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Fluid MechanicsFluid MechanicsChapter 13Chapter 13

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FluidFluid• Anything that can flow• A liquid or a gas

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DensityDensity• Mass per unit volume

• Where (rho) is the density, m is mass, and V is volume

• A homogeneous material has the same density throughout

• The SI unit of density is the kg/m3.

V

m

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Specific gravitySpecific gravity• Should be called relative density, but

we are stuck with the traditional term

• The ratio of its density to the density of water.

water

material

gravity specific

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PressurePressure• Pressure is force per unit area,

expressed in Pascals (Pa). 1 Pa = 1 N/m2

A

Fp

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Pressure changesPressure changes• Pressure increases with depth

– Atmospheric pressure is greater at sea level than on top of a mountain

– Water pressure is greater in deeper water

• When y2 is greater, p2 is less.

1212 yygpp

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Open containersOpen containers• The pressure at the surface is

atmospheric pressure, or p0. If we are at a depth, h, below the surface,

ghpp

ghpp

0

0

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Pascal’s LawPascal’s Law• If we increase the pressure at the

surface, the pressure at any depth increases by the same amount.

• The pressure is transmitted throughout the fluid – if it has a uniform density – this is a fairly safe assumption for most liquids and for gases over small distances.

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Pascal’s LawPascal’s Law• Used in hydraulics to use a small

force over a small area to exert a large force over a large area.– See page 304

2

2

1

1

A

F

A

Fp

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Gauge pressureGauge pressure• If the pressure in your tire equals

atmospheric pressure, the tire is flat.• When your pressure gauge reads

32 psi, that means the pressure in the tire is 32 psi above the atmospheric pressure.

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Absolute pressureAbsolute pressure• Total pressure

• Atmospheric pressure at sea level is 101.3 kPa or 14.7 psi. Also called 1 atm

ga ppp 0

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ExampleExample• A residential hot water heating

system has an expansion tank in the attic, 12 m above the boiler. If the tank is open to the atmosphere, what is the gauge pressure in the boiler? What is the absolute pressure?

• pg= 118 kPa, pa=219 kPa

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Open-tube manometerOpen-tube manometer

• Measures the pressure in a container.

gaugepghpp 0

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BarometersBarometers• Long glass tubes full of mercury used

to measure atmospheric pressure.

h

ghp 0

ghpp 0

ghp 00

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BuoyancyBuoyancy• When an object is less dense than

water, it floats.

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Archimedes's principleArchimedes's principle• When an object is completely or

partially immersed in a fluid, the fluid exerts an upward force on the object equal to the weight of the fluid displaced by the object.

• We call this upward force the buoyant force.

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Weight of water displacedWeight of water displaced

V

m

Vm

mgw

gVw

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ExampleExample• A cork has a density of 200 kg/m3.

Find the fraction of the volume of the cork that is submerged when the cork floats in water.

0, corkbuoyantynet wFF

0 corkwater ww

corkwater ww

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Example continuedExample continued

totalcorksumbergedwater gVgV

totalcorksumbergedwater VV

water

cork

total

sumberged

V

V

5

1

kg/m 1000

kg/m 2003

3

total

sumberged

V

V

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ExampleExample• An ore sample weighs 14.00 N in air.

When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 9.00 N. Find the total volume and the density of the sample.

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Example continuedExample continued0 wFTF Bnet

TwFB

TwgVwater

g

TwV

water

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Example continuedExample continued

23 m/s 8.9kg/m 1000

N 00.9N 00.14 V

34 m 1010.5 V

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Example continuedExample continued

gVw

gV

w

342 m 1010.5m/s 8.9

N 00.14

3kg/m 1280

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Fluid FlowFluid Flow• An ideal fluid is incompressible and

has no internal friction.• We will only deal with laminar flow,

which has a steady-state pattern.• We will not deal with turbulent flow,

which is chaotic.

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Continuity equationContinuity equation• The mass of a moving fluid doesn’t

change as it flows.

tvAtvA 2211

2211 vAvA

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Volume flow rateVolume flow rate• The rate at which volume crosses a

section of the tube:

Avt

V

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ExampleExample• Blood flows from an artery of radius

0.3 cm, where it’s speed is 10 cm/s into a region where the radius has been reduced to 0.2 cm. What is the speed of the blood in the narrower region?

• 22.5 cm/s

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Bernoulli’s EquationBernoulli’s Equation• Relates pressure, flow speed, and

height for flow of ideal fluids.• Derived in book by applying the work

energy theorem to a flowing fluid

2222

2111 2

1

2

1vgypvgyp

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Bernoulli’s EquationBernoulli’s Equation• Make sure that your units are

consistent.– Always use pascals, kg/m3, and m/s

• Always use either all absolute pressures or all gauge pressures.

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ExampleExample• A large tank of

water has a small hole a distance h below the water surface. Find the speed of the water as it flows from the tank.

• Vb=sqrt(2gh)

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On your ownOn your own• Water enters a house through a

pipe with an inside diameter of 2.0 cm at an absolute pressure of 4.0 x 105 Pa. A 1.0 cm diameter pipe leads to the second-floor bathroom 5.0 m above. When the flow speed at the inlet pipe is 1.5 m/s, find the pressure and volume flow rate in the bathroom.

• 3.3 x 105 Pa 4.7 x 10-4 m3/s