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 [SHIVOK SP211]
October 22, 2015 CH 14 Fluids
I.
WhatisaFluid?
A. Afluid,incontrasttoasolid,isasubstancethatcan_________________.
B. Fluidsconformtotheboundariesof________________________________in
whichweputthem.Theydosobecauseafluidcannotsustainaforce
thatistangentialtoitssurface.Thatis,afluidisasubstancethatflows
becauseitcannotwithstandashearingstress.
C. Itcan,however,exertaforceinthedirection__________________________
toitssurface.
II.
DensityandPressure
A. Tofindthedensityofafluidatanypoint,weisolateasmall
volumeelementVaroundthatpointandmeasurethemassmofthe
fluidcontainedwithinthatelement.Ifthefluidhasuniformdensity,
then
1.
Densityisascalarproperty;itsSIunitisthekilogrampercubicmeter.
B. IfthenormalforceexertedoveraflatareaAisuniformoverthat
area,thenthepressureisdefinedas:
1.
TheSIunitofpressureisthenewtonpersquaremeter,whichisgivena
specialname,thePascal(Pa).
2.
1atmosphere(atm)=
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October 22, 2015 3.
Exampleproblem:
a)
Apartiallyevacuatedairtightcontainerhasatightfittinglidof
surfacearea77cm2andnegligiblemass.Iftheforcerequiredtoremove
thelidis480Nandtheatmosphericpressureis1.0x105Pa,whatisthe
internalairpressure?
(1)
Solution:
III.
FluidsatRest
A. Thepressureatapointinafluidinstaticequilibriumdependson
thedepthofthatpointbutnotonanyhorizontaldimensionofthe
fluidoritscontainer.
1.
Above:Atankofwaterinwhichasampleofwateriscontainedinan
imaginarycylinderofhorizontalbaseareaA.
2.
Below:Afree‐bodydiagramofthewatersample.
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October 22, 2015 3.
Thebalanceofthe3forcesiswrittenas:
4.
Ifp1andp2arethepressuresonthetopandthebottomsurfacesofthe
sample,
5.
Sincethemassmofthewaterinthecylinderis,m=V,wherethe
cylinder’svolumeVistheproductofitsfaceareaAanditsheight(y1‐y2),then
6.
Therefore,
7.
Ify1isatthesurfaceandy2isatadepthhbelowthesurface,then
(wherepoisthepressureatthesurface,andpthepressureatdepthh).
B. Exampleproblems:
1.
Crewmembersattempttoescapefromadamagedsubmarine100.0m
belowthesurface.Whatforcemustbeappliedtoapop‐outhatch,whichis
1.20mby0.60m,topushitoutatthatdepth?Assumethatthedensityofthe
oceanwateris1024kg/m3andtheinternalairpressureisat1.00atm.
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October 22, 2015 2.
Atadepthof10.9km,theChallengerDeepintheMarianasTrenchof
thePacificOceanisthedeepestsiteinanyocean.Yet,in1960,DonaldWalsh
andJacquesPiccardreachedtheChallengerDeepinthebathyscaphTrieste.
Assumingthatseawaterhasauniformdensityof1024kg/m3,approximate
thehydrostaticpressure(inatmospheres)thattheTriestehadtowithstand.
IV.
Pascal’sPrinciple
A. Achangeinthepressureappliedtoanenclosedincompressible
fluidistransmittedundiminishedtoeveryportionofthefluidandto
thewallsofitscontainer.
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October 22, 2015 B. Pascal’sPrincipleandtheHydraulicLever
1.
TheforceF isappliedontheleftandthedownwardforceF fromthe
i
o
loadontherightproduceachangepinthepressureoftheliquidthatis
givenby
2.
Ifwemovetheinputpistondownwardadistanced ,theoutputpiston
i
movesupwardadistanced ,suchthatthesamevolumeVofthe
o
incompressibleliquidisdisplacedatbothpistons.
3.
Thentheoutputworkis:
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October 22, 2015 C. Sampleproblem:
1.
Apistonofcross‐sectionalareaaisusedinahydraulicpresstoexerta
smallforceofmagnitudefontheenclosedliquid.Aconnectingpipeleadstoa
largerpistonofcross‐sectionalareaA(Fig.below).(a)WhatforcemagnitudeF
willthelargerpistonsustainwithoutmoving?(b)Ifthepistondiametersare
3.80cmand53.0cm,whatforcemagnitudeonthesmallpistonwillbalancea
20.0kNforceonthelargepiston?
a)
Solution:
(a) (b) Page6
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October 22, 2015 V.
ArchimedesPrinciple
A. Whenabodyisfullyorpartiallysubmergedinafluid,abuoyant
forcefromthesurroundingfluidactsonthebody.Theforceis
directedupwardandhasamagnitudeequaltotheweightofthefluid
thathasbeendisplacedbythebody.
1.
Thenetupwardforceontheobjectisthebuoyantforce,F .
2.
Thebuoyantforceonabodyinafluidhasthemagnitude
b
wheremfisthemassofthefluidthatisdisplacedbythebody.
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October 22, 2015 B. FloatingandApparentWeight
1.
Whenabodyfloatsinafluid,themagnitudeF ofthebuoyantforceon
b
thebodyisequaltothemagnitudeF ofthegravitationalforceonthebody.
g
2.
Thatmeans,whenabodyfloatsinafluid,themagnitudeF ofthe
g
gravitationalforceonthebodyisequaltotheweightm gofthefluidthathas
f
beendisplacedbythebody,wherem isthemassofthefluiddisplaced.
f
3.
Thatis,afloatingbodydisplacesitsownweightoffluid.
4.
Theapparentweightofanobjectinafluidislessthantheactualweight
oftheobjectinvacuum,andisequaltothedifferencebetweentheactual
weightofabodyandthebuoyantforceonthebody.
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October 22, 2015 C. Sampleproblem:
1.
Whatistheapparentweightofasphereofradius r  0.020m and
kg
thatisfullysubmergedinwater
m3
kg
(density  f  1000 3 )?
m
density   1500
a)
Solution:
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October 22, 2015 VI.
IdealFluidsinMotion
A. Steadyflow:Insteady(orlaminar)flow,thevelocityofthemoving
fluidatanyfixedpointdoesnotchangewithtime.
B. Incompressibleflow:Weassume,asforfluidsatrest,thatourideal
fluidisincompressible;thatis,itsdensityhasaconstant,uniform
value.
C. Nonviscousflow:Theviscosityofafluidisameasureofhow
resistivethefluidistoflow;viscosityisthefluidanalogoffriction
betweensolids.Anobjectmovingthroughanonviscousfluidwould
experiencenoviscousdragforce—thatis,noresistiveforcedueto
viscosity;itcouldmoveatconstantspeedthroughthefluid.
D. Irrotationalflow:Inirrotationalflowatestbodysuspendedinthe
fluidwillnotrotateaboutanaxisthroughitsowncenterofmass.
VII.
TheEquationofContinuity
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October 22, 2015 A. Theamountofvolume(DeltaV)offluidflowingpastthedotted
lineinthetimeintervaldeltatis:
B. Applyingtheaboveequationforboththeleftandrightdotted
linesweget:
C. Whichyields
1.
Whatitsaysinlaymen’stermsisthatthevolumetricflowrateis
constant;thusthespeedincreasesifwerestrictthearea(i.e.youputyour
thumbovertheendofthegardenhoseandthewatercameoutatahigher
speed):
2.
Ifwemultiplybothsidesoftheequationbydensityofthefluidweget
massflowrate:
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October 22, 2015 D. Example:WaterStream
1.
Whydoesthewaterstreamnarrowasitfallsfromthefaucet?
2.
Solution:
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October 22, 2015 VIII. Bernoulli’sEquation
A. Ifthespeedofafluidelementincreasesastheelementtravels
alongahorizontalstreamline,thepressureofthefluidmustdecrease,
andconversely.
a)
Fig.14‐19FluidflowsatasteadyratethroughalengthLofa
tube,fromtheinputendatthelefttotheoutputendattheright.From
timetin(a)totimet+tin(b),theamountoffluidshowninpurple
enterstheinputendandtheequalamountshowningreenemergesfrom
theoutputend.
2.
Thatmeansthat
Or
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October 22, 2015 B. Proof
1.
Thechangeinkineticenergyofthesystemistheworkdoneonthe
system.
2.
Ifthedensityofthefluidis,
3.
Theworkdonebygravitationalforcesis:
4.
Thenetworkdonebythefluidis:
5.
Therefore,
6.
Finally,
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October 22, 2015 C. Sampleproblems:
1.
AcylindricaltankwithalargediameterisfilledwithwatertoadepthD
=0.30m.Aholeofcross‐sectionalareaA=6.5cm2inthebottomofthetank
allowswatertodrainout.(a)Whatistherateatwhichwaterflowsout,in
cubicmeterspersecond?(b)Atwhatdistancebelowthebottomofthetankis
thecross‐sectionalareaofthestreamequaltoone‐halftheareaofthehole?
where h1 is the height of the water in the tank, p1 is the pressure there, and v1 is the speed of the water there; h2 is the altitude of the hole, p2 is the pressure there, and v2 is the speed of the water there.  is the density of water. The pressure at the top of the tank and at the hole is atmospheric, so p1 = p2. Since the tank is large we may neglect the water speed at the top; it is much smaller than the speed at the hole. The Bernoulli equation then becomes (b) We use the equation of continuity: A2v2 = A3v3, where A3  12 A2 and v3 is the water speed where the area of the stream is half its area at the hole. Thus The water is in free fall and we wish to know how far it has fallen when its speed is doubled to 4.84 m/s. Since the pressure is the same throughout the fall, Page
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October 22, 2015 2.
Figurebelowshowsastreamofwaterflowingthroughaholeatdepth
h  10cm inatankholdingwatertoheight H  40cm .(a)Atwhatdistancex
doesthestreamstrikethefloor?(b)Describetheprocessyouwoulduseto
findthevalueofhwhichmaximizesthedistancex.
a)
Solution
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