = # moles of gas

Where one mole = 6.02 x 1023 particles

R = 8.31 J/(mol * K)

IDEAL GAS LAW

• If the # of gas particles remains constant, then…

P1V1 = P2V2

T 1 T2

Sample

• A sealed tank with a volume of 0.10 m3 contains air at 27 degrees Celsisus under pressure 18000 Pa. The valve can withstand pressures up to 50000. Will the valve hold if the air inside the tank is heated to 227 degrees C?

Sample• A sealed tank with a volume of 0.10 m3 contains air at 27

degrees Celsius under pressure 18000 Pa. The valve can withstand pressures up to 50000. Will the valve hold if the air inside the tank is heated to 227 degrees Celsius?

P1V1 = P2V2

T 1 T2

Cont…18000 (0.10) = (P2) (0.10)

300 500

900,000 = 30 (P2)

PP22 = 30,000 Pa = 30,000 Pa

Thus, yes the valve will hold because 30,000 Thus, yes the valve will hold because 30,000 Pa is LESS THAN 50,000 Pa max that it can Pa is LESS THAN 50,000 Pa max that it can

withstand.withstand.

Homework

• Page 341: # 1-2

• Page 341 Review: #1 and 5

• STUDY FOR YOU EXAM!!!

• Make sure you know the summary and Key Ideas found on page 342 in your text.