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Lecture One
CENG 6601- Hydrodynamics
Department of Civil EngineeringEi-!" !e#elle $niversity
1
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Lecture 1Content%ntroduction
&luid !ec'anics (pplications
)roperties of &luid!at'ematical )reliminaries and ensor
(nalysis*ector (nalysis
Dot productCross product
+inematics
,
http://var/www/apps/conversion/tmp/scratch_5/Hydrodyamics%20Course%20outline.docxhttp://var/www/apps/conversion/tmp/scratch_5/Hydrodyamics%20Course%20outline.docx7/21/2019 Hydrodynamics Lecture 1
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%ntroduction( &luid is a sustance .'ic' deforms continuously
under t'e action of a s'earing stress/( olid is a sustance t'at resist a s'ear stress y
static deformation
&luids- li2uid and gas
3
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'e study of t'e motion of
4uids t'at are practicallyincompressile 5suc' as li2uids"
especially .ater" and gases atlo. speeds is usually referredto as hydrodynamics.
( sucategory of'ydrodynamics is hydraulics,.'ic' deals .it' li2uid 4o.s in
7
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&luids in )ure ciences
ag o a c rcu a on ong-range.eat'er prediction8 analysis ofclimate c'ange 5gloal.arming
mesoscale .eat'er patterns8
s'ort-range .eat'er prediction8tornado and 'urricane.arnings8 pollutant transport
,/Oceanograp'yaocean circulation patterns
causes of El Nino" e9ects ofocean currents on .eat'er andclimate
e9ects of pollution on livingorganisms
3/Geop'ysicsaconvection 5t'ermally-driven4uid motion in t'e Eart':smantle understanding of platetectonics" eart'2ua#es"volcanoes
convection in Eart':s molten
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&luids in )ure ciences7/ (strop'ysicsagalactic structure and
clusteringstellar evolution=fromformation ygravitational collapse
to deat' as asupernovae" from.'ic' t'e asicelements are
distriuted t'roug'outt'e universe" all via4uid motion
iological sciences
acirculatory and respiratorysystems in animals6
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&luids in tec'nology1/ %nternal comustionengines=all types of
transportation systems,/uro?et" scram?et" roc#et
engines aerospacepropulsion systems
3/ @aste disposalac'emical treatment
incineration
cse.age transport and treatment
7/ )ollution dispersal in t'e
atmosp'ere 5smog8 inrivers and oceans
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Contd/5a crude oil and natural gas
transferral
5 irrigation facilities
5c oBce uilding and'ouse'old pluming
A/ &luidstructureinteraction5a design of tall uildings
5 continental s'elf oil-drillingrigs
5c dams" ridges" etc/
5d aircraft and launc' ve'icleairframes and controlsystems
/ Heating" ventilatingand air conditioning5H*(C systems
/ Cooling systems for'ig'-densityelectronic devices
digital computers
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Contd/
10/olar 'eat and geot'ermal 'eatutiliFation
11/(rti;cial 'earts" #idney dialysismac'ines" insulin pumps
1,/!anufacturing processes5a spray painting automoiles" truc#s" etc/
5 ;lling of containers" e/g/" cans of soup" cartonsof mil#" plastic ottles of soda
5c operation of various 'ydraulic devices5d c'emical vapor deposition" dra.ing of synt'etic
;ers" .ires" rods" etc/
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Continuum (ssumption
!icroscopic approac' (nalyFe molecularstructure and associated collisions 5e/g/pressure is due to t'e net ec'ange ofmomentum at a solid surface
!acroscopic 5continuum approac' (nalyFeul# e'avior of 4uid 5e/g/ pressure is forceeerted y 4uid per unit area of solid surface
@'ile a ody of 4uid is comprised of molecules"
most c'aracteristics of 4uids are due toaverage molecular e'avior/
'at is" a 4uid often e'aves as if it .erecomprised of continuous matter t'at is in;nitely
divisile into smaller and smaller parts/10
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11
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&luid )roperties
&luid properties a9ect 'o. a 4uid reacts toapplied forces" t'us t'e 4uid:s motion
%ntensive and etensive properties%ntensive properties are independent of siFe or
volume
e/g/" density" pressure" temperatureEtensive properties are dependent of siFe or
volume and are additive for susystems/
e/g/" mass" volume" area" force%ntensive properties are otained from t'e ratio
of etensive properties/
e/g/" densitymassvolume" pressure1,
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Density 5mass%t is a ratio of t'e mass of 4uid element to
its volume
Density of a 4uid a9ects its 4o. in t.o.ays%t determines t'e inertia of a unit volume
of 4uid t'us its
acceleration .'en su?ected to a givenforcefor t'e same force" lo. density 4uids
accelerate more readily t'an 'ig' density
4uids
13
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>ul# !odulus E CoeBcient of compressiility
CoeBcient of 'ermal Epansion >
&or .ater E ,/1 10E
Nm3
> 1/
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)ressure&luids do not support s'ear stresses)ressure is 5compression stress at a point
in a static 4uid/
Net to velocity " it is t'e most dynamicvariale in 4uid dynamics
Di9erence in pressure causes a 4uid to 4o.
(tmosp'eric pressure at m/s/l 101/3 #pa"
.ill e set to Fero for convenience
16
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emperature
emperature is t'e measure oft'e internal energy of a 4uid/
Generally" temperaturedi9erences cause 'eat transfer/
%n t'is course" .e treat
isot'ermal situation/
1A
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*iscosity*iscosity 5also called dynamic viscosity" or
asolute viscosity is a measure of a 4uidsresistance to deformation under s'earstress/
&or eample" crude oil 'as a 'ig'erresistance to s'ear t'an does .ater/
'e symol used to represent viscosity is
5mu and its unit is 5#gm/s/%t is given y t'e ratio of t'e s'earing stress
to rate of deformation/
Ne.tonian and Non Ne.tonian &luids1
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Shear strain angle will grow as f(t)
For fluids such as water, oil, air
stress strain rate
Viscosity = Resistance to shear
1
y
u tt
u
x
t
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However,
As , , 0
But
here dyna!ic viscosity" #his is a constitutive
relation, which relates forces to !aterial $fluid% &ro&erties"
For fluids'
(Stress is &ro&ortional to
strain rate("
For solids'(Stress is &ro&ortional to
strain( $=)%
,0
y
tu
=tan
dy
du
dt
d=
t y
dt
d
t
dydu=
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Notes on shear stress
$i% Any shear stress, however small, &roduces relative!otion"
$ii% *f =0, du+dy=0, ut -0"$iii% Velocity &rofile cannot e tangent to a solid
oundary . #his re/uires an infinite shear stress"
(o.sli&( condition' u=0 at solid oundary"
y
10,1
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1
Bingha! 2lastic
Real 2lastic
Shear.#hinning Fluid
Newtonian
Shear.#hic3ening Fluid
Types of fluids
ewtonian fluid' Stress is linearly &ro&ortional to strain rate"
Shear.thinning' 4etchu&, whi&&ed crea!
Shear.thic3ening' 5orn starch in water
,,
dydu +
=
dy
du
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Units
6yna!ic Viscosity
e"g" S*'
,3
dydU +
=
7+87878 dydU =
99978
LT
M
LT
ML
Area
Force==
=
dy
dU
TLT
L ::==
T
ML
LT
M
TLT
M
=== 978s2as+!9
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e"g" S*'
4ine!atic Viscosity
,7
7+87878 =v
T
L
M
L
LT
Mv
9;
78 ==
LT
M=78
;78
L
M=
Sto3es:+:0 9< = sm
=v
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athe!atical &reli!inaries and tensor analysis
,6
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Di9erentiation%f a function is di9erentiale at a point "
t'e derivative value at t'at point is"
One sided limit
,A
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Vector AnalysisRight-handed, Cartesian coordinate system
1nit vectors
2osition vector
here are co!&onents of
,
;xz
:xx
9xy;a
:a
9
a
>x
ka
ja
ia
?%:,0,0$
?%0,:,0$
?%0,0,:$
;
9
:
=
=
=
;
;
9
9
:
: xaxaxax ++=
x%,,$ ;9: xxx
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Vector: 4undu. @Any /uantity whose co!&onents
change li3e the co!&onents of a &osition
vector under the rotation of the coord"
syste!"
Scalar: Any /uantity that does # change withrotation or translation of the coord" syste!
e"g" density $% or te!&erature $#%
,
: 9 ;>
$ %x x x x
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Vector AnalysisConsider vectors and
Dot )roduct
Cross )roduct
30
AB
A
B B
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'e 5PDelQ" PGradientQ" or PGradQOperator
(/'e Gradient 5Grad and DirectionalDerivatives
'e rate of c'ange of a scalar function in an aritrary direction ise2ual to t'e scalar product of t'e gradient .it' t'e unit vector" " int'at direction/ince
@$ % is &er&endicular to
lines and gives !agnitude
and direction of !ai!u! s&atial
rate of change of
31
zk
yj
xi
+
+
=
???
zk
yj
xi
++=???
:C=
;C=
9C=
C=
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'e 5PDelQ" PGradientQ" or PGradQOperator
>/ Divergence and Divergence'eorem (pplication of to vector 2uantity
Consider vector ;eld
'e dot product of .it' is called t'edivergence
%n volume terms" t'e net volume 4u out oft'e volume d* e2uals t'e product of and d*/
3,
kfjfiff zyx ??? ++=
z
f
y
f
x
f
f zyx
+
+
=
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Divergence 'eorem 5Gauss'eorem
Given an aritrary volume "enclosed y t'e surface " .it'out.ard unit normal vector at allpoints on " t'e divergence t'eorem
states
$outward unit nor!al vector
to surface ele!ent%
$infinitesi!al surface area%
$infinitesi!al volu!e%
33
dSnfdVfSV =
>n
dV
dA
AAreaSurface
VVoume
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*f is a scalar, vector, or any order tensor
S&ecifically, if is a vector
or
6ivergence #heore!' *ntegral over volu!e of divergence
of flu = integral over surface of the flu itself37
V A >i
dV n dAC C
=
%$C x
%$C x !
CC
=
i
i iV A
i
dV n dAx
> = R RV A!dV ! n dA
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)a!&les@ $a%
$%
$c% *f a scalar is $advectively% trans&orted
y the velocity>
U
or
6ivergence of flu within volu!e et flu across
3/ dV / n dA =
V A ># dV # n dA =
%$>>
ii
advadvUFUF ==
=
A advVi
i
advdAnFdV
x
F
>>
= AV adv dAnFdVF >>>
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C/ Curl and to#es: 'eorem'e cross product of .it' t'e
vector is called t'e curl
'e integral of around t'e contourenclosing e2uals t'e component of int'e direction normal to multiplied y
36
ky
f
x
fj
x
f
z
fi
z
f
y
f
fffzyx
kji
f xyzxyz
zyx
???
???
+
+
=
=
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5 $ounding curve%
A $o&en surface%
to#es: 'eorem
Let e a t.o-dimensional surfaceenclosed y C" t'en
@Surface integral of the curl of a vector,>
U , e/uals
the line integral of>
U along the ounding curve3A
>
n
ds
dA
= CS dxfndSf
> > >$ % = A Cu n dA u ds
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'e LaplacianDivergence of a gradient for scalar
Descries t'e net 4u of t'e scalar2uantity into volume
3
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ensors and %nde Notation
ome .ays of multiplying scalars or vectorsresult in tensor
calar is a tensor of ran# 5order Fero
*ector is a tensor of order one
%f order is not speci;ed" a tensor implies oforder t.o
Order of tensor speci;es t'e numer of
indices to descrie it
3
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Tensor:Assigns a vector to each direction in s&ace $ 9nd
order%
e"g"
Rows 5olu!ns
$a% *sotro&ic D 5o!&onents are unchanged y a rotation of
fra!e of reference $i"e" inde&endent of direction . e"g"4ronec3er 6elta iE%
$% Sy!!etric ' AiE = AEi $in general AiE = A#Ei%
$c% Anti.sy!!etric' AiE = .AEi
$d% 1seful result' AiE = :+9 $AiEAiE%:+9 $AEi.AEi%
= :+9 $AiEAEi%:+9 $AiE.AEi%=
70
11 12 13
21 22 23
31 32 33
A A A
A A A A
A A A
RRijA
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!instein summation convention
A% *f an inde occurs twice in a ter! a su!!ation over the
re&eated inde is i!&lied
B% Higher.order tensors can e for!ed y !ulti&lying
lower order tensors'
a% *f 1iand VEare :st
.order tensors then their &roduct1i VE= iE is a 9
nd.order tensor" Also 3nown as
vector outer &roduct $ %"
% *f AiEand B3lare 9nd.order tensors then their &roduct
AiE B3l = 2iE3l is a
= U
0
9
=
=
ji
k
ijkj
k
ijki xx
U
x
U
x
> > > > > >$ % $ 9% = a a a a a a
>> &aLet =
( )
( )
mjkmijk
mkm
kk
kjijki
x
aaaa
x
aa&And
&a&a
=
==
=
>>
>
>>
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: if : if : if : if
7
( )
mjjim
mjjmi
x
aa
x
aaaa
= >>
i= mj= mi= j=
>>
>>
>
9
%$%9+$
aaaa
aaaax
x
aa
x
aa
jjmm
i
j
ij
i
mm
=
=
=
ja%$>
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*elocity Gradient ensor
Laplacian *elocity
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*f is a scalar, vector, or any order tensor
S&ecifically, if is a vector
or
6ivergence #heore!' *ntegral over volu!e of divergence
of flu = integral over surface of the flu itselfi
dV n dAC C
=
%$C x
%$C x !
CC
=
i
i iV A
i
dV n dAx
> = R RV A
!dV ! n dA
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C'ain Sule 5di9erentiation
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C'ain Sule
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Curvilinear Coordinates
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Cylindrical
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p'erical coordinate
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p'erical
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tudies 4uid motion .it'out concernfor t'e force causing t'e 4o.
+inematics
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De;nition
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*elocity &ield
&ield p'ysical 2uantity associated .it' every point in inspace-time continuously de;ned in t'e region/
'e velocity at point C is otained y ta#ing t'e averagevelocity of t'e molecules in *T
'e velocity ;eld at any point can e de;ned li#e .ise
leading to *elocity &ield $/$ is continuous function of space and time" i/e/ $5"y"F
$ is a vector function/ (t any point 5"y"F at any instanttime 5t it 'as t'ree component" u" v" ./
u5"y"F"t- along -direction
v5"y"F"t- along y-direction
@5"y"F"t- along F direction
60
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&lo. *isualiFations%f t'e 4o. ;eld 5velocity is #no.n eit'er y
solving t'e motion e2uations or measuring itla;eld" t'e 4o. ;eld can e displayed usingpat'lines" strea#lines" streamlines/
Pathlines
( pat' t'at a 4uid particle traces over time/
'e position of a 4uid particle is given y t'reenumers 5" y" F/ 'e position is related to t'e
instantaneous velocity y Given t'e velocity ;eld" t'e successive positions 51"
y1" F1 of a 4uid particle from its initial position 50" y0"
F0 can e determined8 and
at'lines are generated eperimentally61
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Streaklines
6,
( line t'at connects at some instant of time
all particles t'at 'ave gone t'roug' a ;edposition or point/
trea#lines are generated eperimentally/ (dye is in?ected continously at a c'osen
point
St li
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Streamline%t is a line in a 4o. ;eld t'at is every.'ere
tangent to t'e velocity vector $ at eac'point along t'e streamline for any instant oftime t/
treamlines display a snaps'ot of t'e entire
4o. ;eld at a single instant in time"63
s
ds
c
o flu By definition' $i%
*elocity is tangent to ds'ence parallel
%,,$>
'vuU=
%,,$> dzdydxds=
0>
= dsU
;
>
9
>
:
>
;
>
9
>
:
>
000%$
%$%$
aaaaudyvdx
a'dxudzavdz'dy
++=++
udyvdx'dxudzvdz'dy === KK
'
dz
v
dy
u
dx==
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treamtue
treamlines never intersect eac' ot'erecause" at any point" t'ere can e onlyone direction of t'e velocity/
( streamtubeis a surface in t'e 4o.
formed from streamlines and closed uponitself to form a tue of variale cross-section/
67
)article pat's and streamlines are not in
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)article pat's and streamlines are not ingeneral coincident/ >ut"teady &lo. streamlines" pat'lines and
strea#lines coincide)at'line 5
(rc lengt' along t'e pat'line is
Di9erential e2uation for pat' lines and streamlines
are t'e same
$nsteady 4o. .'ic' direction doesnotc'ange .it' time
6x
>>xx +
>x
>u
>>duu+
L
>x
%,$ 0>>
txu
%,N$O 0>>>>
txxudu ++
#herefore after ti!e t
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#herefore, after ti!e
Relative !otion of two &oints de&ends on the velocity
gradient, , a 9nd.order tensor"
to first order
I
'
2I'
#aylor series e&ansion of
.. $A%
$% !eans order of =
&ro&ortional to
0
t
ttxuxtx(tuxttxuxxx %,$%PP$PP%,$%$ 0>>>
9
>>>0
>>>>
L
> ++++=
t
x
uxtuxxxse)arationinC*an+e
j
ij
===
>>>
L
>
tuxxx >>>
L
>+=
j
i
x
u
ttxux %,$ 0>>>
+
tx(uxtxuxx
ttxxuxx
%NPP$PP%,$O%$
%,$%$
9
>>>0
>>>>
0>>>>>
++++=+++
%,$ 0>>>
txxu +
$9% 6eco!&osition of otion
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$ % &
@Any tensor can e re&resented as the su! of a sy!!etric
&art and an anti.sy!!etric &art@
=O rate of strain tensorN Orate of rotation tensorN
ote'
$i% Sy!!etry aout
diagonal
$ii% Q uni/ue ter!s
inear angular straining1
+
+
=
i
j
j
i
i
j
j
i
j
i
x
u
x
u
x
u
x
u
x
u
9
:
9
:
iE iEe r= +
+
+
+
+
+
+
=
;
;
;
9
9
;
;
:
:
;
9
;
;
9
9
9
9
:
:
9
:
;
;
:
:
9
9
:
:
:
9
9
9
9
:
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
eij
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ote'
$i% Anti.sy!!etry aout
diagonal
$ii% ; uni/ue ter!s $r:9, r:;, r9;%
Rotation
#er!s in
,
=
0
0
0
9
:
;
9
9
;
;
:
:
;
9
;
;
9
9
:
:
9
:
;
;
:
:
9
9
:
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
rij
: :iE 9 9
>$ % = ijk k ijk k r u
; 9
; :
9 :
0
: 09
0
=
ijr
: ::9 :9; ; ;9 9
: : :;9 ;9: : : :9 9 9
" "
$ %
= =
= = =
e + r
r
I h 3 hi i
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etIs chec3 this assertion aout rij
#he reci&e'$a% ! = i and l =E
$% l = i and ! = E
gives
3
9
=
=
=
=
=
mijk k ijk km
mjik km
mim j i jm
ji
j i
ij
u
x
u
x
ux
uu
x x
r
*nter&retation tu
xxx ij
=L
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*nter&retation
Relative velocity due
to defor!ation of
fluid ele!ent
Relative velocity due
to rotation of
ele!ent at rate :+9
7
>
$ %
:
9
:9
:$ %
9
= +
=
= +
= + R
i j ij ij
ij j ijk k j
ij j ikj k j
i ij j i
u x e r
e x x
e x x
u e x x
tx
xxxj
j >>L
> >
= =
ii j
j
x x uu x
t x
$;% #y&es of !otion and defor!ation "
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$i% 2ure #ranslation
M9
M:
t:
t0
$ii% inear 6efor!ation . 6ilatation
M9
M:
t:
t0
a
>
=V
dVm
'e La. of Conservation of!
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!ass
@it'in some prolem domain de;ned y acontrol volume" t'e net mass of 4uidpassing from outside to inside t'roug' t'econtrol surfaces e2uals t'e net increase of
mass in t'e control volume/%n its most general form t'e la. stat t'at
P mass is neit'er created not destroyed ina closed systemQ
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