Chapter 14

FLUID MECHANICS

Chapter 14Chapter 14 1st semester, AY 2009-20101st semester, AY 2009-2010

Chapter 14Chapter 14 Classification of matter 2Classification of matter 2

• Solids– tightly packed, usually in a regular pattern– retains a fixed volume and shape – not easily compressible – doesn’t easily flow

Liquids close together with no regular arrangement assumes the shape of the part of the container

which it occupies not easily compressible, flow easilyflow easily

Gas well separated with no regular arrangement assumes the shape of the part of the container easily compressible, flow easilyflow easily

Images from:http://www.chem.purdue.edu/gchelp/liquids/character.html

Chapter 14Chapter 14 What’s in store for us? 3What’s in store for us? 3

Fluid statics

• Density

• Pressure

• Buoyancy

Fluid dynamics

• Continuity equation

• Bernoulli’s equation

Chapter 14Chapter 14 Properties of fluids 4Properties of fluids 4

Density

M

V

Units :

1 kg/m3 = 10-3 g/cm3

Material Density, kg/m3

Air (1 atm, 200 C) 1.21

Water 0.998 x 103

Ice 0.917 x 103

Blood 1.060 x 103

Seawater 1.024 x 103

Styrofoam 1 x 102

Gold 19.3 x 103

• Density may vary from point to point

• Higher ρ sinks under lower ρ

• Solids and liquids: ρ independent of T & P

Gases: strongly dependent on T & P


Specific gravity/relative density

water

materialSG

• Specific gravity is dimensionless.

• ρ > 1 object sinks under water

ρ < 1 object floats over water


Pressure

A

Fp

Useful Units :

1 Pa = 1 N/m2

1 atm = 101 325 Pa = 760 Torr =1013 mbar

• Fluid exerts a force at each point on the surface of an object in

contact with it.

• Force is perpendicular to the object surface

• Pressure has no preferred direction (scalar)

Chapter 14Chapter 14 Fluid pressure 7Fluid pressure 7

At equilibrium, the pressure in a fluid of uniform density depends depends only on the depthonly on the depth, NOT THE SHAPE, of the container.

ghpp 0

p = pressure at a some depth h

po = pressure at the surface (or the atmosphere)

= density of the fluid

gh = gauge pressure

* With the assumption that g is uniform all throughout the fluid.

• Pressure below > Pressure above

• Pressure is the same at all points at the same depth of the fluid.


For a homogeneoushomogeneous fluid in an open container, the pressure is the same at a given depth independent of the container’s shape.

p(y)

y

Differences in fluid pressure at the same elevation will arise only if the densities are different.


Examples:• Ear-popping

• SCUBA

• Using a sphygmomanometer

• Bath tub vs. Pitcher


Example:• A U tube contain immiscible liquids of

density 1 and 2. Compare the densities of the liquids.

1

2

d

dhgpp 10

At the bottom, both liquids have the same pressure.At the top, both are in equilibrium with the atmosphere.At the interface, both have the same pressure as well.So from the Pressure-Depth relation:

ghpp 20

h

dhgpghp 1020

dhh 12 12 h

dh 12

hdh

Chapter 14Chapter 14 Fluid pressure variation w/ altitude 12Fluid pressure variation w/ altitude 12

WHY?(i) The gravitational force on air

molecules is greater for those near the earth’s surface, dragging them closer together and increasing the pressure between them.

(ii) Molecules further away from the earth have less weight but exert compressive force on those below them. In turn, those lower down have to support more molecules above them and are further compressed in the process.

o

o

g

hp

epp

0

Mt. Everest

Commercial jet cruising altitude


Problem set 11.1:(a) What water pressure would a diver experience at a depth of 200

m? Express your answer in atmospheres.

(b) Find the total weight of water on top of a nuclear submarine at this depth, assuming that its horizontal cross-sectional hull area is 3000 m2.

(Assume that the density of sea water is 1.03 g/cm3)

(a) What water pressure would a diver experience at a depth of 200 m? Express your answer in atmospheres.

(b) Find the total weight of water on top of a nuclear submarine at this depth, assuming that its horizontal cross-sectional hull area is 3000 m2.

(Assume that the density of sea water is 1.03 g/cm3)

Chapter 14Chapter 14 Pressure applied to fluid 14Pressure applied to fluid 14

Pascal’s Principle

“PPressure applied to an enclosed fluid is transmitted

undiminished to every portion of the fluid and to the walls of the

containing vessel.”

By adding more weight at the top, the pressure also increases proportionally within the fluid.

Chapter 14Chapter 14 Pascal’s principle 15Pascal’s principle 15

Applications:• Heimlich maneuver

• Squeezing the end of toothpaste tube

• Hydraulic lift


Application: Hydraulic lift

Small applied force, F1

A1

1

1

A

Fp

p is transmitted through the larger piston

11

22 F

A

AF

Larger than F1

2

2

A

Fp

1

1

A

F


Example:

A1A2 = 20m2

200 kg40 kg

Chapter 14Chapter 14 Measuring pressure 18Measuring pressure 18

Pressure measurements are always done with respect to the pressure of the surroundings.

atmabsolutegauge PPP

Pabsolute = total pressure

Patm = atmospheric pressure

Pgauge = pressure excess of atmospeheric


Open-tube manometer• Measures gauge pressure directly

21 gyPgyP atm

ghyygPP atm )( 12

ghPgauge


Example:A manometer tube is partially filled with mercury. Water is then

poured into the left arm of the tube until the mercury-water interface is at the midpoint. Both arms of the tube are open to air. Find the relationship between hH2O and hmercury.

hH2O hmercury


Example:A barrel contains a 0.120 m-layer of oil floating on water that is 0.250 m-deep. The density of the oil is 600 kg/m3.

(a)What are the gauge and absolute pressures at the oil-water interface?

(b)What are the gauge and absolute pressures at the bottom of the barrel?


Buoyancy• Apparent weight loss of an object when totally/partially

immersed in a fluid

FB

Lower Pressure

Higher Pressure

mg

Chapter 14Chapter 14 Buoyancy 23Buoyancy 23

Archimedes’ Principle

““WWhen a body is fully or partially submerged in a fluid, a hen a body is fully or partially submerged in a fluid, a buoyant force from the surrounding fluid acts on the buoyant force from the surrounding fluid acts on the body.” body.”

““TThe buoyant force is directed he buoyant force is directed UPWARDUPWARD and has a and has a magnitude equal to the magnitude equal to the WEIGHTWEIGHT of the of the displaced FLUIDdisplaced FLUID by the body.”by the body.”

• The line of action of FB passes through the CoG of the displaced fluid, which doesn’t necessarily coincide with the CoG of the submerged object


Example: Compare the magnitude of tension on the string for the three cases.

A B C

m

m

m

ρf


Question:Based on the summation of forces, therefore, what makes an object sink, float or hover?

Sink(accelerate downwards)

Float(accelerate upwards)

Hover(stay at the same level)

objB WF

objB WF

objB WF

objf

objf

objf


Examples:• Fishes and their air sacs

• Life vests

• Ice cubes

• Boats/ships


Example:What fraction of the iceberg afloat in seawater is visible from

the surface?

VVicebergiceberg = total volume of iceberg = total volume of iceberg

VVfluid,disp.fluid,disp. = equal to the submerged portion of the iceberg = equal to the submerged portion of the iceberg

= V= Vsubsub


Example:You have found a treasure chest afloat

at sea! To keep it, though, from other

pirates coming your way, you jumped on

the water and stood on top of the chest.

What should your mass be to be able

to keep the chest totally submerged

in water while keeping you afloat?


Problem set 11.2:Three children, each of weight 36 kg, make a log raft by lashing together logs of diameter 0.3 m and length 1.8 m. How many logs will be needed to keep them afloat? Consider the density of wood to be 842 kg/m3.

Density of water = 1 x 103 kg/m3

Minimum requirement is for the logs to be completely submerged, but the children standing on them are not.

Chapter 14Chapter 14 What’s in store for us? 30

What’s in store for us? 30

Fluid dynamics• We will only consider fluids that are:

• Non-viscous

(no internal friction)

• Incompressible

(constant density)

• Steady/ non-turbulent

(P, V and flow velocity are

constant in time)

Chapter 14Chapter 14 Fluid dynamics 31Fluid dynamics 31

Continuity equationIdeal fluids obey continuity equation. Ideal fluids obey continuity equation. Conservation of Mass: “What goes in comes out”Conservation of Mass: “What goes in comes out”

222111 vAvA (Mass Flux of (Mass Flux of Compressible Fluids)Compressible Fluids)

2211 vAvA (Incompressible Fluids)(Incompressible Fluids)

21

21 v

A

Av

The narrower the The narrower the constriction (area), the constriction (area), the

faster a fluid flows through faster a fluid flows through it!it!

Volume Volume flow rateflow rate

Chapter 14Chapter 14 Continuity equation 32Continuity equation 32

Examples:• “ Still waters run deep”

• Necking down of water from faucets

• A housing contractor saves some money by reducing the size of a

pipe from 1” diameter to 1/2” diameter at some point in your

house.

Assuming the water moving in the pipe is an ideal fluid (incompressible),

relative to its speed in the 1” diameter pipe, how fast is the water going in

the 1/2” pipe? v1 v1/2

Chapter 14Chapter 14 Fluid dynamics 33Fluid dynamics 33

Bernoulli’s equationAn ideal fluid flowing through a pipe may change its motion

depending on the:(i) cross-section area of the pipe, (ii) elevation of the inlet and outlet, and

(iii) variation in pressure between inlet and outlet

y1

y2

A1

A2v1

v2

P1, V1

P2, V2

Chapter 14Chapter 14 Bernoulli’s equation 34Bernoulli’s equation 34

Based on Energy Conservation:Based on Energy Conservation:

22222211

211 2

1

2

11

gyvpgyvp

constant2

1 2 gyvp (For ideal (incompressible) fluid)


Case 1. Static pressure: No flow (but with elevation change)Case 1. Static pressure: No flow (but with elevation change)

22221

211 2

1

2

1gyvpgyvp

1212 yygpp

Same with what we have derived before for static fluid.Same with what we have derived before for static fluid.

i.e. since the pressure at a higher elevation yi.e. since the pressure at a higher elevation y22 is less than is less than

at lower depth yat lower depth y11

=0


Case 2. Dynamic pressure: with flow (but no elevation change)Case 2. Dynamic pressure: with flow (but no elevation change)

22221

211 2

1

2

1gyvpgyvp

Implication: Implication: Where the Where the SPEEDSPEED is is LARGELARGE, the , the PRESSUREPRESSURE must be must be SMALLSMALL!!!!!!

222

211 2

1

2

1vpvp

y2 – y1 =0


Examples:• Blowing on top of paper

• Roofs flying off houses during storms

• MRT’s yellow line

• Airplane wings

higher velocity lower pressure

lower velocityhigher pressure


Example:

A B C D

Vol flow rate, y, ρ, g

Cross-sectional area

Speed

Pressure

DCBA DCAB

BCAD

DCAB


Example:Water circulates throughout a house in a hot-water heating system. If water is pumped out at a speed of 0.50 m/s through a 4.0 cm diameter pipe in the basement under a pressure of 3.0 atm, what will be the flow speed and flow speed and pressure pressure in a 2.6 cm diameter pipe on the second floor 5.0 m above assuming that the pipes do not divide into branches.


Problem set 12.1: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 out of the hole.

h

yB

yA

Hints:

Cross-sectional area of the flow tube at A << area at B

Velocity at A is negligible

A

B

Chapter 14

Documents

fluid pressure p

water pressure

measuring pressure

total pressure

atmospheric pressure

pressure measurements

pressure f p

object surface pressure