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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
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dt
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elementtheofndeformatioangularx
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flowalIrrotation0
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only when)(i.e.blockundeformedanasaxis-zaboutRotation
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6.2.1 Differential Form of Continuity Equation
inout
cv
vAvA
dAnv
zyx
t
d
t
d
t
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)()(
elementtheofsurfacesthethroughflowmassofrateThe)(
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0
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s
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orderhighneglecting--expansionseriesTaylor
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)24.6()(direction-yinrateNet
similarly
)23.6()(
]2
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direction-in xoutflowmassofrateNet
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.formaldifferentiinequationcontinuityThe
)27.6(0
outflowmassofrateNet:
0][
0)(Since
z
w
y
v
x
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vzyx
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uNote
zyxz
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t
dAnvd
t cscv
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)31.6(0
)30.6(0
0
flowibleincompressFor--
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fluidlecompressibofflowsteadyFor--
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formIn vector
mechanicsfluidofequationslfundamentatheofOne--
z
w
y
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uor
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const
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uor
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vt
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equation.continuityhesatisfy ttorequired,w:Determine
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flowibleincompressanFor
26.Example
222
w
zyzxyv
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),(2
3nIntegratio
3)(2
0)()(
0
continuityofequationthefrom:Solution
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222
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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
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y
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and
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.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
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1)(
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equationEulersForm
2 vvzgvp
vvvvpzg
vvvvvv
zgg
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t
v
vvt
vpg
1p
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28.202
1
)]([
./)(0)]([
)()(
)(
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state.)steadyif(
,,,,
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streamlinealonglengthaldifferentiaLet
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1
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gdzdv
dp
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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--
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zyx
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kji
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flow.potentialacalledcommonlyisflowoftypeThis
.fieldflowalIrrotation
,ibleincompress,Inviscid
)66.6(00
thatfollowsit,)(flowalirrotationand
)0(fluidibleincompressanFor
2
2
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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
;
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),,(
)68.6(1
)67.6((.)(.)1(.)
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2
zrr
r
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evevevv
zrwhere
ez
er
er
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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
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conditonslcylindricainpotentialVelocityflowalIrrotat ion
rrr
v
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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/)416(/102
11030
(3)EqFrom
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5.0(2),pointAt
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fluid,ibleincompress,nonviscousaofflowalirrotationanFor(b)
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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
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bzvw
yvv
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tv
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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
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e
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rz
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dV
rr
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c
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br
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ar
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rzzr
z
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r
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zz
r
r
rr
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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
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x
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