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Chapter 6

Flow Analysis Using Differential Methods(Differential Analysis of Fluid Flow)

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In the previous chapter-- Focused on the use of finite control volume for the

solution of a variety of fluid mechanics problems. The approach is very practical and useful since it

doesnt generally require a detailed knowledge of thepressure and velocity variations within the control

volume. Typically, only conditions on the surface of the

control volume entered the problem.

There are many situations that arise in which thedetails of the flow are important and the finitecontrol volume approach will not yield thedesired information

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For example --

We may need to know how the velocity varies over the cross

section of a pipe, or how the pressure and shear stress vary

along the surface of an airplane wing.

we need to develop relationship that apply at a point,

or at least in a very small region ( infinitesimal volume)within a given flow field.

involve infinitesimal control volume (instead of finite

control volume)

differential analysis (the governing equations are

differential equation)

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In this chapter

(1) We will provide an introduction to the differential

equation that describe (in detail) the motion of fluids.

(2) These equation are rather complicated, partial differential

equations, that cannot be solved exactly except in a few

cases.

(3) Although differential analysis has the potential for

supplying very detailed information about flow fields, the

information is not easily extracted.

(4) Nevertheless, this approach provides a fundamental basis

for the study of fluid mechanics.

(5) We do not want to be too discouraging at this point,

since there are some exact solutions for laminar flow that

can be obtained, and these have proved to very useful.

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(6) By making some simplifying assumptions, many otheranalytical solutions can be obtained.

for example , small 0 neglected

inviscid flow.

(7) For certain types of flows, the flow field can be conceptuallydivided into two regions

(a) A very thin region near the boundaries of the system inwhich viscous effects are important.

(b) A region away from the boundaries in which the flow isessentially inviscid.

(8) By making certain assumptions about the behavior of the fluidin the thin layer near the boundaries, and

using the assumption of inviscid flow outside this layer, a large

class of problems can be solved using differential analysis .

the boundary problem is discussed in chapter 9.

Computational fluid dynamics (CFD) to solve differential eq.

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)(.)((.)(.)(.)(.)(.)(.)

.

.

problem.particularafort,z,y,on x,dependlyspecificalcomponents

velocitythesehowdeterminetoisanalysisaldifferentiofgoalstheofOne

vtz

wy

vx

utt

Da

a

z

uw

y

uv

x

uu

t

ua

dt

vd

z

vw

y

vv

x

vu

t

va

z

y

x

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elementtheofndeformatioangularx

v

y

u

elementtheofndeformatiolinear

z

w

y

v

x

uNote

v

ratedilationvolumetricvz

w

y

v

x

u

dt

d

,

,,:

fluid,ibleincompressanfor0

)(1

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flowalIrrotation0

zeroisaxis-zthearoundRotation

only when)(i.e.blockundeformedanasaxis-zaboutRotation

)(2

1assuch(6.12)Eq.From

2Define

vor

y

u

x

v

x

v

y

uww

y

u

x

vw

Vcurlvwvorticity

OBOA

z

Vcurlv

wvuzyx

kji

k

y

u

x

vj

x

w

z

ui

z

v

y

w

kwjwiw

vectorrotationtheW

zyx

2

1)(

2

1

2

1

})()(){(

2

1

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9

6.2.1 Differential Form of Continuity Equation

inout

cv

vAvA

dAnv

zyx

t

d

t

d

t

zyxd

)()(

elementtheofsurfacesthethroughflowmassofrateThe)(

)(

0

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s

yx

xx

xxx

x

uuu

y

x

uuu

)(,)(assuch,terms

orderhighneglecting--expansionseriesTaylor

2

)(|

2

)(|

direction-xtheinflowThe

2

2

2

zyxz

w

zyxyv

zyxx

uzy

x

x

uuzy

x

x

uu

)(direction-zinrateNet

)24.6()(direction-yinrateNet

similarly

)23.6()(

]2

[]2

[

direction-in xoutflowmassofrateNet

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.formaldifferentiinequationcontinuityThe

)27.6(0

outflowmassofrateNet:

0][

0)(Since

z

w

y

v

x

u

t

zyxz

wzyx

y

vzyx

x

uNote

zyxz

wzyx

y

vzyx

x

uzyx

t

dAnvd

t cscv

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12

)31.6(0

)30.6(0

0

flowibleincompressFor--

)29.6(0

0)(

fluidlecompressibofflowsteadyFor--

)28.6(0

formIn vector

mechanicsfluidofequationslfundamentatheofOne--

z

w

y

v

x

uor

vt

const

z

w

y

v

x

uor

v

vt

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equation.continuityhesatisfy ttorequired,w:Determine

?

flowibleincompressanFor

26.Example

222

w

zyzxyv

zyxu

),(2

3nIntegratio

3)(2

0)()(

0

continuityofequationthefrom:Solution

2

222

yxczxzw

zxzxxz

w

zwzyzxy

yzyx

x

z

w

y

v

x

u

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6.2.2 Cylindrical Polar Coordinates

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01)(1

)flowunsteadyorsteady(flowibleincompressFor

0)()(1

)(1

flowlecompressibsteady,For

scoordinatelcylindricain

equationycontinualltheofformaldifferentitheisThis

)33.6(0

)()(1)(1

z

vv

rr

rv

r

vz

vr

vrrr

z

vv

rr

vr

rt

zr

zr

zr

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6.2.3 The Stream Function

)36.6(0

)2(0)(0

flowD-2&plane,ible,incompresssteady,ofequationcontinuityFor the

0

equationContinuity

y

v

x

u

flowDz

wcte

twhere

z

w

y

v

x

u

t

0)()(

eq.continuitythesatisfiesitthatso;where

function,streamthe),(functionaDefine

xyyxy

v

x

u

xv

yu

yx

satisfiedbewillmassofonconservati

unknowoneunknowstwofunction

streamusing

v

u

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6.3.Example

)42.6(1

01)(1

0)()(1)(1

flow.D-2place,,ibleIncompress

forequationycontinuallthe,scoordinatelcylindricaIn

rv

rv

v

rr

rv

rz

vv

rr

vr

rt

r

rzr

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Examp le 6.3 Stream Func t ion

The velocity component in a steady, incompressible, twodimensional flow field are

Determine the corresponding stream function and show on asketch several streamlines. Indicate the direction of glow alongthe streamlines.

4xv2yu

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Example 6.3 Solution

(y)fx2(x)fy 2212

Cyx222

From the definition of the stream function

x4x

vy2y

u

For simplicity, we set C=0

22yx2

=0

01

2/

xy22

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6.3 Conservation of Linear Momentum

amF

amFordt

vdmFEq

VCsmallFor

FdAnvvdvtdt

vmD

dmvdvP

Pdt

Ddv

dt

D

dt

vmD

sys

cvcv

csvc

cv

sys

syssys

sys

sys

systemaforlaw2ndNewtonsThe

)44.6(

..

)44.6()()(

momentumlinearfor thet theoremtransporReynoldstheFrom

where

)(

momentumlinearFor the

..

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Figure 6.9 (p. 287)Components of force acting on an arbitrary differential area.

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Figure 6.10 (p. 287)Double subscript notation for stresses.

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Figure 6.11 (p. 288)Surface forces in the x direction acting on a fluid element.

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6.3.2 Equation of Motion

Velocitiesstresses-----Unknowns

.restatormotioninfluid)or(solidcontinuumanytoapplicablealsoareThey

fluid.aformotionofequationaldifferentiGeneral

)50.6()(

)50.6()(

)50.6()(

using

czww

ywv

xwu

tw

zyxg

bz

vw

y

vv

x

vu

t

v

zyxg

azuw

yuv

xuu

tu

zyxg

dzyxm

zzyzxzz

zyyyxy

y

zxyxxxx

zszbzzz

ysybyxx

xsxbxxx

maFFmaF

maFFmaF

maFFmaF

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6.4 Inviscid Flow

stressnormalecompressiv

0&

0flowinvisicidFor

.ssfrictionleor,nonviscous,inviscidbetosaidare

negligiblebetoassumedarestressesshearingthein whichfieldFlow

zzyyxxP

0&0&&,waterandairassuch,fluidcommonSome

waterair

waterairsmall

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6.4.1 Eulers Equations of Motion

)51.6()(

)51.6()(

)51.6()(

00with(6.50c)&(6.50b)(6.50a)EqFrom

cz

ww

y

wv

x

wu

t

w

z

Pg

bzvw

yvv

xvu

tv

yPg

az

uw

y

uv

x

uu

t

u

x

Pg

P

and

z

y

x

zzyyxx

(6.52)Eqsolveto)(usingSimplify

.solvetoDifficulty

)52.6(])([

motionofequationsEulersastoreferredCommonlyareequationsThese

vvt

vPg

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6.4.2 The Bernoulli Equation

equationEulersequationBernoullisectionIn this

law2ndNewtonsequationBernoulli23.sectionIn

)(

2

1

)()(2

becomes(6.53)Eq

)()(

2

1)(

identityvectorandngUsi

)53.6()(

)statesteady(0where

)52.6(])([

equationEulersForm

2 vvzgvp

vvvvpzg

vvvvvv

zgg

vvpg

t

v

vvt

vpg

1p

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28.202

1

)]([

./)(0)]([

)()(

)(

Similarly

state.)steadyif(

,,,,

)]([)()(2

1

streamlinealonglengthaldifferentiaLet

)(2

1

22

22

2

2

constgz

vdp

gdzdv

dp

sdvv

papeyofoutinvvbecausesdvv

dzkdzjdyidxkz

zdsz

dvsdv

dpdzzpdy

ypdx

xp

dzdydxz

p

y

p

x

pdsp

sdvvsdzgsdvsdp

kdzjdyidx

ds

dsvvzgvp

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streamlineaalongFlow

flowibleIncompress

flowSteady

flowInviscid

2

fluidibleincompressInviscid,For

2

constgzvp

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6.4.3 Irrotational Flow

flowalirrolationanofexampleAn

0

0

.)(

does.flowuniformaHowever,

equations.threehesesatisfy t

notcouldfieldflowgeneralA

Vorticity)(0Vorticity00)(2

1

flowalIrrotation

w

v

constUu

x

w

z

uzv

yw

y

u

x

v

VorVorVw

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6.4.5 The Velocity Potential

mass.ofonconservatiofeconsequenca--

functionstreamThe

.fieldflowtheofallyirrotationtheofeconsequenca--

potentialvelocityThe:Note

flowD-2torestrictedis

flowD-3generalafordefinedbecan

potentialvelocityfunctionscaleais),,(where

,,

0

00)(2

1flowalirrotationFor--

kwjviuv

zyx

zw

yv

xu

x

w

z

uz

v

y

wyu

xv

wvuzyx

kji

v

vorvw

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flow.potentialacalledcommonlyisflowoftypeThis

.fieldflowalIrrotation

,ibleincompress,Inviscid

)66.6(00

thatfollowsit,)(flowalirrotationand

)0(fluidibleincompressanFor

2

2

2

2

2

22

equationLaplace

zyxor

v

v

pressurescalculateToequationBernoulliwith

determinedbecan

conditionsboundary

withEq.(6.66)from

knownisIf

vor

w

v

u

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)71.6(01

)(1

)70.6(;1

;

)69.6(Since

),,(

)68.6(1

)67.6((.)(.)1(.)

(.)

,,s,Coordinate

2

2

2

2

2

2

zrr

r

rr

zv

rv

rv

evevevv

zrwhere

ez

er

er

ez

er

er

zrlCylindricaIn

zr

zzrrr

zr

zr

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2133

1

22

,/10,30if

(2)pointatpressure(b)

potentialvelocity(a)

:eDetermin

rightonreFigu

&/2sin2:Given

functionstream6.4Example

zzmkgkpaP

mrsmr

)1()(2cos2)(2cos4

;1

;

2sin42sin2

2cos42)2(cos21

2sin211

)(:

12

1

2

22

CrCdrrdrvdrv

zv

rv

rv

conditonslcylindricainpotentialVelocityflowalIrrotat ion

rrr

v

rrr

rr

v

rv

rv

massofonconservati

conditionslcylindricainfunctionstreamaSolution

rr

zr

rr

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)(2cos2(2)&(1)Eq

)2(.)(0)(

)(2)2sin(22sin4

)](2cos2[12sin4

1Since

)1()(2cos2)1(

2

11

1

22

1

2

1

2

AnsCr

constCCorC

Crr

Crr

r

rv

CrEqFrom

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36

)(36/)416(/102

11030

(3)EqFrom

/245.01616

5.0(2),pointAt

/41616

1(1),pointAt

16)2sin4()2cos4(

sin

)3(2

1

2

1

)(

2

1

2

1

aswrittenbecanequationBernoullithe

fluid,ibleincompress,nonviscousaofflowalirrotationanFor(b)

22333

2

2

222

2

1

22

1

2222

222

2

2

2

112

212

2

22

1

2

11

Anskpasmmkgpap

smvrv

mr

smvrv

mr

rrrv

vvvevevvce

vvpp

zzgzvp

gzvp

rrr

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6.5 Some Basic, Plane Potential Flows

00

0)()(0

flowalIrrotation,ibleincompressfor(6.66)EqFrom

0)()(

(6.74)EqUsingflow)nal(Irrotatio

(6.72)EqFrom

2

2

2

2

2

2

2

2

2

2

2

2

2

2

yxzyx

yyxxy

v

x

u

yxxxyy

x

v

y

u

flowplaneyx

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6.8 Viscous Flow

es.velociti&stresseshebetween tiprelationshaestablishtonecessaryisIt

equations.thanunknownsmoreareThere

.fluidaformotionofequationaldifferentiGeneral

)50.6()(

)50.6()(

)50.6()(

.assuch

Eq.6.50,motion,ofequationsgeneralderivedpreviouslythereturn tomustwe

motion,fluidofanalysisaldifferentitheintoeffectsviscouseincorporatTo

cz

ww

y

wv

x

wu

t

w

zyxg

bzvw

yvv

xvu

tv

zyxg

az

uw

y

uv

x

uu

t

u

zyxg

zzyzxz

z

zyyyxyy

zxyxxx

x

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6.8.1 Stress - Deformation Relationships

)125.6(2

)125.6(2

)125.6(2

2

2

2

n.deformatioofratethetorelatedlinearlyare

stressest theknown thaisit,fluidsNewtonian,ibleincompressFor

cz

wP

b

y

vP

ax

uP

z

wP

y

vP

x

uP

zz

yy

xx

zz

yy

xx

)125.6()(

)125.6()(

)125.6()(

fx

w

z

u

ey

w

z

v

dx

v

y

u

xzzx

zyyz

yzxy

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)126.6(][

)126.6(]

1

[

)126.6(]1

)([

)126.6(2

)126.6()1

(2

)126.6(2

fluidsibleincompressNewtonian,forstressesThe

scoordinatepolarlcylindricaIn

fr

V

z

V

e

V

rz

V

dV

rr

V

rr

c

z

VP

br

VV

rP

ar

V

P

zr

rzzr

z

zz

r

rr

z

zz

r

r

rr

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41

6.8.2 The NavierStokes Equations

)127.6()()(

direction-z

)127.6()()(

direction-y

)127.6()()(

direction-x(6.31),Eq,continuityofEq.and

(6.125f)~(6.125a)with(6.50c)~Eq.(6.50a)From

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

cz

ww

y

wv

x

wu

t

w

z

w

y

w

x

w

z

Pg

bz

vw

y

vv

x

vu

t

v

z

v

y

v

x

v

y

Pg

az

uw

y

uv

x

uu

t

u

z

u

y

u

x

u

x

Pg

z

y

x