11

Chapter 14Chapter 14FluidsFluids

流體流體

22

14.2 What is a Fluid?14.2 What is a Fluid?

•A fluid, in contrast to a solid, is a substance that can flow.

•Fluids conform to the boundaries of any container in which we put them. They do so because a fluid cannot sustain a force that is tangential to its surface. That is, a fluid is a substance that flows because it cannot withstand a shearing stress.

•It can, however, exert a force in the direction perpendicular to its surface.)

33

Stresses in Solids

The level of stress required to obtain a givenThe level of stress required to obtain a given

deformation ※ ※Tensile stress(Tensile stress( 拉伸應力拉伸應力 )) ，， Tensile strain (Tensile strain ( 拉伸應變拉伸應變 ))

and and Young’s modulus (Young’s modulus ( 楊氏模數楊氏模數 ))

※ ※Shear stress(Shear stress( 剪應力剪應力 )) ， ， Shear strain (Shear strain ( 剪應變剪應變 )) and and

Shear modulus (Shear modulus ( 剪力模數剪力模數 ))

※ ※ Volume stress(Volume stress( 體積應力體積應力 )) ，， Volume strain (Volume strain ( 體積應變體積應變 ))

and Bulk modulus (and Bulk modulus ( 體積彈性模量體積彈性模量 ))

12.7: Elasticity

A stress is defined as deforming force per unit area, which produces a strain, or unit deformation.

Stress and strain are proportional to each other. The constant of proportionality is called a modulus of elasticity.

55

Strain unit deforTensile = matio n

= L

L

2

Normal forcstre

e NTensile = =

Ass ( S )

rea m

F

A

Tensile stressTensile stress(( 拉伸應力拉伸應力 )) ，， Tensile strain Tensile strain (( 拉伸應拉伸應變變 ) and ) and Young’s modulusYoung’s modulus ( ( 楊氏模數楊氏模數 ))

Tensile stress Young's modulus ( E ) =

Tensile strain

F /AE =

ΔL/L

66

Stress-strain relationship

A stress–strain curve for a steel test specimen. The specimen deforms permanently when the stress is equal to the yield strength of the specimen’s material. It ruptures when the stress is equal to the ultimate strength of the material.

77

Thin cylinder under internal pressure

Resisting force F S 2 { S Longitudinal Tensile stress }R 1 1

2Pressure force F { Pressure in the tube along the longitudinal axis }

p

Rt

p R p

{ Shear stress at wall }

2 S 2 { For equilibrium }1

S 1 2

p

Rt p R

pRt

88

FR = S2 Lt

F = p L2R

FR = S2 Lt

p

p

Resisting force F S 2 { S Transverse Tensile stress }R 2 2

Pressure force F 2

S 2 2 { For equilibrium }2

S 2

Lt

p RL

Lt p RL

pRt

99

Shear stressShear stress(( 剪應力剪應力 )) ，， Shear strain Shear strain (( 剪應變剪應變 )) and and Shear modulus Shear modulus (( 剪力模數剪力模數 ))

Shear stress

Shear st ( G )

i=

ra n

F/AShear modulus

/Lx

2

Tangential force N = =

Area m

FShear stress

A

= Fractional change in length

=

Shear strain

L

xL

1010

血液動力學中的剪力血液動力學中的剪力

3

4 Q

R

血管栓塞

動脈硬化

1111

Volume stress(( 體積應力體積應力 )) ， Volume strain (( 體積應體積應變變 )) and Bulk modulus (( 體積彈性模量體積彈性模量 ))

2

force NVolume stress ( change in pressure ) =

Area m

P = F

A

Voulme strain = Fractional change in volume

= V

V

Volume stressBulk modulus ( B ) =

Volume strain

> 0 Generally < 0 /

- P VB

V V P

Note : Compressibility ( k ) = 1 / B

1212

YYoung’soung’s modulusmodulus ( ( 楊氏模數 )) ，， SShear modulushear modulus ( ( 剪力模數 ) ) ，，BBulk modulusulk modulus ( ( 體積彈性模量 ) )

1313

Stresses in FluidsStresses in Fluids Normal stress ( pressure )Normal stress ( pressure ) Tangential or shearing stressTangential or shearing stress shear the fluid particle and deform its shapeshear the fluid particle and deform its shape

1414

The Nature of FluidsThe Nature of Fluids

The fluids The fluids cannot cannot support support Tensile stressesTensile stresses and and Shear stresses Shear stresses ..

The fluids flow and The fluids flow and deform continuously deform continuously and permanently under and permanently under Shear stresses Shear stresses ..

1515

Fluid Mechanics Fluid Mechanics (Liquids and Gases)(Liquids and Gases)

Fluid StaticsFluid Statics Fluid PropertiesFluid Properties Density (ρ=m/vDensity (ρ=m/v ）） Pressure Pressure （（ P=F/AP=F/A ）） Variation of Pressure with DepthVariation of Pressure with Depth Pascal’s PrinicplePascal’s Prinicple Archimedes’ PrincipleArchimedes’ Principle

20A

2

m

N

A

FlimP

1m

1N)1Pa(Pascal

14.3 Density and Pressure

To find the density of a fluid at any point, we isolate a small volume element V around that point and measure the mass m of the fluid contained within that element. If the fluid has uniform density, then

Density is a scalar property; its SI unit is the kilogram per cubic meter.

If the normal force exerted over a flat area A is uniform over that area, then the pressure is defined as:

The SI unit of pressure is the newton per square meter, which is given a special name, the pascal (Pa).

1 atmosphere (atm) = 1.01x105 Pa =760 torr =14.7 lb/in.2.

2

1N 1Pa(Pascal

m )

1

2A 0

NFP lim

A m

1717

Fig. 15-3, p.466

14.4: Fluids at Rest

Above: A tank of water in which a sample of water is contained in an imaginary cylinder of horizontal base area A.

Below: A free-body diagram of the water sample.

The balance of the 3 forces is written as:

If p1 and p2 are the pressures on the top and the bottom surfaces of the sample,

Since the mass m of the water in the cylinder is, m =V, where the cylinder’s volume V is the product of its face area A and its height (y1 -y2), then m =A(y1-y2). Therefore,

If y1 is at the surface and y2 is at a depth h below the surface, then

(where po is the pressure at the surface, and p the pressure at depth h).

The pressure at a point in a fluid in static equilibrium depends on the depth of that point but not on any horizontal dimension of the fluid or its container.

F2

F1

1919

14.5: Measuring Pressure: The Mercury Barometer

Fig. 14-5 (a) A mercury barometer. (b) Another mercury barometer. The distance h is thesame in both cases.

A mercury barometer is a device used to measure the pressure of the atmosphere. The long glass tube is filled with mercury and the spaceabove the mercury column contains only mercury vapor, whose pressure can be neglected.If the atmospheric pressure is p0 , and is the density of mercury,

(Pascal) Pa101.013

m

N101.013

0.76ms

m9.8

m

Kg1013.6

m

N10760g

760mmHg

e)(atmospher 1atm

5

25

233

23

Hg

2020

14.5: Measuring Pressure: The Open-Tube Manometer

An open-tube manometer measures the gauge pressure pg of a gas. It consists of a U-tube containing a liquid, with one end of the tube connected to the vessel whose gauge pressure we wish to measure and the other end open to the atmosphere.

If po is the atmospheric pressure, p is the pressure at level 2 as shown, and is the density of the liquid in the tube, then

2121

Blood pressure is measured using an inflatable cuffBlood pressure is measured using an inflatable cuff

systolic

diastolic

Pgauge (mmHg)

t

2222

2323

Pressure Change within a Gas Change within a Gas

Mt.Everest

10km

20km

30km

40km

90% of the Atom is below here

99% of the Atom is below here

Ozone layer

50% of the atmosphere is below 5.6km

Atmospheric pressure as Atmospheric pressure as a function of a function of elevation?elevation?

2424

P＋△ P

y＋△ y△ y

Py

P0

0 0

0

0

PV nRT

mRT M

Mm RT

PV M

RTP air density

MIf T constant P

P sea level

y P yP

空氣之分子量

Ideal gasIdeal gas

PP00≡Atmospheric ≡Atmospheric

Pressure at sea levelPressure at sea level

2525

P

y

P0

0

0

0

air

air

air

0

0

P y y0

0P y 0

gyP

0

P g y

uniform

P g y

P y g yP

dPg dy

P P

P y P e

2626

14.6: Pascal’s Principle

A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.

2727

14.6: Pascal’s Principle and the Hydraulic Lever

The force Fi is applied on the left and the downward force Fo from the load on the right produce a change p in the pressure of the liquid that is given by

If we move the input piston downward a distance di, the output piston movesupward a distance do, such that the same volume V of the incompressible liquid isdisplaced at both pistons.

Then the output work is:

2828

Fig. 15-5, p.468

2929

1 1

2 2

2 2 1 1

2 1

b

Vertical direction

F P A

F P A

P gy P gy

(Net force) B P -P A

g hAV

f f

f

blockBuoyant Force B gVf

浮力

Net Force on the block = ?Net Force on the block = ?

P1

P2

h

A

A

y2y1

1F

2F

h

Vb : Volume of the fluid that has been displaced by the body

14.7: Archimedes Principle: Floating and Apparent Weight

When a body floats in a fluid, the magnitude Fb of the buoyant force on the body is equal to the magnitude Fg of the gravitational force on the body.

That means, when a body floats in a fluid, the magnitude Fg of the gravitational force on the body is equal to the weight mfg of the fluid that has been displaced by the body, where mf is the mass of the fluid displaced.

That is, a floating body displaces its own weight of fluid.

The apparent weight of an object in a fluid is less than the actual weight of the object in vacuum, and is equal to the difference between the actual weight of a body and the buoyant force on the body.

3131

HomeworkHomework

Chapter 14 ( page 378 )Chapter 14 ( page 378 )

11 , 18, 24, 27, 29, 32, 47, 49, 62, 63, 65, 6611 , 18, 24, 27, 29, 32, 47, 49, 62, 63, 65, 66

67, 71, 82, 83, 8467, 71, 82, 83, 84

3232

Fluid DynamicsFluid Dynamics

Fluid FlowFluid Flow Laminar Flow (Steady Flow)Laminar Flow (Steady Flow)

Flow RateFlow Rate Bernoulli’s Equation and ApplicationBernoulli’s Equation and Application Viscosity and Laminar FlowViscosity and Laminar Flow

(Poiseuille’s Law)(Poiseuille’s Law) Flow in the Circulatory SystemFlow in the Circulatory System

Turbulent FlowTurbulent FlowThe Onset of TurbulenceThe Onset of Turbulence

3333

Fluid in MotionFluid in Motion (Fluid Dynamics) (Fluid Dynamics)

Fluid velocity Fluid velocity

Steady flow (laminar flow) : Steady flow (laminar flow) : the fluid velocity at any point is independent of the fluid velocity at any point is independent of timetime

Non-steady flow (turbulent) : not steady flowNon-steady flow (turbulent) : not steady flow

tz,y,x,v

3434

x y z

yx zx y z

v ( x + x , y + y , z + z ) - v (x, y, z)Acceleration a =

t

But x = v t , y = v t , z = v t

vv vv ( x + x , y + y , z + z ) = v (x, y, z) + ( )(v t) + ( )(v t) + ( )(v t)

x y z

yx zx y z

v + ( )( t)

t vv v v

a = ( )(v ) + ( )(v ) + ( )(v ) + ( )x y z t

v a = ( v ) v +

t

3535

Stream lineStream line

““No two stream lines No two stream lines ever cross.”ever cross.”

Stream lines are Stream lines are parallel to each otherparallel to each other

Fluid does not cross Fluid does not cross the side of boundarythe side of boundary

Stream line: Path of the fluid particle in motionStream line: Path of the fluid particle in motion（流線）（流線）

Steady flow (laminar flow)

3636

Fig. 15-15, p.476

3737

Fig. 15-13, p.475

3838

Fig. 15-14, p.476

3939

The steady The steady ( ( laminar ) flow patternlaminar ) flow pattern

4040

Compressible and Incompressible FlowCompressible and Incompressible Flow Incompressible: density of fluid is independent Incompressible: density of fluid is independent

of both position and time (of both position and time (ex : ex : liquid)liquid) Compressible: GasesCompressible: Gases

Rotational and Irrotational FlowRotational and Irrotational FlowRotational: non-zero angular velocityRotational: non-zero angular velocity

4141

Circulation and Vortex lines

4242

Steady flowSteady flow Incompressible → density of fluid is constantIncompressible → density of fluid is constant IrrotationalIrrotational Nonviscous flow → no frictionNonviscous flow → no friction

(no relative velocity between flow tubes)(no relative velocity between flow tubes)

Ideal Fluid Flow

4343

time

volumeconstantAv rate Flow

The fluid flow The fluid flow into the tube at into the tube at “1” with Δt“1” with ΔtΔmΔm11=ρ=ρ11ΔvΔv11=ρ=ρ11vv

11ΔtAΔtA11

Steady flowSteady flowΔmΔm11=Δm=Δm22

ρρ11vv11AA11Δt=ρΔt=ρ22vv22AA22ΔtΔt

(incompressible fluid (incompressible fluid ρρ11=ρ=ρ22))

AA11vv11=A=A22vv22

Equation of continuityEquation of continuity

The fluid flow The fluid flow out the tube at out the tube at “2” with Δt“2” with ΔtΔmΔm22=ρ=ρ22ΔvΔv22=ρ=ρ22vv

22ΔtAΔtA22

4444

dt vsd

APF

22

222

dt vsd

APF

11

111

Bernoulli’s Equation

4545

時間時間 dtdt 期間，流體柱 期間，流體柱 ac (t) → bd (t+dt)ac (t) → bd (t+dt) SteadySteady IncompressibleIncompressible IrrotationalIrrotational NonviscousNonviscous

Bernoulli’s EquationBernoulli’s Equation

22222

211111 v

2

1gyPv

2

1gyP

4646

1 2F F

1 1 1 1 2 2 2 1 1 1 2 2 2

2 2 1

1 1 1 2 2 2

t dt t bd ac

2 22 2 1 1

2 2 2 2 1 1 1 1

W F d Δ K E

i W P A ds P A ds P A v dt P A v dt

ii W mg y y

m A v dt A v dt

iii Δ K.E K.E K.E K.E K.E

1 1 m v m v

2 2 m A v dt ; m A v dt

i

由 1 1 1 2 2 2 2 2 2 2 1 1 1 1

2 22 2 2 2 1 1 1 1

ii iii P A v dt P A v dt A v dtgy A v dtgy

1 1 A v dt v A v dt v

2 2

Work-Energy TheoremWork-Energy Theorem

4747

Bernoulli’s Equation

22222

211111 v

2

1gyPv

2

1gyP

TheThe Venturi Tube Venturi Tube ( 文氏管 )

4848

2 21 2 L 2 1

1P P (v v ) ;

2 1 1 2 2A v = A v

2 22 1 2

1 2 L 2 21

A A1P P v ( )

2 A

4949

vL > vH PL < PH

VH

VL

2 2H H H L L L

1 1P gy v P gy v

2 2 H H L L

AHvH = ALvL

AH > AL vL > vH

Venturi effect

5050

High speed , Low Pressure regions

Low speed , High Pressure regions

AirfoilStream (Flow) lines are closer together

5151

Motion of air layer due to the viscosityMotion of air layer due to the viscosity

Spinning ballSpinning ball

Low speed , High Pressure regions

High speed , Low Pressure regions

5252

Measurement of fluid velocity based on Measurement of fluid velocity based on Bernoulli’s principleBernoulli’s principle

5353

C D C 1 air 1 D 2 air 2 Hg

1 air 1 2 air 2 Hg

1 2 Hg air

2 21 air 1 air 1 2 air 2 air 2

21 2 air 2 air 2 1

1 222

air

Hg22

air

P P ; P P gh ; P P gh gh

P gh P gh gh

P P gh gh

1 1P v gy P v gy

2 20

1P P v g y y

2 0

2 P Pv

2 ghv 2

gh

5454

2atm D

1P + 0 + v constant

2

2A A

1P gd v

2 constant

2B 1 B

1P gh v

2 constant

D

2C 2 C

1P g(d+h ) v

2 constant

C atm

C D

C 2

B

P P

v

v v >> v

2g(d+h )

maxC atm 1

2atm B

1

atm1,max

2v = (P gh )

0

P vh

g 2g

Ph

g

BP

Siphon( 虹吸管 )

maxc

max1B h ?

v

?

點高度

5555

Real fluid has a certain amount of internal friction, which is called viscosity.

Viscosity exists in both liquids and gases, and is essentially the frictional force between the adjacent layers of fluid as the layers move past one another.

In liquids, viscosity appears due to the cohesive forces between the molecules. In gases, it arises from collisions between the molecules.

黏滯力 黏滯力 ( ( Viscosity )

5656

b c

A a

y1 y2

h

P1 P2

ρ

ρg

1 2 gP P ( )gh

2 21 2 g 2 1

1P P (v v ) ;

2 1 2Av = av

黏滯力 黏滯力 ( ( Viscosity )

5757

Experiment setup for obtaining of viscosity coefficient

5858

A thin layer of fluid is placed between two flat plates. One plate is static and the other is made to move . The fluid directly in contact with each plate is held to the surface by the adhesive force between the molecules of the liquid and those of the plate. Thus the upper surface of the fluid moves with the same speed v as the upper plate, whereas the fluid in contact with the stationary plate remains stationary. The stationary layer of fluid retards the flow of the layer just above it, which in turn retards the flow of the next layer, and so on.

Experiment setup for obtaining of viscosity coefficient

5959

For a given fluid, it is found that the required force F, is proportional to the area of a fluid in contact with each plate A, and to the speed v, but is inversely proportional to the separation , of the plates, what comes down to the following relation

Experiment setup for obtaining of viscosity coefficient

Velocity gradient v

Av

F

6060

The Nature of FluidsThe Nature of Fluids

The fluids The fluids cannot cannot support support Tensile stressesTensile stresses and and Shear stresses Shear stresses ..

The fluids flow and The fluids flow and deform continuously deform continuously and permanently under and permanently under Shear stresses Shear stresses ..

6161

x

x

x

dvF /A )

dy

dv F/A

dy

A:area of moving plate

:coefficient of viscosity

F A

dv dy

切液體 剪應 ( 力 剪應變 率

剪應 力

流體黏滯性

6262

2

2

v AvF A ; F ; F

y y

A:area of moving plate

:coefficient of viscisity

F A

v y

NN-sm SI unit

m 1 ms m

6363

dvdy

Fluid Temperature [oC] Viscosity [Pa.s]

Water 0 1.8 .10-3

Water 20 1.0.10-3

Water 100 0.3.10-3

Ethyl alcohol 20 1.2.10-3

Engine oil 30 200.10-3

Air 20 0.018.10-3

Hydrogen 0 0.009.10-3

Water vapour 100 0.013.10-3

The viscosity coefficients of various fluids at the specified temperatures 玉米粉漿、

矽酸鉀溶液

紙漿、乳膠漆、血漿、糖漿、髮膠

6464

Viscosity Shear Stress ≡ Change in momentum in bulk fluids

Viscosity Stresses tend to decrease the velocity of the flow on the high speed side of the layer, increase the velocity on the low speed side.

黏滯應力 黏滯應力 ( ( Viscosity Stress )

Momentum exchange by molecular mixing

6565

A shear layer near a solid wallA shear layer near a solid wall

Velocity profile in the region near a solid surface.

uyyxdd

du

6666

黏滯性黏滯性

1)1) What is the velocity What is the velocity distribution across the distribution across the radius?radius?

2)2) What is the flow rate? What is the flow rate? (Poiseuille’s Law)(Poiseuille’s Law)

3)3) Human & Biological Human & Biological ApplicationApplication

Laminar flow of viscous fluids in a Laminar flow of viscous fluids in a cylindrical pipecylindrical pipe

6767

Laminar flow of viscous fluids in a Laminar flow of viscous fluids in a cylindrical pipecylindrical pipe

0vWall

6868

6969

0dr

dvrL 2rF0

dr

dvdr

dvrL 2

dr

dvArF

f

f

Rr r PPrF 221p

Viscous force on the fluid of the cylindrical section Viscous force on the fluid of the cylindrical section by the surrounding fluid isby the surrounding fluid is

External “driving force” on this cylindrical sectionExternal “driving force” on this cylindrical section

7070

dr

dvrL 2r PP ; rFrF 2

21fp In steady flow:In steady flow:

1 2

1 2

v r' r r'1 2

v R 0 r R

r r'21 2

0r R

P Pdv 1 r

dr 2 L

P P1 dv - rdr

2 L

P P1dv - rdr

2 L

P P1 rv r' v R -

2 L 2

1 2 2 2P P1 v r' R r'

4η L

7171

1 2 2 2

1 2 2max

Q Av

dQ v r 2 rdr

P P1v r R r

4 L P P1

v v r 0 R4 L

π

η

η

Flow rateFlow rate

r

dr

7272

r

R dr

1 2 2 2

r R r R1 2 2 2

r 0 r 0

r R r R1 2 2 3

r 0 r 0

P P1dQ R r (2 r dr)

4 L

π P P Q dQ R r rdr

2 L

P P R rdr r dr

2 L

π

π

4 4R R

2 4

Poiseuille's Law

4

1 2P P1 RQ

8 LηV

IR

7373

Q PP Av PP Fvt

Fx

t

WP 2121ower

421 R

1PP

1P

2P

R

Power of the driving sourcePower of the driving source

For constant Flow rate QFor constant Flow rate Q

7474

Show that wall shear stress Show that wall shear stress

3

4 Q

R