Fluid mechanics

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Fluid Mechanics Notes

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Fluid:A fluidisa substance whichconformscontinuouslyunderthe actionof
shearingforces. OR A flowingisa substance whichiscapable of flowing.OR
A fluidisa substance whichdeformscontinuouslywhensubjectedtoexternal shearingstress.
FluidMechanicsisthat branch of science whichdealswith behaviourof the fluidsatrest as well asin
motion.
Fluid:Fluidsare substance whichareacapable of flowingand conformingthe shapesof container.Fluids
can be ingas or liquidstates.
Mechanics:Mechanicsis the branch of science thatdealswiththe state of restor motionof bodyunder
the action of forces.
FluidMechanics:Branch of mechanicthatdealswiththe response orbehaviorof fluideitheratrestor
inmotion.
Branches of Fluid Mechanics
FluidStatics: It is the branch of fluidmechanicswhichdealswiththe response/behaviorof fluidwhen
theyare at rest.
Fluidkinematics:It dealswiththe response of fluidwhentheyare inmotionwithoutconsideringthe
energiesandforcesinthem.
Hydrodynamics: It dealswiththe behaviorof fluidswhentheyare inmotionconsidering energiesand
forcesinthem.
Hydraulics: It isthe mostimportantandpractical/experimentalbranchof fluidmechanicswhichdeals
withthe behaviorof waterandother fluideitheratrestor in motion.
Significance of Fluid Mechanics: Fluidisthe mostabundantavailable substance e.g.,air,gases,
ocean,riverand canal etc. It providesbasisforothersubjectse.g., Publichealth/environmental
engineering,HydraulicEngineering,IrrigationEngineering, Coastal engineering, etc
What is fluid:A substance whichdeformscontinuouslyunder the actionof shearingforces,however
small theymaybe. If a fluidisat rest,there can be no shearing forces
All forces inthe fluidmustbe perpendicularto the planesuponwhichtheyact.
Solids & Fluids (liquids & gases).
Matter existintwo principal forms:• Solid, •Fluids.
Fluidsare furthersub-dividedinto:•Liquid, •Gas.
For all practical purposes,the liquidsandsolidscanbe regardedas incompressible. Thismeansthat
pressure andtemperature have practicallynoeffectonthem. Example, Water,Kerosene, petrol etc. But

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Gasesare readilycompressiblefluids. Theyexpandinfinitelyinthe absenceof pressure andcontract
easily underpressure.Example: air, ammoniaetc.
YOU WOKEUP IN THE MORNINGAND THE ROOMIS COOL. CoolantcirculatingInside itandcool Air
whichitgives isFluid.Afterthatyouwashedyourface at the sink.The waterwhichcomesat yourtap is
fluidandhas come throughthe pipingsystemwhich alsocomesunderfluidmechanics. A mixture of fuel
like petrol andairis forced byatmospheric(orgreater) pressure into the cylinderthroughthe intake
port. All physical quantitiesare givenbyafew fundamental quantitiesortheir combinations.The units
of suchfundamental quantitiesare calledbase.
Units and dimensions:Units,combinationsof thembeingcalledderivedunits.The systeminwhich
length,massandtime are adoptedasthe basic quantities,andfromwhich the unitsof otherquantities
are derived,iscalledthe absolute systemof units.
Absolute systemof units
MKS systemof units: Thisis the systemof unitswhere the metre (m) isusedforthe unitof length,
kilogram(kg) forthe unitof mass, andsecond(s) for the unitof time asthe base units.
CGSsystem of units:Thisis the systemof unitswhere the centimetre(cm) isusedforlength, gram(g)
for mass,and second(s) fortime as the base units.
International system ofunits (SI):SI, the abbreviationof LaSystemInternationald’Unites,isthe system
developedfromthe MKSsystemof units.Itis a consistentandreasonable systemof unitswhichmakes
it a rule to adoptonlyone unitfor eachof the variousquantitiesusedinsuchfieldsasscience,
educationand industry.
There are sevenfundamental SIunits,namely:metre (m)forlength, kilogram(kg) formass,second(s)
for time,ampere (A) forelectriccurrent,kelvin(K) forthermodynamictemperature,mole (mol) formass
quantityandcandela(cd) for intensityof light.Derivedunitsconsistof these units.
Dimension:All physical quantitiesare expressedincombinationsof base units.The index
numberof the combinationof base unitsexpressingacertainphysical quantityiscalledthe dimension,
as follows. Inthe absolute systemof unitsthe length,massandtime are respectivelyexpressedbyL,M
and T. PutQ as a certainphysical quantityandc as a proportional constant,andassume thattheyare
expressedasfollows: Q = cLà
Mß
T
where a, ß and  are respectivelycalledthe dimensionsof QforL, M, T.
Physical properties of Fluid
1: Density: The massof a liquidperunitvolume atstandardtemperature &pressure (STP) iscalledits
density.Itisalsotermedas mass densityor specificmass of the liquid.Thus
Density= ρ = Mass / Volume = M/ V
2: Specific Weight: The weightof a liquidperunitvolumeatstandardtemperature &pressure (STP) is
calleditsSpecificWeight.Itisalsotermedas weightdensityof the liquid.Thus
SpecificWeight=weight/Volume =W / V
3: Specific Volume: The volume of a liquid occupiedbyunitmassiscalledspecificvolume of the liquid.
SpecificVolume=volume of liquid/massof liquid ORV = 1/density= 1/ρ.
4: Specific Gravity:The ratioof the specificweightof aliquidtothatof the specificweightof the water
at standard temperature &pressure (STP) iscalledthe SpecificGravityof the liquid.
It isalso termedas Relative densityof the liquid.Thus
SpecificGravity(sp.gr.) =S = sp. Weightof liquid/sp.Weightof water

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5: Surface tension:Whentwoliquidsof differentdensitiesor whenaliquid&a gas are in contact,
thenthe surface of contact will be intensiondue topressure differencedue tocohesionwhichiscalled
surface tension.
6: Capillary Action:Whena tube of small diameteropentothe atmosphere isinsertedinaliquid,the
liquidrisesorfallsinside the tube.Thisbehaviourof the liquidsistermedasCapillaryActionof the
Liquid.
7: Compressibility: The reductioninvolume of aliquidonincreasingpressure,iscalledcompressibility
of the liquid.The value of compressibilityissosmall thatforall practical purposesitisneglected.
8: Viscosity: The propertyof a liquidwhichoffersresistance tothe movementof one layerof the liquid
overthe overadjacentlayerof the liquidiscalledViscosity.Itsunitiscalledpoise &
1 poise = p = dyne – sec/cm2 or p= 1/10 N – sec/m2.
Unitsof Viscosity:N.s/m2or kg/m/s
Mollases, tar,glycerine are highlyviscous fluids.
Water, air, petrol have verysmall viscosity andare calledthinfluids
Newton’sLaw or equationof Viscosity
τ= µ (du/dy)
Where,du/dy= velocitygradient
µ= coefficientof viscosity, absolute viscosityordynamicviscosity.
Measurement of VISCOSITY
The viscosityof a liquidismeasuredusinga viscometer,andthe bestviscometersare those whichare
able to create and control simple flow fields.The mostwidelymeasuredviscosityisthe shearviscosity,
and here we will concentrate onitsmeasurement,althoughitshouldbe notedthatvariousextensional
viscositiescanalsobe definedandattemptscanbe made to measure them, althoughthisisnoteasy.
Most modernviscometersare computer- ormicroprocessor-controlledandperformautomatic
calculationsbasedonthe particulargeometrybeingused.We donottherefore needtogointoa great
deal of discussionof calculationprocedures,ratherwe willconcentrate ongeneral issuesandartifacts
that intrude intomeasurements. Or
Viscosityisthe measure of the internal frictionof afluid.Thisfrictionbecomesapparentwhenalayerof
fluidismade tomove in relationtoanotherlayer.The greaterthe friction,the greaterthe amountof
force requiredtocause thismovement,whichiscalledshear.Shearingoccurswheneverthe fluidis
physicallymovedordistributed,asinpouring,spreading,spraying,mixing,etc.Highlyviscousfluids,
therefore,require more force tomove thanlessviscousmaterials.
Isaac Newtondefinedviscositybyconsideringthe model representedinthe figure above.Twoparallel
planesof fluidof equal areaA are separatedbya distance dx and are movinginthe same directionat
differentvelocitiesV1andV2. Newtonassumedthatthe force requiredtomaintainthisdifference in

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speedwasproportional tothe difference inspeedthroughthe liquid,orthe velocitygradient.To
expressthis,Newtonwrote:
The velocitygradient,dv/dx,isa measure of the change in speedatwhichthe intermediatelayersmove
withrespecttoeach other.It describesthe shearingthe liquidexperiencesandisthuscalledshearrate.
Thiswill be symbolizedasSinsubsequentdiscussions.Itsunitof measure iscalled the reciprocal second
(sec-1).
The term F/A indicatesthe force perunitarea requiredtoproduce the shearingaction.Itis referredto
as shearstressand will be symbolizedbyF′.Itsunitof measurementisdynesper square centimeter
(dynes/cm2).
Usingthese simplifiedterms,viscositymaybe definedmathematicallybythisformula:
The fundamental unitof viscositymeasurementisthe poise.A material requiringashearstressof one
dyne persquare centimetertoproduce ashear rate of one reciprocal secondhasa viscosityof one
poise,or100 centipoise.Youwill encounterviscositymeasurementsexpressedinPascal-seconds(Pa·s)
or milli-Pascal-seconds(mPa·s);these are unitsof the International Systemandare sometimesusedin
preference tothe Metricdesignations.One Pascal-secondisequal totenpoise;one milli-Pascal-second
isequal to one centipoise.
Newtonassumedthatall materialshave,atagiventemperature,aviscositythatisindependentof the
shearrate. Inotherwords,twice the force wouldmove the fluidtwice asfast. Aswe shall see,Newton
was onlypartlyright.
NEWTONIAN FLUIDS:Thistype of flow behaviorNewtonassumedforall fluidsiscalled,not
surprisingly,Newtonian.Itis,however,onlyone of several typesof flow behavioryoumay encounter.A
Newtonianfluidisrepresentedgraphicallyinthe figure below.GraphA showsthat the relationship
betweenshearstress(F′) andshearrate (S) isa straightline.GraphB showsthat the fluid'sviscosity
remainsconstantas the shearrate is varied.Typical Newtonianfluidsinclude waterandthinmotoroils.
What thismeansinpractice is that at a giventemperature the viscosityof aNewtonianfluidwill remain
constantregardlessof whichViscometermodel,spindleorspeedyouuse tomeasure it.Brookfield
ViscosityStandardsare Newtonianwithinthe range of shearratesgeneratedbyBrookfieldequipment;
that's whytheyare usable withall ourViscometermodels.Newtoniansare obviouslythe easiestfluids
to measure - justgrab your Viscometerandgo to it.Theyare not,unfortunately,ascommonas that

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much more complex groupof fluids,the non-Newtonians,whichwill be discussedinthe nextsection.
NON-NEWTONIAN FLUIDS:A non-Newtonianfluidisbroadlydefinedasone forwhichthe
relationshipF′/Sisnota constant.Inotherwords,whenthe shearrate isvaried,the shearstressdoesn't
vary inthe same proportion(orevennecessarilyinthe same direction).The viscosityof suchfluidswill
therefore change asthe shearrate isvaried.Thus,the experimentalparametersof Viscometermodel,
spindle andspeedall have aneffectonthe measuredviscosityof anon-Newtonianfluid.Thismeasured
viscosityiscalledthe apparentviscosityof the fluidandisaccurate onlywhenexplicitexperimental
parametersare furnishedandadheredto.
Non-Newtonianflowcanbe envisionedbythinkingof anyfluidasa mixture of moleculeswithdifferent
shapesandsizes.Astheypassby eachother,as happensduringflow,theirsize,shape,andcohesiveness
will determine howmuchforce isrequiredtomove them.Ateachspecificrate of shear,the alignment
may be differentandmore or lessforce may be requiredtomaintainmotion.
There are several typesof non-Newtonianflow behavior,characterizedby the waya fluid'sviscosity
changesinresponse tovariationsinshearrate.The most commontypesof non-Newtonianfluidsyou
may encounterinclude:
Psuedoplastic:Thistype of fluidwill displayadecreasingviscositywithanincreasingshearrate,as
showninthe figure below.Probablythe mostcommonof the non-Newtonianfluids,pseudo-plastics
include paints,emulsions,anddispersionsof manytypes.Thistype of flow behaviorissometimescalled
shear-thinning.
Dilatant: Increasingviscositywithanincrease inshearrate characterizesthe dilatantfluid;see the figure
below.Althoughrarerthanpseudoplasticity,dilatancyisfrequentlyobservedinfluidscontaininghigh
levelsof deflocculatedsolids,suchasclayslurries,candycompounds,cornstarchinwater,and
sand/watermixtures.Dilatancyisalsoreferredtoasshear-thickeningflow behavior.

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Plastic: Thistype of fluidwill behaveasa solidunderstaticconditions.A certainamountof force must
be appliedtothe fluidbefore anyflowisinduced;thisforce iscalledthe yieldvalue.Tomatocatsupisa
goodexample of thistype fluid;itsyieldvaluewill oftenmake itrefusetopourfromthe bottle until the
bottle isshakenorstruck, allowingthe catsuptogushfreely.Once the yield value isexceededandflow
begins,plasticfluidsmaydisplayNewtonian,pseudoplastic,ordilatantflow characteristics.See the
figure below.
So far we have onlydiscussedthe effectof shearrate on non-Newtonianfluids.Whathappenswhenthe
elementof time isconsidered?Thisquestionleadsustothe examinationof twomore typesof non-
Newtonianflow:thixotropicandrheopectic.
THIXOTROPY AND RHEOPEXY : Some fluidswill displayachange inviscositywithtime underconditions
of constantshearrate.There are two categoriestoconsider:
Thixotropy: Asshowninthe figure below,athixotropicfluidundergoesadecrease inviscositywith
time,while itissubjectedtoconstantshearing.
Rheopexy:Thisisessentiallythe opposite of thixotropicbehavior,inthatthe fluid'sviscosityincreases
withtime as itis shearedata constant rate.See the figure below.
Both thixotropyandrheopexymayoccurin combinationwithanyof the previouslydiscussedflow
behaviors,oronlyat certainshearrates.The time elementisextremelyvariable;underconditionsof
constantshear,some fluidswill reachtheirfinalviscosityvalueinafew seconds,whileothersmaytake
up to several days.
Rheopecticfluidsare rarelyencountered.Thixotropy,however,isfrequentlyobservedinmaterialssuch
as greases,heavyprintinginks,andpaints.
Whensubjectedtovaryingratesof shear,a thixotropicfluidwill reactasillustratedinthe figure below.
A plotof shear stressversusshearrate wasmade as the shearrate was increasedtoa certainvalue,
thenimmediatelydecreasedtothe startingpoint.Note thatthe up and downcurvesdonot coincide.
Thishysteresisloopiscausedbythe decrease inthe fluid'sviscositywithincreasingtime of shearing.
Such effectsmayormay not be reversible;some thixotropicfluids,if allowedtostandundisturbedfora
while,will regaintheirinitial viscosity,while othersneverwill.

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The rheological behaviorof afluidcan,of course,have a profoundeffectonviscositymeasurement
technique.Laterwe will discusssome of these effectsandwaysof dealingwiththem.
FollowingObservationscan be
made from Newton’sviscosity
Equation:
• Max. shearstressoccur whenvelocity gradientislargestandshearstress disappearswherevelocity
gradientiszero.
• VelocityGradientbecomessmall with distance fromthe boundary. Consequently the max value of
shearstressoccurs at the boundaryandit decreasesfromthe boundary.
Considerfluidsare full of twoparallel walls.A shearstress,τ, isapplied
to the upperwall.Fluidsare deformedcontinuouslybecause fluidscannotsupportshearstresses.The
deformationrate,however,isconstant.
Furthermore,if the deformationrate orthe so-calledrate of strainisproportional tothe shearstress,
thenthe fluidwill be classifiedasaNewtonian
fluid,i.e. τ ∝ dγ / dt , where γ is shearangle or
τ = µ dγ / dt . Inaddition, dγ / dt = du / dy . Hence, τ = µ du / dy.
Again,the relationshipbetweenshearstressactingona Newtonianfluid
and rate of strain(or velocitygradient) islinear.If itisnot linear,then
1.9 Speedof sound·9 · the fluidwill be calledanon-Newtonianfluid.µisthe so-calleddynamic
viscosity.Itsunitsare dyne·cms2 or Poise (cP).
Hydrostatics: thatstudiesthe mechanicsof fluidsatabsolute andrelative rest. or
The study of pressure exertedbyliquidsatrestistermedas hydrostatics.
Kinematics:dealswithtranslation,rotationand deformationof fluidwithoutconsideringthe force and
energycausingsucha motion.
Hydrodynamics: thatprescribesthe relationbetween velocitiesandaccelerationandthe forceswhich
are exertedbyoruponthe movingfluids. Or
Studyof flowingliquids &forcescausingtheirmotioniscalledashydrodynamics.
Hydraulics: The engineeringscience of liquidpressure andflow. Hydraulicengineeringisthe Science of
waterin motionanditsinteractionswiththe surroundingenvironment.Waterplaysamajorrole in
humanperceptionof the environmentbecauseitisanindispensableelement.
The term 'Hydraulics'isrelatedtothe applicationof the FluidMechanicsprinciplestowaterengineering
structures,civil andenvironmentalengineeringfacilities:e.g.,canal,river,dam, reservoir,water
treatmentplant.Hydraulicengineeringis the science of waterinmotion,andthe interactionsbetween
the flowingfluidandthe surroundingenvironment.Hydraulicengineersare concernedwith

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application of the basicprinciplesof fluidmechanicstoopenchannel flowsandreal fluidflow
hydrodynamics.Examplesof openchannelsare natural streamsandrivers.Man-made channelsinclude
irrigationandnavigationcanals,drainage ditches,sewerandculvertpipesrunningpartiallyfull,and
spillways.
Fluid Statics
Pressure:The perpendicularforce exertedbyafluidperunitarea.P= P/A.
Pressure Intensity:The force exertedbythe liquidonthe unitareaof bottom& the sidesof the vessel is
calledintensityof pressure.
Pressure Head: Pressure influidsmayarise frommanysources,forexample pumps,gravity,
momentumetc.Since p= ρgh,a heightof liquidcolumncanbe associatedwiththe
pressure parisingfromsuch sources.Thisheight,h,isknownasthe pressure head.
The vertical distance (infeet) equal tothe pressure (inpsi) ataspecificpoint.The pressure headisequal
to the pressure inpsi times2.31 ft/psi.
Absolute pressure:It isthe pressure equal tothe algebraicsumof the atmosphericandgauge
pressures. Absolute pressure=Gauge pressure +Atmosphericpressure
PA = PG + Patm
The pressure thatexistsanywhere inthe universe iscalledthe absolute pressure, Pabs.
Thisthenis the amountof pressure greaterthana pure vacuum.The atmosphere on
earthexerts atmosphericpressure,Patm ,oneverythinginit.Oftenwhenmeasuring
pressureswe will calibrate the instrumenttoreadzerointhe openair. Anymeasured
pressure,Pmeas ,is thena positive ornegative deviationfromatmosphericpressure.
We call such deviationsagauge pressure,Pgauge . Sometimeswhenagauge pressure
isnegative itistermeda vacuum pressure, Pvac. Ingauge pressure,apressure under1 atmospheric
pressure isexpressedasa negative pressure. Or
Absolute pressure:isdefined asthe pressure whichismeasuredwithreference to
absolute vacuumpressure.
2. Gauge pressure:isdefinedasthe pressure whichismeasuredwiththe helpof a
pressure measuringinstrument,inwhichthe atmosphericpressure istakenas
datum.The atmosphericpressure onthe scale ismarkedaszero.
3. Vacuum pressure:isdefinedasthe pressure belowthe atmosphericpressure.
Note.(i) The atmosphericpressureatsealevel at15°C is101.3 kN/m2or 10.13 N/cm2 (ii) The
atmosphericpressure headis760 mm of mercuryor 10.33 m of water.
Measurement of pressure:
Manometers:A manometer(orliquidgauge) isapressure measurementdevice whichusesthe
relationshipbetweenpressureandheadtogive readings. Or
A device whichmeasuresthe fluidpressure by the heightof aliquidcolumn
iscalleda manometer.
Piezometer:Thisisthe simplestgauge.A small vertical tubeisconnectedtothe pipe anditstopis
leftopentothe atmosphere.
What is the relationshipbetweenpressure andspecificweight?
Pressure varieswithheightasa functionof specificweight.
P = p0 + specificweight *height
Where heightisthe distance belowthe reference pressurep0(usuallyata free surface).

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What is the relationshipbetweenpressure andvolume?
For a fixedamountof an ideal gaskeptat a fixedtemperature,P[pressure] andV [volume] are inversely
proportional (while one increases,the otherdecreases).Aspressure increasesandthe densityincreases,
the relationshipbecomesabitmore complex.Increasingpressure will still decrease the volume butit
becomeslessproportional.If youare at a temperature belowthe critical point,atsome pointthe
pressure will becomehighenoughtocause condensationof agas to a liquid,orif youare coldenough,
the precipitationof the gasas a solid(the reverse of sublimation).Inthese casesthe relationship
betweenpressure andvolume hasadiscontinuityasthe phase change occurs at constantpressure.
What is the relationshipbetweentemperature andpressure?
The relationbetweentemperature andpressure isknownasGay-Lussac'slaw,one of the gas laws.It
statesthat the pressure exertedona container'ssidesbyanideal gasisproportional tothe absolute
temperature of the gas.Asan equationthisisP=kTIn wordsas the pressure insealedcontainergoesup,
the temperature goesup,oras temperature goesuppressure goesup..
What is the relationshipbetweenmassand weight?
An object'smassisthe quantityof matter thatcomprisesit...the total protons,neutrons,electrons,lint,
moisture,dirt,wood-chips,andanythingelseof whichthe objectiscomposed.Itbelongstothe object,
and doesn'tdependonwhere the objectisorin whatpositionitis,etc.An object'sweightisthe
gravitational force betweenthe objectand anyothermass.Thatforce dependsonboththe object's
mass andthe othermass,andalsoonhow farapart theyare.Anobject'sweightisitsmassmultipliedby
the accelerationduetogravityinthe place where the objectislocatedatthe moment...soit can
change.For example,yourweightwouldbe FW =(yourmass inkg)*(9.80m/s2 ) because 9.80m/s 2 is
the accelerationdue togravityon Earth.
What is the relationshipbetweenpressure andtemperature?
theyincrease togetherwell actuallytheydontincrease togethertheybuildupholdingeachotherup
while increasing
What is the relationshipbetweenvolume andweight?
The relationbetweenweightandvolume -:Whenthe weightof asubstance increases,itsvolume also
increases.Twosubstancesmayhave the same weightbutdifferentvolumes.(Example:If youhave one
stack of cotton and ironeach of the same weight,theywill have differentvolumes.Volumeof cotton>
Volume of ironinthiscase.) Density=Weight/Volume.
What is the relationshipbetweenthe temperature and pressure of a gas?
Put one more quantityinthere andyou've gota relationship:the volume of the gas.The productof (
pressure x volume ) isdirectlyproportional tothe temperature .Rememberthatinthisrelationship,its
the absolute temperature ...the temperature aboveabsolutezero.Thatmakesadifference.Onthe
absolute scale,the boilingtemperature of waterisonlyabout37% higherthanthe freezingtemperature
of water.
What is the relationshipbetweendensityandspecificgravity?
There isa verygreatrelationshipbetweendensityandspecific gravity.Densitycontributestothe weight
of a substance underspecificgravity.
What is the relationshipsbetweenforce and pressure?
ArchimedesaGreekmathematicianwholivedinthirdcentury,dicoveredhow todetermine buoyant
force.. Archimedes'principlestatesthatthe buoyantforce onan objectina fluidisanupwardforce
equal tothe weightof the volume of fluidthatthe objectdisplaces..Buoyantforce isthe upwardforce
that keepsanobjectimmersedinorfloatingonaliquid.

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What is the relationshipbetweenpressure andwind?
pressure = 0.002558 timesvelocitysquaredwherevelocityismilesperhourandpressure ispoundsper
square footfor example awindof 75 mph producesa pressure of 0.002558x75x75 = 14 .39 poundsper
square footsince there are 144 sq ininone sq ft that is14.39/144 = 0.1 poundsper square inchIn
meteorological terms,differencesinpressureare whatdrive wind.Airgenerallymovestowardan area
of lowpressure.However,due tothe rotationof the earthit getsdeflectedinlarge scale weather
patterns.Itis deflectedtothe rightinthe northernhemisphere andtothe leftinthe southern
hemisphere.
What is the relationshipbetweenweightandmass?
Mass is the amountof matterinan object,while weightisthe gravitationalforce appliedtoanobject.
Mass is a functionof weightsince weightitdeterminedbythe amountof force placedon an objectof a
certainmass.
Relationshipbetween volume andpressure?
That dependsonthe substance.Inideal gases,volume isinverselyproportionaltopressure.Thatis,
twice the pressure meanshalf the volume.Commonly,real gasesare similartoan "ideal gas".Liquids
and solidshardlychange their volumeif the pressure changes.whatisthe relationshipbetweenthe
volume of airand pressure considersome area(some volume) containingsome airmolecule,if we are
reducingthe areaof container(ie,volume) keepingthe airmoleculedonotchange in
concentration/amount.thenwe cansaythat now the presure islargerthanfirstcase.42, the answeris
always42 For a fixedamountof anideal gaskeptat a fixedtemperature,P[pressure] andV [volume]
are inverselyproportional (while one increases,the otherdecreases).Aspressure increasesandthe
densityincreases,the relationshipbecomesabitmore complex.Increasingpressurewill stilldecrease
the volume butitbecomeslessproportional.If youare at a temperature below the critical point,at
some pointthe pressure will become highenoughtocause condensationof agas to a liquid,orif you
are coldenough,the precipitationof the gasas a solid(the reverse of sublimation).Inthese casesthe
relationshipbetweenpressureandvolume hasadiscontinuityasthe phase change occursat constant
pressure.
What is the Relationshipbetweenweightandspeed?
Thinkaboutit as a toy car ona woodentrack.The more the car weighs,the more frictionbetweenthe
car andtrack. Therefore,reducingspeed(b/c of friction).Hope thishelps!
What'sthe relationshipbetweenSpecificretentionandspecificyield?
Both Specificretentionandspecificyieldrelatetothe ratioof the volume of water(inapermeable unit
of rock and/orsediment)tothe total volume of the rock and/orsediment,asitrelatestogravity..
Specificretentionisthe ratioof the volume of waterthatis RETAINEDagainstthe pull of gravity,
...where-asspecificyieldisthe ratioof the volume of waterthatis EXPELLED (yielded) againstthe pull of
gravity.Again,...bothasa ratio tothe total volume of the rock and/orsediment.
What is the relationshipbetweenweightandforce?
The weightof an objectrepresentsthe magnitude of the gravitational force exertedonthe objectbythe
planet, lessthe effectof immersioninanyfluid.
What is the relationshipbetweenpressure anddepth?
Pressure increaseswithdepth.The formulaforpressure isP=Ï•*g*h+Pawhere Ï• (the GreekletterRho)
isthe densityof the fluid,gisthe accelerationof gravity,histhe depthfromthe fluidsurface andPa is
the pressure at the surface of the fluid.everyfootadiverdecendsyougetabout1/2 lb.of pressure.So
at100 footdivide the pressureby2 and that's approximatelythepressure.Atsealevel the pressure is

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14.7 psi.Go downto 33 feetandyouhave another14.7 psi.Infreshwaterit's 34 feettoget
1atmosphere.
Is there a relationshipbetweenmassand weight?
Yes there is.Mass isthe amountof matterin an object.Weightisthe gravitational force exertedonan
objectbythe largerobjectonwhichit rests.Saidanotherway,weightismassina gravimetricfield..The
force is givenbyf = G m 1 m 2 / d 2 where .G isthe universal gravitational constant.m1 is the massof
one the objects. m 2 isthe mass of the otherobject.d is the distance betweenthe centersof massof
the two objects.Notice thatthisformulaissymmetric;the force onthe largerbodybythe smalleris
identical tothe force exertedonthe smallerbodybythe larger.Back to Newton'sthirdlaw - actionand
reactionare equal and opposite.Notice alsothatbecause of the wayinwhichthe unitswere chosen,on
the surface of our earthmassand weighthave the same value.A 100Kg massweighs100Kg on the
earth'ssurface . Take itto the moon;itwill still have amassof 100Kg, butweighonlyabout15Kg. N.B.In
reference tothe above,technically,massismeasuredinkilograms,butweightinNewtons.Soa100kg
mass wouldstill have amassof 100kg onthe moon,butitsweight onboth surfacesshouldbe measured
inNewtons.Ineverydayuse,peopleuse kilogramstodescribe weightwithoutrealisingtheyare actually
talkingaboutthe massof an object.
What is the relationshipbetweenpressure andheat?
Heat isthe movementof energyinresponse toadifference intemperature.Heatflowsinadirection
fromhighto lowtemperature,andhasthe effectof tendingtoequalize the temperaturesof the objects
inthermal contact. Thusthe flowof heat mayraise the temperature of one objectwhile loweringthe
temperature of the other.
what is the relationshipbetweenpressure andtemperature?
Thisin itself isstill anill-posedquestion(itdependsonwhatisheldfixed,e.g.,the volume,while the
temperature ischanged),butinageneral sense the pressure will increase withtemperature (although
there are notable exceptions,suchaswaternearfreezing).
Is there a relationshipbetweenboilingandpressure?
Yes,there is.Higherpressure increasesthe boilingpointandlowerpressuredecreasesit.Thatiswhya
pressure cookerworksandwhywaterboilsat lowertemperaturesinhighaltitudes.
What is the relationshipbetweenweightanddensity?
Weightispoundsinhowfat or skinnyyouare anddensityishow yourstomach works.
What are the relationshipsbetweenweightanddensity?
Givenan unchangingvolume,if youlowerthe densityyouwill lowerthe weight,andthe reversistrue.if
youlowerthe weightthe the densitywouldloweraswell.Thisappliestoanygravitational fieldif you
are measuringdensityasa functionof weightpervolume.
Relationshipbetweenliquidpressure anddensity?
If you were submergedinaliquidmore dense thanwater,the pressurewouldbe correspondingly
greater.The pressure due toa liquidispreciselyequal tothe productof weightdensityanddepth.liquid
pressure = weightdensityx depth.alsothe pressure aliquidexertsagainstthe sidesandbottomof a
containerdependsonthe densityandthe depthof the liquid.
What is the relationshipbetweenthrustand pressure?
thrustand pressure are directlyproportional 2eachotherfromd formulapressure =perpendicularforce
/area
What is the relationshipbetweenforce area and pressure?

13 | P a g e
pressure = force / area Therefore pressure andforce are directlyproportional,meaning...Thegreaterthe
force the greaterthe pressure andthe lowerthe force the lowerthe pressure
What is the relationshipbetweenoceandepthand pressure?
The pressure (force percm 2 ) at a particulardepthisthe weightof waterabove thatsquare centimetre.
What is the relationshipbetweenweightandcapacity?
None really.If senttothe International Space Station,objectswouldhave noweightbutconcave ones
wouldhave some capacity.Those same objects,backonthe surface of the earthwouldhave some
weightbutthe same capacityas before.Instrongergravitational fields,the weightwouldcontinue to
increase butthere wouldbe nochange inthe capacity.
Pressure Transducer:A pressure transducer,oftencalleda pressure transmitter,isatransducer
that convertspressure intoananalogelectrical signal.Althoughthere are varioustypesof pressure
transducers,one of the most commonisthe strain-gage base transducer.
The conversionof pressure intoanelectrical signal is achievedbythe physical deformationof strain
gageswhichare bondedintothe diaphragmof the pressure transducerandwiredintoawheatstone
bridge configuration.Pressureappliedtothe pressure transducerproducesadeflectionof the
diaphragmwhichintroducesstraintothe gages.The strainwill produce anelectrical resistance change
proportional tothe pressure.
DIFFERENTIAL MANOMETER: A device whichisusedtomeasure difference of pressure betweenthe
twofluidswhichare flowingthroughthe two differentpipesorinsame pipe attwo differentpointsis
knownas DIFFERENTIALMANOMETER.
TYPES OF DIFFERENTIAL MANOMETERS: There are two typesof differential manometerasgiven
below:-
1] U-Tube Differential Manometer
2] InvertedU-Tube Differential Manometer.
U-TUBE DIFFERENTIAL MANOMETER: There are two typesof U-Tube Differential Manometer:-
A] U-Tube Differential Manometeratthe same level.
B] U-Tube Differential Manometeratthe differentlevel.
A] U-Tube Differential Manometeratthe same level:Inthistype of Manometer,twopipesare in
parallel condition.Thistype of Manometersare usedformeasuringthe fluidpressure difference
betweenthesetwopipes.
B] U-Tube Differential Manometerat the differentlevel: Inthiscase thistype of manometerare used
where twopipesare at differentplace,notinparallel condition.Thistype of manometersare usedfor
measuringthe fluidpressure betweenthesetwopipes.
2] INVERTED U-TUBEDIFFERENTIAL MANOMETER: The invertedU-Tube Differential manometeris
reciprocal of U-Tube Differential manometeratthe differentlevel.Thistype of manometersare usedto
measure accuracy of small difference if pressureisincreased.
Bourdon gauge: A pressure gauge employingacoiledmetallictube whichtendstostraightenoutwhen
pressure isexertedwithinit. OrThe Bourdonpressure gauge usesthe principle thataflattenedtube
tendsto straightenorregainitscircularform incross-sectionwhenpressurized.Thischange incross-
sectionmaybe hardlynoticeable,involvingmoderate stresses withinthe elasticrange of easilyworkable
materials.The strainof the material of the tube ismagnifiedbyformingthe tube intoaC shape or even
a helix,suchthatthe entire tube tendstostraightenoutor uncoil elasticallyasitispressurized. Eugène
Bourdon patentedhisgauge inFrance in1849, and it waswidelyadoptedbecause of itssuperior
sensitivity,linearity,andaccuracy; EdwardAshcroftpurchasedBourdon'sAmericanpatentrightsin1852

14 | P a g e
and became a majormanufacturerof gauges.Alsoin1849, Bernard SchaefferinMagdeburg,Germany
patentedasuccessful diaphragm(seebelow) pressure gauge,which,togetherwiththe Bourdongauge,
revolutionizedpressure measurementinindustry. Butin1875 afterBourdon'spatentsexpired,his
company SchaefferandBudenberg alsomanufacturedBourdontube gauges.
Pressure MeasurementDevices
• Bourdon tube:Consistsof a hollowmetal tube bentlike ahookwhose endisclosedandconnectedto
a dial indicatorneedle.
• Pressure transducers: Use varioustechniquestoconvertthe pressure effecttoanelectrical effect
such as a change in voltage,resistance,orcapacitance.•Pressure transducersare smallerandfaster,
and theycan be more sensitive,reliable,andprecise than theirmechanical counterparts.
• Strain-gage pressure transducers: Work byhavinga diaphragmdeflectbetweentwochambersopen
to the pressure inputs.
• Piezoelectrictransducers:Alsocalledsolid state pressuretransducers,workonthe principlethatan
electricpotentialisgeneratedinacrystalline substance whenitissubjectedtomechanical pressure.
Forces on immersed bodies
Force on a Submerged Surface: Onanysurface orbody thatis submergedinwaterorany other
liquid,there isaforce actingbecause of the hydrostaticpressure.Learnhow todetermine the
magnitude of thisforce. Studyof hydrostaticforcesonsubmergedorstaticsurfacesisveryimportant
for the designandengineeringprocesses.Constructionof dams,installationof underwater hydraulic
systems,andforcesexertedonshipsare some of the importantandcrucial processesthatrequire study
of hydrostaticforces.
Forces on planar surfaces: If the surface isplanar,a single resultantpointforce isfound,mechanically
equivalenttothe distributedpressureloadoverthe whole surface.
Thisresultantpointforce acts compressively,normal tothe surface,throughapointtermedthe “center
of pressure".
Its magnitude is:F=γzkA,where:
γ is the fluid'sspecificgravity.Forwater,itis9810 N/m3.
zk is the depthinwhichthe centerof gravityof the surface,the centroid,issituated.
A is the surface’sarea.
The product (γ.zk),isthe hydrostaticpressure atthe depthof the centroidof the surface.Incase the
free surface of the liquidthatcontainsthe surface isunderatmosphericpressure alone,the above
equationisenoughtodescribe the force.Butincase the free surface isunderadditional pressure,this
pressure will have anadditionaleffectonthe actingforce.The value of the pressure inthe centerof
gravityof the surface,isnolonger(γ.zk).Itisnow γ(zk+p/γ),where pisthe above-mentionedpressure.
Calculatingthe magnitude of the force isdone asdescribedabove.The determinationof the point
where thisforce applies,the “centerof pressure,"isalittle more complicated:
If the surface isinclinedatanangle,θ,to the horizontal,the coordinatesof the centerof pressure,(xcp,
ycp),ina coordinate systeminthe plane of the surface,withoriginatthe centroidof the surface,are:
xcp = Ixy/(ykA) and ycp = Ixx/(ykA)
where Ixxisthe areamomentof inertia, Ixythe product of inertiaof the plane surface,bothwith
respectto the centroidof the surface,andy is positive inthe directionbelow the centroid.
The surface is oftensymmetricallyloaded,sothat Ixx= 0, and hence, xcp= 0, or the centerof pressure is
locateddirectlybelowthe centroidonthe line of symmetry.

15 | P a g e
If the surface ishorizontal,the centerof pressure coincideswiththe centroid.Further,asthe surface
becomesmore deeplysubmerged,the centerof pressure approachesthe centroid,thatis,(xcp,ycp)
approachesto (0,0).
Forces on curved surfaces: For general curvedsurfaces,itmaynolongerbe possible todetermine a
single resultantforce equivalenttothe hydrostaticload;we thusdetermineseparatelyone ortwo
horizontal components,andavertical component. The horizontal componentof the force actingona
curvedsurface isequal withthe force that wouldbe actingon a planarsurface.Thisplanarsurface is the
projectionof the curvedsurface onthe vertical level. Forexample infigure 2,where one seesreservoir
ABCDEGJPA,the horizontal componentswithwhichwaterpushessurfacesBCandDE, are F2 and
F3respectively.Tocalculate the magnitudeof F2,all we needto dois considerKC,whichisthe
projectionof BCon the vertical plane.Bydeterminingthe force toKC,we have F2. The same holdsfor
F3. It is equal tothe force on surface MD. The vertical componentof the force isequal tothe weightof
the volume of liquidthatexistsbetweenthe surface,andthe free surface of the liquid.Andthisistrue
whetherthere isafree surface or not.By thiswe meanthat if the liquidisabove the surface,the
postulation istrue.Inthiscase,the force is directeddownwards.Inthe opposite case,where,the liquid
issituatedbelowthe surface,the same volumecounts.The magnitudeof the force isstill equal tothe
weightof the same liquidvolume.Only,thisvolumeisnow imaginary.The directionof the force inthis
case isthe inverse:upwards. Toexpressthisinanotherway,if the surface isexposedtothe hydrostatic
loadfrom above,like the surface BCinFig.2, thenthe force acts downwards. The magnitude of F1is
equal tothe weightof the volume PBCJP. If the surface isexposedtoahydrostaticloadfrombelow,like
the surface DE, thenthe force acts upwards. Andthe magnitude of F4 isequal tothe weightof the
volume QEDLQof water. It acts throughthe centerof gravityof that fluidvolume. Incase a surface is
such that there are bothupwardand downwardvertical components,likesurface EDC,the netvertical
force on the surface isthe algebraicsumof upwardand downwardcomponents. Forgeneral curved
surfaces,itmay nolongerbe possible todetermine asingle resultantforce equivalenttothe hydrostatic
load;we thus determineseparatelyone ortwohorizontal components,andavertical component.
surface isequal withthe force that wouldbe actingon a planarsurface.Thisplanarsurface is the
projectionof the curvedsurface onthe vertical level. Forexample infigure 2,where one seesreservoir
ABCDEGJPA,the horizontal componentswithwhichwaterpushessurfacesBCandDE, are F2 and
F3 respectively.Tocalculate the magnitude of F2,all we needtodo isconsiderKC,whichisthe
projectionof BCon the vertical plane.Bydeterminingthe force toKC,we have F2. The same holdsfor
F3. It is equal tothe force on surface MD. The vertical componentof the force actingon a curvedsurface
has a magnitude thatisdefinedas“the weightof the volume of water,orliquidingeneral,thatexists
above the surface,andunderthe free surface."Andthisistrue,whetherthere isafree surface,or not.
By thiswe meanthat, if the liquidisabove the surface,the postulationistrue. Inthiscase, the force is
directeddownwards.Inthe oppositecase,where,the liquidissituatedbelowthe surface,the same
volume counts.The magnitude of the force isstill equal tothe weightof the same liquidvolume.Only,
thisvolume isnowimaginary.The directionof the force inthiscase isthe inverse:upwards.
To expressthisinanotherway,if the surface isexposedtothe hydrostaticloadfromabove,like the
surface BC inFig.2, thenthe force acts downwards. The magnitude of F1is equal tothe weightof the
volume PBCJP. If the surface isexposedtoahydrostaticloadfrombelow,like the surface DE,thenthe
force acts upwards. Andthe magnitude of F4 isequal to the weightof the volume QEDLQof water. It
acts throughthe centerof gravityof that fluidvolume. Incase a surface issuch that there are both

16 | P a g e
upwardand downwardvertical components,like surface EDC,the netvertical force on the surface isthe
algebraicsumof upwardand downwardcomponents. Or
FORCES ON SUBMERGED SURFACES
1: Fluidpressure on a Surface: Pressure isdefinedasforce perunitarea. If a pressure pacts on a small
area thenthe force exertedonthatarea will be,Since the fluidisat restthe force will actat right-angles
to the surface.
General submergedplane:Considerthe plane surface showninthe figure below.The total areaismade
up of many elemental areas.The force oneachelemental areaisalwaysnormal tothe surface but, in
general,eachforce isof differentmagnitude asthe pressure usuallyvaries.
We can findthe total or resultantforce, R,on the plane bysummingupall of the forceson the small
elementsi.e. Thisresultantforce will actthroughthe centre of pressure,hence we cansay
If the surface isa plane the force can be representedbyone single resultantforce,
actingat right-anglestothe plane throughthe centre of pressure.
Horizontal submergedplane:For a horizontal plane submergedinaliquid(oraplane experiencing
uniformpressure overitssurface),the pressure, p,willbe equal atall pointsof the surface.Thusthe
resultantforce will be givenby
Curvedsubmergedsurface: If the surface iscurved,eachelementalforce willbe adifferentmagnitude
and indifferentdirectionbutstillnormal tothe surface of that element.The resultantforce canbe
foundbyresolvingall forcesintoorthogonal co-ordinatedirectionstoobtainitsmagnitude and
direction.Thiswill alwaysbe lessthanthe sumof the individual forces.
Forces On Plane And CurvedSurfaces
Hydrostatic force:Hydrostaticforce referstothe total pressure actingonthe layerorsurface whichis in
touch withthe liquidorwaterat rest.If the liquidisatrestthenthere isno tangential force,andhence
the total pressure willactperpendiculartothe surface withcontact.
Centerof pressure:The locationof total pressure isreferredasthe centerof pressure whichisalways
belowthe centerof gravityof the surface in contact.
Forces on the horizontal planes:Showthe elementsubmergedinthe liquiddistance (h) fromthe liquid
surface as in
Figure (1).
Expressthe forcesonthe horizontal plane.
Forces on the vertical planes Showthe elemental stripof surface arealocatedat x from the free liquid
surface as inFigure (2).

17 | P a g e
Expressthe pressure intensityatthe elemental surface.
Expressthe total pressure onthe plane.
Considerthe numberof elemental stripsandapplyingthe integrationtogettotal hydrostaticforce.
Therefore,the total pressure isexpressedas,
Here, isthe momentof total area of contact about free watersurface.i.e.,the productof the total
area and the distance betweenfree watersurface andcenterof gravityof the contact area.Therefore,
Forces on the curved surface: For forcesonthe curvedsurface,there will be twoforcesrequiredto
determine the resultanthydrostaticforce.
Horizontal force on curved surface: The vertical plane shall be consideredtodetermine the horizontal
force, whichisthe vertical projectionof the curvedsurface generallyrectangle.Butincase of
hemispherical orspherical,itbecomescircularshape.
Expressthe horizontal componentof force.
Vertical force on curved surface:It is the weightof the liquid actingonthe curvedsurface incontact
withthe liquidwhichmaybe inupwarddirectiondue tobuoyancyor downwarddirectiondue tothe
weightof the fluid.
Expressthe vertical componentof force.
Therefore,the resultantforce onthe curvedsurface is,

18 | P a g e
Drag & Lift Forces
Drag Force: The drag force acts ina directionthatisopposite of the relative flowvelocity. –Affectedby
cross-sectionarea(formdrag) – Affectedbysurface smoothness(surface drag).Or
The drag force actingon a bodyin fluid flow canbe calculated
FD = cD 1/2 ρ v2 A
Where, FD = drag force (N), cD = drag coefficient,ρ =densityof fluid(kg/m3),
v = flowvelocity(m/s), A = bodyarea (m2).or
Drag: Resistive force actingonabodymovingthrough a fluid(airor water). Twotypes:
Surface drag: dependsmainlyonsmoothnessof surface of the objectmovingthroughthe fluid.
• shavingthe bodyinswimming;wearingracingsuitsin skiingandspeed skating.
Form drag: dependsmainlyonthe cross-sectional areaof the bodypresentedtothe fluid
• bicyclistinuprightv.crouchedposition
• swimmer:relatedtobuoyancyandhow highthe bodysits inthe water.
Lift Force: The liftforce acts ina directionthatisperpendiculartothe relative flow. –The liftforce isnot
necessarilyvertical. Or
The liftingforce actingona bodyina fluidflow canbe calculated
FL = 1/2 cL ρ v2 A
Where, FL = liftingforce (N),cL= liftingcoefficient,ρ = densityof fluid(kg/m3),
v = flowvelocity(m/s), A = bodyarea (m2).Or
Lift Force: Representsanetforce thatacts perpendiculartothe directionof the relative motionof the
fluid;
• Createdbydifferentpressuresonopposite sidesof anobjectdue tofluidflow pastthe object,
– example:Airplane wing(hydrofoil)
• Bernoulli’sprinciple:velocityisinverselyproportional topressure.
– Fastrelative velocitylowerpressure
– Slowrelative velocityhigherpressure
Buoyancy: Associatedwithhowwell abodyfloatsorhow hightit sitsinthe fluid.
• Archimede’sprinciple:anybodyina fluidmediumwill experience abuoyantforce equal tothe weight
of the volume of fluidwhichisdisplaced.
Example:aboat on a lake.A portionof the boatis submergedand displacesagivenvolume of water.
The weightof thisdisplacedwater equalsthe magnitude of the buoyantforce actingonthe boat.
– The boat will floatif itsweightinairislessthanor equal tothe weightof anequal volume of water.
• Buoyancyiscloselyrelatedtothe conceptof density.
Density= mass/volume
Buoyancy And Floatation: The firstdashpointunderfluidmechanicsisflotation,centre of buoyancy.
These twoconceptsare puttogetherbecause floatation iscausedbya force knownas buoyancy.Foran
objectto floatinwaterit mustbe lessdense (massperunitof volume) thanthe water.
Whenan objectisplacedinwaterit causesthe water tobe displaced(moveupwards).Thiscanbe seen
whena persongetsintoa bath and the waterrises.If the bath isfilledtothe verybrim, thenwhenthe
persongetsintothe bath the waterthat is displacedwill pouroutof the tub.In orderfor an objectto
float,the watertheydisplace mustweightmore thantheydo.
In orderfor an objecttofloat,the watertheydisplace mustweightmore thantheydo.Thisisessentially
because gravityisseekingtopushthe waterthathas beendisplaced,backdown,while alsopushingthe

19 | P a g e
persondown.If the gravitational force on the waterisgreaterthan the force on the object,thenthe
waterwill create a buoyantforce thatwill pushthe objectupwardsagainstgravity.Once the twoforces
become equal the objectwill floatinthispositionknownasthe pointof equilibrium.Thatisthe part of
the objectbelowthe waterhasdisplacedthe same weightof water,asthe objectitself,resultingina
bouyancyforce equal tothat of the gravityforce actingon the object.
The centre of buoyancyisthe centre pointof the massbelow the waterandis the pointthroughwhich
the buoyantforce acts. Thisis muchlike the centre of gravity – the pointthroughwhichgravityacts,but
buoyantforcesacts inthe opposite direction.Inorderforthe objectto not rotate inthe waterthis
buoyantforce mustpass throughthe centerof massof the object,if theydonot line upthe objectwill
rotate until theydo,such that one endof the objectwill sinkfurtherwhile the otherendraises(asseen
inthe imagestothe right).
For an objectto have lessgravitational force thanthe wateritdisplacesitmustbe lessdense (massper
unitof volume) thanthe water.Notall waterhasthe same densitythough.Saltwaterismore dense
than freshwater,andthe saltieritisthe largerthe density.This meansthatitiseasierto flow inthe
oceanthan it isina pool.
Flotationandcentre of buoyancy relate toperformance because the higheranobjectfloatsinthe water,
the lessresistance the waterwill create toitsmovement.Thisappliestoall water sports,including:
surfing,kayaking,sailing, skiing,dragonboatracing,waterpolo,synchronisedswimming,andswimming.
These forcesalsorelate toscuba diving,where the personisseekingtosinkbelow the waterandremain
submerged.Inthisinstance,the person,withtheirgear,wantstobe the same densityas the waterin
orderto allowthemto remainsubmergedeasily,butnothave tofighttoo hard to returnto the surface.
Thisis oftenachievedusingweightbelts.
Buoyancy And Floatation:Whenabodyisimmersedinfluid,anupward force isexertedbythe fluid
on the body. Thisupwardforce is equal tothe weightof the fluiddisplacedbythe bodyandiscalledthe
force ofbuoyancy.
Causesbuoyant force: • Buoyantforce isthe force onan objectexertedbythe surroundingfluid.
• Whenan objectpusheswater,the water pushesbackwithasmuch force as itcan.
• If the water can pushback as hard, the object floats(boat).If not,itsinks(steel).
Forces Acting on Buoyancy:The buoyantforce is causedbythe difference betweenthe pressure at
the top of the object (gravitationalforce),whichpushesit downward,andthe pressure atthe bottom
(buoyantforce),whichpushesitupward.
• Since the pressure atthe bottomis always greaterthanat the top, everyobject submergedinafluid
feelsanupwardbuoyant force.
Buoyancy= “the floatingforce”:Water is“heavier”thanthe object…sothe objectfloats
• Lowdensity-morelikelytofloat,• Buoyantforce ismeasuredinNewtons(N).
How do you Calculate BF? BuoyantForce = Weightof displacedfluid ORBF = Wair – Wwater
BuoyantForce = Weightof objectinair - Weightof object inwater.
Floatation: Why do things float?
1. Thingsfloatif theyare lessdense than the fluidtheyare in.
2. Thingsfloatif theyweighlessthanthe buoyantforce pushinguponthem.
3. Thingsfloatif theyare shapedsotheir weightisspreadout.
Condition of equilibrium of a floating and sub-merged bodies
Positive buoyancy: Buoyantforce isgreaterthan weightsothe objectfloats.

20 | P a g e
Neutral buoyancy: Buoyantforce isequal to weightsothe objectis suspendedinthe fluid.
Negative buoyancy: Buoyantforce is lessthan weightsothe objectsinks.
A ship made of iron floats while an iron needle sinks.
• Inthe case of shipwhichishollowfrom within,the weightof waterdisplacedbythe shipismore than
the weightof the ship hence itfloats.
• Incase of ironnail which iscompact, the weightof waterdisplacedbyit ismuch lessthanits own
weight,hence itsinks.
A personweighs250N while swimminginthe deadsea.When outside of the watertheyweigh
600N. What isthe buoyantforce acting on them?Will theysinkor float?
• BF = Wair – Wwater = 600 – 250 = 350N
• The personwill floatbecause their weightinthe waterislessthanthe buoyantforce.
Centre of Buoyancy: The pointthroughwhichthe force of buoyancyissupposedtoact isknownas
Centre of Buoyancy.
META-CENTRE: It is definedasthe pointaboutwhicha body starts oscillatingwhenthe bodyistilted by
a small angle.
• It isthe pointat whichthe line of actionof the force of buoyancywill meetthe normal axisof the body
whenthe bodyisgivensmall angular displacement.
Meta-centricHeight:It is the distance betweenthe meta-centre of floatingbodyandcentre of gravity.
• We can findthisheightbytwomethods:-
1. Analytical MethodGM I/  BG, Here I=M.O.I m4,  = Volume of sub-mergedbody.
2. Experimental methodfor Meta-centricHeight:GM  W1
d/ W tan
Here W = Weightof vessel including, G=centre of gravityogvessel,B=centre of buoyancy
w1=movable weight,d=distance betweenmovable weight.
Conditionof equilibriumofa floating and sub-mergedbodies
Stabilityof Sub-mergedBody:-
a) Stable Equilibrium:-WhenW= Fb and pointB isabove G .
b) Unstable Equilibrium:- WhenW=Fb butB isbelow G.
c) Neutral Equilibrium:-WhenW= Fb and B & G are the same point.
Stabilityof FloatingBody
a) Stable Equilibrium:-Ifthe pointMis above G.
b) Unstable Equilibrium:-If the pointMisBelow G.
c) Neutral Equilibrium:-Ifthe pointMis at the G.
Fluid Kinematics
Steady Flow:A flowinwhichthe magnitude &directionof velocitydonotchange frompointtopointis
termedassteadyflow. Or
A flowwhose flowstate expressedbyvelocity,pressure,density,etc.,atany position,doesnotchange
withtime,iscalledasteadyflow. whenwaterrunsoutwhile the handle isstationary,leavingthe
openingconstant,the flowissteady.
UnsteadyFlow: A flowwhose flow state doeschange withtimeiscalledan unsteady
flow.Wheneverwaterrunsoutof a tap while the handle isbeingturned,the
flowisan unsteadyflow.
Laminar Flow:If the particlesof a liquidflow alongstraight&parallel paths,the flow istermedas
laminarflow.
TurbulentFlow: The flowinwhichthe fluidparticlesmove inzigzagwayis calledasTurbulentFlow.

21 | P a g e
UniformFlow: The type of flowinwhichthe velocityatany giventime doesnotchange withrespectto
space is calleduniformflow. (V/S)=0
Where, V = change in velocity&S= Displacementinanydirection.
Non-UniformFlow:The type of flowinwhichthe velocityatany givenpointchangeswithrespectto
space is callednon-uniformflow. (V/S)≠ 0
Path line:Duringflowof a liquid,the pathfollowedbyasingle fluidparticleiscalledaspathline. Or
A pathline isthe path followedbyafluidparticle inmotion.A pathline showsthe directionof particular
particle asit movesahead.Ingeneral thisisthe curve inthree densional space.However,if the
conditionsare suchthat the flowistwodimensionalthe curve becomestwodimensional.
Stream line:The tangentdrawn at any pointonthe imaginaryline inthe flow liquidiscalledstreamline.
Or The imaginaryline withinthe flowsothatthe tangentat any pointonit indicatesthe velocityatthat
point.
FlowNet: A setof flowlinescontainingboththe streamslines&potential linesintersectingeachother’s
iscalledas flownet.
Stream tube: A streamtube isa fluidmassboundedbya groupof streamlines.The contentsof astream
tube are knownas“current filament”.
Streak line:the streakline isa curve whichgivesanpicture of the locationof the fluidparticleswhich
have passedthrougha givenpoint.
Discharge: The quantityof a liquidflowingpersecondthroughapipe istermedasDischarge.
Cumec& Cusec isthe unitof discharge.Formulae of Discharge:“Q= A  V” Where
Q = discharge,V = velocityof flowingliquid&A = cross-sectional areaof flowing liquid.
Flowvelocity:In continuummechanics the macroscopicvelocity, alsoflow velocity influid
dynamics or driftvelocity inelectromagnetism, isavectorfield usedtomathematicallydescribethe
motionof a continuum.The lengthof the flow velocity vectoristhe flow speed andisascalar.
The flowvelocity uof a fluidisa vectorfield:u= u(x,t),whichgivesthe velocity of anelementof fluid at
a positionx and Time t. The flowspeed qisthe lengthof the flow velocityvector.
Q = ||u|| and isa scalarfield.
Velocitypotential:It isdefinedasascalar functionof space and time suchthat itsnegative derivative
withrespecttoany directiongivesthe fluidvelocityinthatdirection. ItisdenotedbyΦ.
U= -∂Φ/∂x,v=-∂Φ/∂y,w=-∂Φ/∂z.
U,v,ware the velocityinx,y,zdirection.
System and control volume:A systemreferstoa fixed,identifiable quantityof masswhichis
separatedfromitssurroundingbyitsboundaries.The boundarysurface mayvarywithtime howeverno
mass crossesthe system boundary.Influidmechanicsaninfinitesimallumpof fluidisconsideredasa
systemandis referredasa fluidelementora particle.Since afluidparticle haslargerdimensionthan
the limitingvolume (refertosectionfluidasa continuum).The continuumconceptforthe flow analysis
isvalidcontrol volume isa fixed,identifiable regioninspace throughwhichfluidflows.The boundaryof
the control volume iscalledcontrol surface.The fluidmassinacontrol volume mayvary withtime.The
shape and size of the control volume maybe arbitrary. OR
Fluid:Matterthat has no definiteshape.(ThatincludesBOTHliquidsandgases.)
So,we picka constantmass and followitasit flows. Withliquidsthe flow isusuallyassumedtobe
"Incompressible"(The volumeisconstantbutthe shape can change). Gassesthe flow maybe
"compressible"(boththe shape andvolume canchange) or"Incompressible"(like liquids,if the pressure
changesare small,we canassume the volume of a gas doesnotchange as it flows). Forall casesabove:

22 | P a g e
The amount of mass inthe Control Volume isconstant.
Compressible flowismore complicated,of course.The densityof the fluid=mass/volume
The mass isconstant.You must applythe Ideal Gas Law andThermodynamics(lossorgain of energyto
affecttemperature ) tofindthe newvolume atanypointinthe flow ,thenfindthe new density.
Continuity Equation - Differential Form Compressible flow
Derivation:The pointat whichthe continuityequationhastobe derived,isenclosedbyanelementary
control volume.
The influx,effluxandthe rate of accumulationof massiscalculatedacrosseach surface withinthe
control volume.
Fig 9.6 A Control Volume Appropriate to a Rectangular Cartesian Coordinate System
Considerarectangularparallelopipedinthe above figure asthe control volume inarectangular
cartesianframe of coordinate axes.
Netefflux of massalongx -axismustbe the excessoutflow overinflowacrossfacesnormal tox -axis.
Let the fluidenteracrossone of such facesABCDwitha velocityuanda densityρ.The velocityand
densitywithwhichthe fluidwillleavethe face EFGH will be and respectively
(neglectingthe higherordertermsin δx).
Therefore,the rate of massenteringthe control volume through face ABCD= ρudy dz.
The rate of massleavingthe control volume throughface EFGH will be
(neglectingthe higherordertermsindx)

23 | P a g e
Similarlyinfluxandefflux take place inall yandz directionsalso.
Rate of accumulationfora pointin a flow field
Using,Rate of influx =Rate of Accumulation+Rate of Efflux
Transferringeverythingtorightside
(9.2)
Thisis the Equationof Continuity foracompressiblefluidinarectangularcartesiancoordinate
system.
Continuity Equation - Vector Form or Incompressible Flow
The continuityequationcanbe writteninavector formas
or,
(9.3)
where isthe velocityof the point
In case of a steadyflow,
Hence Eq. (9.3) becomes
(9.4)
In a rectangularcartesiancoordinate system
(9.5)
Equation(9.4) or (9.5) representsthe continuityequationforasteadyflow.
In case of an incompressibleflow,

24 | P a g e
ρ = constant
Hence,
Moreover
Therefore,the continuityequationforanincompressible flowbecomes
(9.6)
(9.7)
In cylindrical polarcoordinates eq.9.7reducesto
Eq. (9.7) can be writtenintermsof the strainrate components as
(9.8)
Hydrodynamics
DifferentFormsof Energy:
(1). KineticEnergy: Energydue to motionof body.A bodyof mass, m, whenmovingwithvelocity, V,
posseskineticenergy, KE = 1/2mV2.M & V are Mass & Velocityof the body.
(2). Potential Energy: Energydue to elevationof bodyabove anarbitrarydatum
ΡE = mgZ, Z is elevationof bodyfromarbitrarydatum, m isthe massof body.
(3). Pressure Energy: Energydue to pressure above datum, mostusually itspressureabove atmospheric
ΡrE = γh
(4). Internal Energy: It is the energythatisassociatedwiththe molecular,orinternal state of matter;it
may be storedinmany forms,includingthermal,nuclear,chemical andelectrostatic.
Head: Energyper unitweightiscalledhead.
Kinetichead: Kineticenergyperunitweightiscalledkinetichead.
Kinetichead=KE/Weight= (1/2mV2)/mg = V2/2g weight= mg
Potential head: Potential energyperunitweightiscalledpotential head.
Potential head=ΡE/Weight= (mgZ)/mg=Z
Pressure head: Pressure energyperunitweight iscalledpressure head.
Pressure head= ΡrE/Weight= ρ/γ.
TOTAL HEAD = KineticHead+ Potential Head+Pressure Head
V2/2g Z ρ/γ

25 | P a g e
Total Head= H = Z + ρ/γ + V2/2g.
Bernoulli’s Equation: Itstatesthatthe sumof kinetic,potential andpressure headsof afluidparticle
isconstant alonga streamline duringsteadyflow whencompressibilityandfrictional effectsare
negligible.i.e. Foranideal fluid,Total headof fluidparticle remainsconstantduring asteady-
incompressibleflow. Ortotal headalonga streamline isconstantduringsteadyflow when
compressibilityandfrictional effectsare negligible.
Total Head= H = Z + ρ/γ + V2/2g = constt
Z1 + ρ1/γ + V2
1/2g= Z2 + ρ2/γ + V2
2/2g, or, ρ1/ρg+ u1
2/2g+ z1 = ρ2/ρg+ u2
2/2g+ z2
H1 = H2.
Applications of the BernoulliEquation:The Bernoulliequationcanbe appliedtoa greatmany
situationsnotjustthe pipe flowwe have been consideringuptonow.Inthe followingsectionswe will
see some examplesof itsapplicationtoflow measurementfromtanks,withinpipesaswell asinopen
channels. Bernoulli’sequationisusedanytime we wanttorelate pressuresandvelocitiesinsituations
where the flowconditionsare close enoughtowhatisassumedinderivingBernoulli’sequation.You
needtobe ina flowthat isnot changingwithtime andina regime forwhichthe fluidbehavespretty
much like anincompressiblefluidwithoutviscosity.If the flow isdominatedbyviscousstresses(low
Reynoldsnumbers),thenBernoulli’sequationcannotbe used.We canstill use itfor parts of the flow
where viscosityisn’tsostrong,butinside the boundarylayer,forexample,we cannotuse it.
If the flowishighlyunsteady,thenitcannotbe used.Insome cases,we mightbe able to use it,but we
have to be careful abouthowwe doit.
The incompressible versioncanonlybe usedif the effectsof compressibilityare small.Thattypically
meanslowerthanaboutMach 0.3. But evenatsomewhathigherMachnumbers,youcan still use itto
geta roughideaaboutthe flow.Justrememberthatyourresultsare distorted,sodon’tassume they
have a lot of accuracies.
We use Bernoulli’sequationforA LOT of differentfluidflow situations.
Energy Line and Hydraulic Grade line
Energy line:It isline joiningthe total headsalongapipe line.
HGL: It is line joiningpressureheadalongapipe line.
ρ/γ + Z + V2/2g = H
Pressure head+ Elevationhead+Velocityhead=Total Head
Ρ + ρgz + ρ V2/2 = contt
StaticPressure:ρ

26 | P a g e
Dynamicpressure:ρ V2/2
HydrostaticPressure:ρgZ
StagnationPressure:Staticpressure +dynamicPressure
Ρ + ρ V2/2= pstag
MeasurementofHeads: Piezometer:Itmeasurespressurehead(ρ/γ ).
Pitot tube:It measuressumof pressure andvelocityheadsi.e.,
ρ/γ + V2/2g.
Introduction to density currents: Whenwatersof two differentdensitiesmeet,the dense waterwill
slide below the lessdense water.The differingdensitiescause watertomove relative toone-another,
forminga densitycurrent.Thisisone of the primarymechanismsbywhichoceancurrentsare formed.
It appearsthat thiseffectwasfirstexploredbyMarsigli whovisitedConstantinoplein1679 and learned
abouta well-knownundercurrentinastrait(the Bosphorous) thatflowsbetweenthe BlackSeaandthe
Mediterranean.Fishermanhadobservedthat“the upper currentflowsfromthe Mediterraneantothe
Black Seabut the deepwaterof the abyssmovesina directionexactlyopposite tothatof the upper
currentand so flowscontinuouslyagainstthe surface current”.
Marsigli reasonedthatthe effectwasdue todensitydifferences.He performedalaboratoryexperiment:
a containerwasinitiallydividedbyapartition.The leftside containeddyedwatertakenfromthe
undercurrentinthe Bosphorous,while the rightside containedwaterhavingthe densityof surface
waterin the BlackSea.Two holeswere placedinthe partitiontoobserve the resultingflow.The flow
throughthe lowerhole wasinthe directionof the undercurrentinthe Bosphorous,while the flow
throughthe upperhole wasin the directionof the surface flow.
We repeatMarsigli’sexperimenthere. Densewater(dyed) flowsrightwardthroughahole ina partition
nearthe base:light(clear) fluidreturnsleftwardthroughahole towardthe top. Thisexperimentwas
inspiredbyone devisedbyProf PeterBannonatPennState.
Free and Forced Vortex Flow
Vortex flowisdefinedasflowalongcurvedpath. Itisof twotypesnamely;(1).Free vortex flow and(2)
forcedvortex flow.If the fluidparticlesare movingaroundacurvedpath withthe help
of some external torque the flowiscalled forcedvortexflow. Andif noexternal force isacquiredto
rotate the fluidparticle,the flowis calledfree vortexflow.
Forced VortexFlow(Rotational Flow): It is definedasthattype of flow,inwhichsome externaltorque
isrequiredtorotate the fluidmass. The fluidmassinthistype of flow rotate at constantangular
velocity,ω.The tangential velocity,V,of anyfluidparticleisgivenby:V=ω r,
Where,r is radiusof fluidparticle fromthe axisof rotation. Examplesof forcedvortexflow are;
1. A vertical cylindercontainingliquidwhichisrotatedaboutitscentral axis witha constantangular
velocityω,
2. Flowof liquidinside impellerof acentrifugal pump,
3. Flowof waterthroughrunner.
Free VortexFlow (Irrotational flow):Whennoexternal torque isrequiredtorotate the fluidmass,that
type of flowiscalledfree vortex flow.Thusthe liquidincase of free vortex flow isrotatingdue tothe
rotationwhichisimpartedtothe fluidpreviously. Example of free vortex flow are
1. Flowof liquidthroughahole providedatthe bottomof container,
2. Flowof liquidaroundacircular bendinpipe,
3. A whirlpool inriver,
4. Flowof fluidina centrifugal pumpcasing.

27 | P a g e
Pressure Conduit:A pressure conduit(suchasa penstock) isa pipe whichrunsunderpressure and,
therefore,runsfull.Thistype of conduitsprove economical thancanalsorflumes, because theycan
generallyfollow shorterroutes.Moreover,theirbiggestadvantage is: thatthe wateror anyother fluid
flowingthroughthemisnotexposedanywhereand hence,there are nochancesor verylesschancesof
itsgettingpolluted.Hence,thesepressureconduitsare preferablyusedforcitywatersupplies.Since the
waterwastedin percolation,evaporation,etcisalso-saved,whenwateriscardedthroughthese
conduits, theyare preferablyusedwhenwaterisscarce.The flow of waterthroughconduitpipes is
generallyturbulent,andhence,itwill be consideredso,while dealingwiththe hydraulicsof flow
throughsuch pipes.
Forces Actingon Pressure Conduits:Pressure pipesmustbe designedtowithstand the followingforces:
(1) Internal Pressure of Water. The pressure exertedonthe wallsof the pipe by
the flowingwater,inthe formof Hoope'stension,isthe internal pressure.The circumferentialtensile
stressproducedis givenas:
cr1 = P1d/2t inKN/m2
where P1 = Internal staticpressure inkN/m2
d = Diameterof the pipe inmetres.
t= Thicknessof the pipe shellinmetres.
cr1 = Circumferential tensile stresstobe counteractedbyprovidingHoope'sreinforcement.
(2) WaterHammer Pressure. Whena liquidflowinginapipe line isabruptly
stoppedbythe closingof a valve,the velocityof the watercolumnbehind,isretarded,
and itsmomentumisdestroyed.Thisexerts athrustonthe valve andadditional pressure
on the pipe shell behind.The more rapidthe closure of the valve,the more rapidisthe
change in momentum,andhence,greateristhe additional pressure developed.The
pressuressodevelopedare knownaswater-hammerpressures andmaybe so highas
to cause burstingof the pipe shell (duetoincreased circumferentialtension)if not
accountedforin the designs.··
. The maximumpressuredevelopedinpipelinesdue towater hammerisgivenby
the formula:
p2 = 14.762.v / e1 + K .d/t. where V=Velocityof 1waterjustbefore the closing
of the valve inm/sec.
d = Diameterof pipe inmetres.
t = Thicknessof pipe shell inmetres.
K= Constant= Modulusof elasticityof pipe material/Bulkmodulusof elasticityof water.
The value of K for steel comesoutto be 0.01, for cast iron= 0.02, andfor cement
concrete=0.1.
(3) Stress due to External Loads. Whenlarge pipesare burieddeepunderthe
ground,the weightof the earth-fillmayproduce large stresses inthe pipe material.The
stressdue to the external earthfill loadisgivenby
F = 22.7
ℎ.𝑑𝟐
𝑡
where h= depth of the earth fill above the crowninmetres.
d = diameterof pipe inmetres.
f= stressproducedinkN/m2.

28 | P a g e
Note.In the above formula,itisassumedthatthe earthto the sidesdoesnotgive
any lateral supportandweighsabout18.4 kN/m3.
(4) TemperaturesStresses. Whenpipesare laidabove the ground,theyare exposedtothe atmosphere
and-are,therefore,subjectedtotemperature changes.They
expandduringdaytime andcontract at night.If thisexpansionorcontraction·is
preventeddue tofixationorfrictionoverthe supports,longitudinal stressesare produced
inthe pipe material.The amountof these stressesmaybe calculatedbythe formula:
F = E. à .T, where E= Modulusof elasticityof the pipe material.
à = Co-efficientof expansionof the pipe material.
T = Change intemperature in°C
(5) Stressesdue to Flowaround Bends and Change inCross-Section:Wheneverthe velocityof aflow
(eithermagnitudeordirection) changes,there isachange in the momentum, andtherefore,by
Newton'sSecondLaw,a force isexerted,whichisproportional tothe rate of change of momentum.The
force requiredtobringthis change in momentumcomes fromthe pressure variation·withinthe fluid
and fromforcestransmittedtothe fluidfromthe pipe walls.
(6) Flexural Stresses. Many a times,steel pipesare laidoverconcrete supports,
builtabove the ground; and sometimes the rainwater,etc.maywashoff the ground
frombelowthe pipesatintervals.Underall suchcircumstances,bendingstressesget
producedinthe pipe,since the pipe thenact &. "like abeamwithloadsresultingfromthe
weightof the pipe,weightof waterinthe pipe andanyothersuperimposedloads:The
stressescausedbythisbeamactionmay be determinedbyusual methodsof analysis
appliedtothe beams.However,thesestressesare generallynegligible exceptforlong
spansor where there are huge superimposedloads.·
Torque in Rotating Machines:A core taskfor a rotatingelectricmachine istoproduce the torque
neededtoachieve the requiredrotationspeedunderload.Inlinearmachines,correspondingly,force
productionisthe keyelement.Torque productionisbasedonforcesaffectingthe statorandthe rotor.
There are several waystostudyforce and torque production.Thischapterexploresthe mostimportant
waysfrom the electrical drive'spointof view.Torque productioncanbe examinedbyanalysingthe
energystoredinthe magneticcircuitof the machine.Ignoringlosses,the torque equationcorrelates
withpower.The voltage of a double‐salientpole reluctance machinecanbe expressedbyapplying
Faraday's inductionlawandOhm'slaw.The saliencyof anelectricmachine producestorque if rotor
movementresultsinareductioninthe reluctance of the mainflux path.Whenapplyingnumerical
methods,Maxwell'sstresstensorisoftenusedforthe calculationof torque.
Bends & Elbows A BEND isthe generictermfor whatis calledinpipingasan"offset" - a change in
directionof the piping.A bendisusuallymeanttomeannothingmore thanthatthere isa "bend" - a
change in directionof the piping.
Pipe Bend: Long radiuspipeline bendsare used influidtransportationlinewhichrequiredpigging.Due
to theirlongradiusandsmoothchange of direction,pipe bendhasverylesspressure drop,andsmooth
flowof fluid&pig ispossible.3Dand5D Pipe bendsare commonlyavailable.Here,Disthe pipe size.
MiterBend: Miter bendsare not standardpipe fittingstheyare fabricatedfrompipes.Usually,theyare
preferredforsize 10” & above because large size elbow isexpensive.Use of miterbendisrestrictedto
the low-pressurewaterline.Miterbendcanbe fabricatedin2, 3, & 5 pieces.
An elbowis a pipe fittinginstalledbetweentwolengthsof pipe ortubingtoallow a change of direction,
usuallya90° or 45° angle ,though22.5° elbowsare alsomade.The endsmaybe machinedforbutt

29 | P a g e
welding,threaded(usuallyfemale),orsocketed,etc.Whenthe twoendsdifferinsize,the fittingis
calleda reducingelboworreducerelbow. Elbowsare categorizedbasedonvariousdesignfeaturesas
below: LongRadius(LR) Elbows –radiusis 1.5 timesthe pipe diameter.
Short Radius(SR) Elbows –radiusis 1.0 timesthe pipe diameter.
90 Degree Elbow – where change indirectionrequiredis90°.
45 Degree Elbow – where change indirectionrequiredis45°.
Couplers & Reducers
Couplers:A couplingconnectstwopipestoeachother.If the size of the pipe isnot the same , the fitting
may be calleda reducingcouplingorreducer,oran adapter.By convention,the term"expander"isnot
generallyusedforacouplerthatincreasespipe size;insteadthe term"reducer" isused.
Reducers:A reducerallowsfora change in pipe size tomeethydraulicflow requirementsof the system,
or to adapt to existingpipingof adifferentsize.Reducersare usuallyconcentricbuteccentricreducers
are usedwhenrequiredtomaintainthe same top- orbottom-of-pipe level.
Pipe Reducers:A pipe reducerchangesthe size of the pipe.There are twotypesof reducerusedin
pipingConcentric&Eccentric.
ConcentricPipe Reducer or Conical Reducer:In Concentricreducerwhichisalsoknownas a conical
reducer,the centerof boththe endsison the same axis.Itmaintainsthe centerlineelevationof the
pipeline.Whenthe centerlinesof the largerpipe andsmallerpipe are tobe maintainedsame,then
concentricreducersare used.
Eccentric Reducer: InEccentric reducer,the centerof boththe endsisondifferentaxisasshowninthe
image.ItmaintainsBOP(bottomof pipe) elevationof the pipeline.Whenone of the outside surfacesof
the pipelineistobe maintainedsame,eccentricreducers are required.
Offset= (LargerID – SmallerID) /2
Swage Reducer:The swage islike reducersbutsmall insize andusedtoconnectpipestosmaller
screwedorsocketweldedpipes.Like reducers,theyare alsoavailableinconcentric&eccentrictype.
Swagesare available indifferentendtypes.Suchasbothplainendsorone plainandone threadedend.
Tees:A tee isthe mostcommonpipe fitting.Itisavailable withall femalethreadsockets,all solvent
weldsockets,orwithopposedsolventweldsockets andaside outletwithfemale threads.Itisusedto
eithercombine orsplitafluidflow.Itisa type of pipe fittingwhichisT-shapedhavingtwooutlets,at90°
to the connectiontothe mainline.Itis a shortpiece of pipe witha lateral outlet.A tee isusedfor
connectingpipesof differentdiametersorforchangingthe directionof pipe runs.
Equal,Unequal.
Whenthe size of the branchis same as headerpipes,equal tee isusedandwhenthe branchsize isless
than that of headersize,reducedtee will be used.Mostcommonare teeswiththe same inletandoutlet
sizes.
Pipe Tee:Pipe tee isusedfordistributingorcollectingthe fluidfromthe runpipe.Itisa short piece of
pipe witha 90-degree branchat center.There are two typesof Tee usedinpiping,Equal /StraightTee
and Reducing/Unequal Tee.
Straight Tee: Instraighttee,the diameterof the branch issame as the diameterof the Run (Header)
Pipe.
ReducingTee: Inreducingtee,diameterof the branchsize issmallerthanthe diameterof the Run
(Header) Pipe
Barred Tee: A barredtee whichisalsoknownas a scrapper tee isusedinpipelinesthatare pigged.The
branch of the tee hasa restrictionbarweldedinternallytopreventthe pigorscrapperto enterthe

30 | P a g e
branch.The bars are weldedinthe branchina waythat it will allow restrictionfree passageof the pig
fromthe runpipe.
Wye Tee / Lateral: It is a type of Tee whichhas the branch at a 45° angle,oran angle otherthan 90°.
Wye tee allowsone pipe tobe joinedtoanotherat a 45° angle.Thistype of tee reducesfrictionand
turbulence thatcouldhamperthe flow.Wye tee isalsoknownasa lateral.
Cross: Crossfittingsare alsocalled4-wayfittings.If abranch line passescompletelythroughatee,the
fittingbecomesacross.A cross hasone inletandthree outlets,orvice versa.Theyoftenhave solvent
weldedsocketendsorfemale threadedends.
Crossfittingscangenerate a huge amountof stresson pipe astemperature changes,because theyare
at the centerof fourconnectionpoints.A tee ismore steadythana cross, as a tee behaveslike athree-
leggedstool,while acrossbehaveslike afour-leggedstool.(Geocentrically,"any3non-colinearpoints
define aplane"thus3 legsare inherentlystable.)Crossesare commoninfire sprinklersystems,butnot
inplumbing,due totheirextracostas comparedto usingtwotees.
Pipe Caps: The cap coversthe endof a pipe.Pipe capsare usedat the dead endof the pipingsystem.It
isalso usedinpipingheadersforfuture connections.
Stub Ends: Stub endsare usedwithlapjointflange.Inthistype of flange,the stubisbuttweldedtothe
pipe,whereasflangeisfreelymovedoverthe stubend.Itisbasicallyflange partbutcoveredunder
ASME B16.9 thatis whyit isconsideredaspipe fittings.
PipingUnion:Unionsare usedas an alternative toflangesconnectioninlow-pressure small bore piping
where dismantlingof the pipe isrequiredmore often.Unionscanbe threadedendorsocketweldends.
There are three piecesinaunion,a nut,a female end,anda male end.Whenthe female andmale ends
are joined,the nutsprovide the necessarypressure toseal the joint.
Pipe Coupling:There are three typesof couplingavailable;
Full Coupling:Full Couplingisusedforconnecting small bore pipes.Itusedtoconnectpipe topipe or
pipe toswage or nipple.Itcan be threadedorsocketendstypes.
Half Coupling:Half Couplingisusedforsmall bore branchingfroma vessel orlarge pipe.Itcan be
threadedorsockettype.It hasa socketor threadendon onlyone side.
ReducingCoupling:Reducingcouplingisusedtoconnecttwodifferentsizesof pipe.Itislike concentric
reducerthat maintainsacenterline of the pipe butsmall insize.
Pipe Nipple:Nipple isashortstubof pipe whichhasa male pipe threadat eachendor at one end.It
usedforconnectingtwootherfittings.Nipplesare usedforconnectingpipe,hoses,andvalves.Pipe
nipples are usedinlow-pressurepiping.
Socket weldand Threaded Pipe Fittings:Socketweld andThreadedPipe Fittingsare forgedproductand
classifiedbasedonitspressure-temperature rating.Socketweld&Threadedendfittingsare available
fromNPS1/8” to 4”. These fittingsare availableinfourpressure-temperature ratingclass.
2000 classfittingsare available inonlyinthreadedtype.
3000 & 6000 class fittingsare availableinbothThreadedandSocketWeldtypes.
9000 classfittingsare available inonlysocketweldtype.
Socketand threadedfittingsare usedforsmall bore andlow-pressurepiping.
Dimensional Analysis & Similitude :Dimensional Analysisisamathematical technique making
use of studyof dimensions. Itdealswiththe dimensionsof physical quantitiesinvolved inthe
phenomenon.Indimensional analysis,one firstpredicts the physicalparametersthatwill influence the
flow,andthenby, groupingthese parametersindimensionlesscombinations abetterunderstandingof
the flowphenomenonismade possible.Itisparticularlyhelpful inexperimental workbecauseit

31 | P a g e
provides aguide tothose thingsthat significantlyinfluence the phenomena;thusitindicatesthe
directioninwhichthe experimental workshouldgo.Thismathematical techniqueisusedinresearch
workfor designandforconductingmodel tests.
TYPES OF DIMENSIONS: There are two typesof dimensions
• Fundamental DimensionsorFundamental Quantities
• SecondaryDimensionsorDerivedQuantities
Fundamental Dimensionsor Fundamental Quantities: These are basic quantities.ForExample;
• Time,T Time,T
• Distance,L Distance,L
• Mass, M Mass, M
Force = Mass * acceleration=MLT-2 .
Secondary Dimensionsor DerivedQuantities:The are those quantitieswhichpossesmore thanone
fundamental dimensions. Forexample;
• Velocityisdenotedbydistance perunittime L/T
• Accelerationisdenotedbydistance perunittime square L/T2
• Densityisdenotedbymassperunitvolume M/L3
Since velocity,densityandaccelerationinvolve more thanone fundamental quantitiessothese are
calledderivedquantities.
Dimensional Analysis: Whenthe dimensionsof all termsof anequationare equal the equationis
dimensionallycorrect.Inthiscase,whateverunitsystemisused,that equationholdsitsphysical
meaning.If the dimensionsof all termsof an equationare notequal, dimensionsmustbe hiddenin
coefficients, soonlythe designatedunitscanbe used.Suchan equationwouldbe voidof physical
interpretation.
Utilisingthisprinciple thatthe termsof physicallymeaningful equations have equal dimensions,the
methodof obtainingdimensionlessgroupsof whichthe physical phenomenonisafunctioniscalled
dimensional analysis.If a phenomenonistoocomplicatedtoderiveaformuladescribingit,
dimensional analysiscanbe employedtoidentifygroupsof variableswhich wouldappearinsucha
formula.Bysupplementingthisknowledgewith experimentaldata,ananalyticrelationshipbetweenthe
groupscan be constructedallowingnumerical calculationstobe conducted.
Modeling and Similitude: A “model”isarepresentationof aphysical systemusedtopredictthe
behaviorof the systeminsome desiredrespect.The physical systemforwhichthe predictionsare tobe
made is called “prototype”.
Usually,a model issmallerthanthe prototype soasto conduct laboratorystudiesanditisless
expensive toconstructandoperate.However,incertainsituations, modelsare largerthanthe
prototype e.g.studyof the motionof bloodcellswhose sizesare of the orderof micrometers.
“Similitude” inageneral senseisthe indicationof aknownrelationshipbetweenamodel andprototype
i.e.model testsmustyielddatathatcan be scaledto obtainthe similarparametersforthe prototype.
Theory of models:A givenproblemcanbe describedintermsof a setof pi termsby usingthe principles
of dimensional analysisas,
 1 2 3, ,.......... n    
(8)
Thisequationappliestoanysystemthatisgovernedbysame variables.So,if the behaviorof aparticular
prototype isdescribedby Eq.(8),a similarrelationshipcanbe writtenforamodel of thistype i.e.
 1 2 3, ,..........m m m nm    
(9)

32 | P a g e
The form of the functionremainsthe same aslongas the same phenomenonisinvolvedinboth
prototype andthe model.Therefore,if the model isdesignedandoperatedunderfollowingconditions,
2 2
3 3
.
.
m
m
nm n
  
  
   (10)
then,itfollowsthat
1 1m  
(11)
Eq. (11) isthe desired“predictionequation”andindicatesthatthe measuredvalue of 1m
obtained
withthe model will be equal tothe corresponding 1
forthe prototype aslongas the otherpi terms
are equal.These are called“model designconditions/similarityrequirements/modelinglaws”.
Flow similarity: In orderto achieve similaritybetweenmodelandprototype behavior,all the
correspondingpi termsmustbe equatedbetweenmodel andprototype.So,the followingconditions
mustbe metto ensure the similarityof the model andthe prototype flows.
1. Geometricsimilarity:A model andprototype are geometricsimilarif andonlyif all bodydimensions
inall three coordinateshave the same linear-scale ratio.Itrequiresthatthe model andthe prototype be
of the same shape andthat all the lineardimensionsof the model be relatedtocorresponding
dimensionsof the prototype byaconstant scale factor.Usually,one or more of these pi termswill
involve ratiosof importantlengths,whichare purelygeometrical innature.The geometricscalingmay
alsoextendtothe finestfeaturesof the systemsuchassurface roughnessorsmall perterbance thatmay
influencethe flowfieldsbetweenmodel andprototype.
2. Kinematicsimilarity:The motionsof two systemsare kinematicallysimilarif homogeneousparticles
lie at homogeneouspointsathomogeneoustimes.Inaspecificsense,the velocitiesatcorresponding
pointsare in the same directionandare relatedinmagnitude byaconstant scale factor.Thisalso
requiresthatstreamlinepatternsmustbe relatedbyaconstantscale factor. The flowsthatare
kinematicallysimilarmustbe geometricsimilarbecause boundariesformthe boundingstreamlines.The
factors like compressibilityorcavitationsmustbe takencare of to maintainthe kinematicsimilarity.
3. Dynamic similarity:Whentwo flowshave force distributionssuchthatidentical typesof forcesare
parallel andare relatedinmagnitude byaconstant scale factorat all correspondingpoints,thenthe
flowsare dynamicsimilar.Fora model and prototype,the dynamicsimilarityexists,whenbothof them
have same length-scaleratio,time-scaleratioandforce-scale (ormass-scaleratio).
 For compressibleflows,the modelandprototype Reynoldsnumber,Machnumberandspecific
heatratio are correspondinglyequal.
 For incompressibleflows,
Withno free surface: model andprototype Reynoldsnumberare equal.
Withfree surface: Reynoldsnumber,Froude number,WebernumberandCavitationnumbersformodel
and prototype mustmatch.
In orderto have complete similaritybetweenthe model andprototype,all the similarityflow conditions
mustbe maintained.Thiswill automaticallyfollow if all the importantvariablesare includedinthe
dimensional analysisandif all the similarityrequirementsbased onthe resultingpi termsare satisfied.

33 | P a g e
Model scales:In a givenproblem,if there are twolengthvariables 1l and 2l ,the resultingrequirement
basedon the pi termsobtainedfromthese variablesis,
1 2
1 2
m ml l
l l

(12)
Thisratio isdefinedasthe “lengthscale”.Fortrue models,there will be onlyone lengthscale andall
lengthsare fixedinaccordance withthisscale.There are other‘model scales’suchasvelocityscale
m
v
V
V

 
 
  , densityscale
m




 
 
  ,viscosityscale
m




 
 
  etc.Eachof thesesscalesisdefined
for a givenproblem.
Distorted models: Inorder to achieve the completedynamicsimilaritybetweengeometricallysimilar
flows,itisnecessarytoduplicate the independentdimensionlessgroupssothatdependentparameters
can alsobe duplicated(e.g.duplicationof Reynoldsnumberbetweenamodel andprototype isensured
for dynamicallysimilarflows).
In manymodel studies,dynamicsimilarityrequiresthe duplicationof several dimensionlessgroupsand
it leadstoincomplete similaritybetweenmodel andthe prototype.If one ormore of the similarity
requirementsare notmet,e.g.inEq. 10, if 2 2m  
, thenitfollowsthatEq.11 will notbe satisfiedi.e.
1 1m  
. Modelsforwhichone or more of the similarrequirementsare notsatisfied,are called
“distortedmodels”.Forexample,inthe studyof open-channel orfree surface flows,bothReynolds
number
Vl

 
 
  and Froude number
V
gl
 
  
  are involved.Then,Froude numbersimilarityrequires,
m
m m
V V
g l gl

(13)
If the model andprototype are operatedinthe same gravitational field,thenthe velocityscale becomes,
m m
l
V l
V l
 
(14)
Reynoldsnumbersimilarityrequires,
. . . .m m m
m
V l V l 
 

(15)
and the velocityscale is,
. .m m
m m
V l
V l
 
 

(16)
Since, the velocityscale mustbe equal tothe square rootof the lengthscale,itfollowsthat
 
 
 
3
32
2
m mm m
l
l
l
 

  
 
   
  (17)

34 | P a g e
Eq. (17) requiresthatbothmodel andprototype tohave differentkinematicsviscosityscale,if atall both
the requirementsi.e.Eq.(13) and (15) are to be satisfied.Butpractically,itisalmostimpossibletofinda
suitable model fluidforsmall lengthscale.Insuchcases,the systemsare designedonthe basisof
Froude numberwithdifferentReynoldsnumberforthe model andprototype where Eq.(17) neednot
be satisfied.Suchanalysiswill resulta“distortedmodel”.Hence,there are nogeneral rulesforhandling
distortedmodelsandessentiallyeachproblemmustbe consideredonitsownmerits.
DIMENSIONAL NUMBERS IN FLUID MECHANICS:Forcesencounteredinflowingfluidsincludethose due
to inertia,viscosity,pressure,gravity,surface tensionandcompressibility.Theseforcescanbe writtenas
2 2
Inertia force . V V
dV dV
m a V V L
dt ds
  
 
    
 
Viscousforce
du
A A V L
dy
    
    2
Pressureforce p A p L   
3
Gravityforce m g g L 
Surface tensionforce L
2
Compressibilityforce ; where is the Bulk modulusv v vE A E L E 
The ratio of any two forceswill be dimensionless.Inertiaforcesare very importantinfluidmechanics
problems.So,the ratioof the inertiaforce to eachof the otherforceslistedabove leadstofundamental
dimensionlessgroups.Theseare,
1. Reynoldsnumber
 eR
: It isdefinedasthe ratioof inertiaforce to viscousforce.
Mathematically,
e
VL VL
R

 
  (1)
where V is the velocity of the flow, L is the characteristics length, , and   are the density,
dynamic viscosity and kinematic viscosity of the fluid respectively. If eR
is very small, there is an
indication that the viscous forces are dominant compared to inertia forces. Such types of flows
are commonly referred to as “creeping/viscous flows”. Conversely, for large eR
, viscous forces
are small compared to inertial effects and flow problems are characterized as inviscid analysis.
This number is alsoused to study the transition between the laminar and turbulent flow regimes.
2. Euler number
 uE
: In most of the aerodynamicmodel testing,the pressure dataare usually
expressedmathematicallyas,

35 | P a g e
21
2
u
p
E
V


(2)
where p isthe difference inlocal pressure andfree streampressure, V isthe velocityof the flow, 
isthe densityof the fluid.The denominatorinEq.(2) iscalled“dynamicpressure”. uE
isthe ratioof
pressure force toinertiaforce andit isalsocalledas the pressure coefficient pC
.Inthe study of
cavitationsphenomena,similarexpressionsare usedwhere p isthe difference inliquidstream
pressure andliquid-vapourpressure.The dimensional parameteriscalled“cavitationnumber”.
3. Froude number  rF
: It isinterpretedasthe ratioof inertiaforce togravityforce.
Mathematically,itiswrittenas,
.
r
V
F
g L
 (3)
where V isthe velocityof the flow, L isthe characteristicslengthdescriptive of the flow fieldand g is
the accelerationdue togravity.Thisnumberisverymuchsignificantforflows withfree surface effects
such as incase of open-channel flow.Insuchtypesof flows,the characteristicslengthisthe depthof
water. rF
lessthanunityindicatessub-critical flow andvaluesgreaterthanunityindicate super-critical
flow.Itisalso usedtostudythe flowof wateraroundshipswithresultingwave motion.
4. Webernumber
 eW
: The ratioof the inertiaforce tosurface tensionforce iscalledWeber
number.Mathematically,
2
e
V L
W



(4)
where V isthe velocityof the flow, L isthe characteristicslengthdescriptive of the flow field,  isthe
densityof the fluidand  isthe surface tensionforce.Thisnumberistakenasa index of droplet
formationandflowof thinfilmliquidsinwhichthere isaninterface betweentwofluids.For
1eW ?
,
inertiaforce isdominantcomparedtosurface tensionforce (e.g.flowof waterina river).
5. Mach number
 aM
: It is the keyparameterthatcharacterizesthe compressibilityeffectsina
fluidflowandisdefinedasthe ratioof inertiaforce tocompressibilityforce.Mathematically,
a
v
V V V
M
c dp E
d 
  
(5)
where V isthe velocityof the flow, c isthe local sonicspeed,  isthe densityof the fluidand vE
is
the bulkmodulus.Sometimesthe square of the Machnumberiscalled“Cauchynumber”
 aC
i.e.
2
2
a a
v
V
C M
E

 
(6)

36 | P a g e
Both the numbersare predominantlyusedinproblemsinwhichfluidcompressibilityisimportant.When
the aM
isrelativelysmall (say,lessthan0.3),the inertial forcesinducedbyfluidmotionare sufficiently
small to cause significantchange influiddensity.So,the compressibilityof the fluidcanbe neglected.
However,thisnumberismostcommonlyusedparameterincompressiblefluidflow problems,
particularlyinthe fieldof gasdynamicsandaerodynamics.
6. Strouhal number  tS
: It isa dimensionlessparameterthatislikelytobe importantin
unsteady,oscillatingflowproblemsinwhichthe frequencyof oscillationis  andisdefinedas,
t
L
S
V


(7)
where V isthe velocityof the flowand L isthe characteristicslengthdescriptive of the flow field.This
numberisthe measure of the ratio of the inertial forcesdue tounsteadinessof the flow (local
acceleration) toinertiaforcesdue tochangesinvelocityfrompointtopointinthe flow field(convective
acceleration).Thistype of unsteadyflowdevelopswhenafluidflowspastasolidbodyplacedinthe
movingstream.
Hydraulic similitude isanindicationof a relationshipbetweenamodel anda prototype. Prototypein
case of hydraulicsimilitudeishydraulicstructure. Or
It is a model studyof a hydraulicstructure.
Model:A “model”isa representationof aphysical systemusedtoforecastthe behaviorof the systemin
some desiredaspect.
Prototype: The physical systemforwhichthe predictionsare tobe made iscalled“prototype”.Behavior
of prototype istobe predictby studyingmodel.
Model analysisisveryfrequentlycarriedoutbefore executingthe designof anyhydraulicstructure.A
model,if properlydesignedgivesthe actual performance of the prototype.Withasmall cost onmodel
analysis,itispossible tosave alotof moneywhichmaybe lostas a resultof faultydesignof prototype.
Model analysisisalwayscarriedoutforhydraulicstructureslike weirs,spillways,reservoirs,pumps,
turbinesandshipsetc.
Fluid Properties Measurement:
Static pressure is the pressure thatisexertedbya liquidorgas,such as wateror air. Specifically,itisthe
pressure measuredwhenthe liquidorgasis still,orat rest.Or
Static pressure,influiddynamics,iswhatyouprobablythinkof aspressure, egthe pressure due tothe
depthof the fluid. The termis usedtodistinguish(static) pressurefromdynamicpressure,whichhas
the same units but meanssomethingdifferent.
Let's considerasimplifiedformof the Bernoulli Equation(where we ignore changesin the heightof the
streamline): p0 =p + q
thiscan put intowordsas: total pressure = static pressure + dynamic pressure
So whatis dynamic pressure? Well, q= 1/2ρv2,or inotherwords the dynamicpressure isthe pressure
due to the velocityof the fluid. Remember,thisisn'treallypressure inthe usual sense (ie youcan't
measure itwithpressure measuringtools). Pressure inthe usual sense isstaticpressure.
Viscosity: Viscosity isapropertyof a fluidwhichoffersresistancetoflow of one layerof a fluidover
anotheradjacentlayerof fluid.Insimple language itisdefinedasthe propertywhichoffersresistanceto
flow. OR

37 | P a g e
Viscosityisfrictionof fluidsanditalsodescribesinternal resistance of flow of fluid.Inthisarticle,we are
goingto discusswhatdynamicviscosityandkinematicviscosityare,theirdefinitions,applicationsof
dynamicandkinematicviscosityandfinallydifferencesbetweenkinematicviscosityand dynamic
viscosity. OR
Viscositycanbe definedasresistance of fluidtoflow.Itisan importantpropertyof fluidandsignof
internal friction.Thisresistance iscausedfromforcesof attractionbetweenfluidmolecules.Insimple
term,viscosityisinternal frictionof fluidandalsoreferredasthicknessof fluid.
Viscosityispropertyof fluidthatfindsoutamountof resistance of fluidtoshearstress.Viscosityis
propertyof fluiditsoffersresistance tomovementof one layerof fluidabove next layer.Insimple
wordsit isdefinedaspropertywhichoffersresistancetoflow.
Examplesof Viscosity:For example,take twobottles,let’shave honeyinone bottleandotherhave
water.If you make small hole atbottomof bottle,sowhichbottle getsemptiedfirst.Here we are talking
aboutis viscosity.Therefore,highviscousfluidsneedmore force tomove thanlessviscousmaterials.In
above example,waterhaslowerviscositythanhoney.
Hence,watergetsemptiedthanhoney.Soviscosityof fluidvarieswithtemperature andpressure.Read
Newton’sLawof ViscosityandEquation.Thisfluidwill onlyflow if enoughenergyisappliedtoovercome
these forces.If youneedtomove throughfluid,fluidhastoflow across itor around it.Hence,energy
neededformovingbodythroughfluidisdirectlyrelatedtofluidresistsflow.
Types ofViscosityof fluid:There are twoways to measure fluid’sviscosity.Itcaneitherbe expressedas
dynamicviscosityorkinematicviscosity.Inreality,theyare twomuchdifferentterms.Thisrelationship
betweenthesetwopropertiesisquite simple.
Dynamic Viscosity:DynamicViscosityisalsoknownasabsolute viscosity.Italsomeasure fluidresistance
to shearflow,whenexternal force isapplied.Itisuseful fortellingbehaviorof fluidsunderstress.
Mainly,itis useful intellingnon-Newtonianfluidsbyhow viscositychangesasshearvelocitychanges.As
a result,these twomaterial propertiesmaynotalwaysbe soeasy.
In otherwords,dynamicviscosityisdefined astangential force perunitareaneedtomove fluidinone
horizontal plane withotherplane whilefluidmoleculeswill maintainunitdistance.Dynamicviscosityis
directlyproportional tothe shearstressandisexpressedbysymbol(µ) andhasthe SIunitsof N s/m2
(Newtonsecondpersquare meter).
Kinematicviscosity: Kinematicviscosityisratioof dynamicviscositytofluidof density.Itismeasure of
fluid’sresistance toshearflowundergravityweight.Here force isappliedweightandmeasure of fluid
resistance toflow,whennoexternalforcesexceptgravityisacting.Butitis more useful intelling
Newtonianfluids.
In some cases,inertial force of fluidisalsoneedwithviscositymeasurement.Onotherhand,inertial
force of fluiddependson densityof fluid.Kinematicviscosityisdividingabsoluteviscosityof fluidwith
fluiddensity. Thus,kinematicviscosityistermedasV and ithas unitsof meterssquareddividedby
seconds.
Where v iskinematicviscosity,µisdynamicviscosityand ρ isdensity.
CoefficientofDynamic Viscosity:Coefficientof dynamicviscosity(π) isdefinedasshearforce perunit
area needtopull one fluidlayerwithunitvelocitypassesanotherlayerunitfromdistance awayfrom
fluid.

38 | P a g e
Unitsof Viscosity:As perNewtonsecondspersquare meteris or kilogramspermeterper
secondis . But note that coefficientof viscosityismeasuredinpoise(P),10P =
1 .
Viscosityof Dimensionsis andvaluesforwaterare or forair
is
Formula for Viscosity:Accordingto Newton’slaw of viscosity,takingdirectionof motionas‘x’direction
and as velocityof fluidin‘x’directionatdistance ‘y’fromboundary,shearstress() in‘x’directionis
givenbyformula.Inthisequation,sheerrate isknownas du/dy. Thisreferstovelocitydividedby
distance.Kinematicviscosityisratioof dynamicviscosityanddensityof fluid.
Where µ – Viscosity,T-ShearStressandDu/Dy – Rate of sheardeformation
Difference betweenKinematicViscosityandDynamic Viscosity
 Dynamicand Kinematicviscosityare twoimportantconceptsinfluidmechanism.These two
conceptshave manyapplicationsinfieldslikefluiddynamics,fluidmechanicsandevenmedical
science.Soyouneedtounderstandconceptsof dynamicviscosityand kinematicviscosityare
needinabove fields.
 Dynamicviscosityalsocalledabsolute viscosityandkinematicviscosityiscalleddiffusivityof
momentum.
 Dynamicviscosityisindependentof the densityof the fluid,butkinematicviscositydependson
the densityof the liquid.
 Kinematicviscosityisequal tothe dynamicviscositydividedbythe densityof the liquid.
 Dynamicviscosityissymbolizedbyeither‘µ’,whileKinematicviscosityissymbolizedby‘v’.
 Dynamicviscosityisquantitative expressionof fluid’sresistance toflow,while Kinematic
viscosityisthe ratioof fluid’sviscousforce toinertial force.
Methodof Viscosity:There are twofactors of viscosityof fluid.Followingare variousmethodsusedto
measure viscosityof fluid.
Cohesionof intermolecularforce:Anylayerin movingfluidtriestodragnextlayertomove withequal
speeddue tostrong forcesbetweenmoleculesandthuseffectof viscosity.Since,cohesiondecreases
withtemperature andliquidviscosityisalsosame.
Molecularexchange:As viscosityincreases,molecularmotionof fluidparticlesincreaseswith
temperature increasesaccordingly.Therefore,exceptforspecial casesviscosityof bothgasesandliquids
will increase temperature.
Difference ofViscosity:Viscosityof fluidvarieswithbothtemperature andpressure anddependson
state of fluidsuchasliquidandgases.
Viscosityof liquids:Forliquids,viscosityincreaseswithincreasingpressurebecause amountof free
volume ininternal structure decreasesdue tocompression.Asaresult,moleculesmovelessfreelyand
internal frictionforcesincrease. Since,Viscosityof liquidsisincompressibleunlesspressureincreaseis
importantbutviscositydoesnotchange much.Below equationfollows-
where T isabsolute temperature andA & B are constants
Gasesof viscosity:Viscosityof anideal gasis independentof pressure andthisisalmosttrue forgases.
In gases,viscosityarisesbecause of transferandexchange of molecularmomentum.Sodoublepressure

39 | P a g e
givesyou double numberof moleculesarrivingatsurface,butonaverage theywill come fromhalf asfar
away andcancel out it effects.
Importance of fluidviscosityin various applications
 Viscosityismeasuredduringvarietyof lubricantsformachines.Forexample,highthickoilsare
chosenforslowlymovingpartswhile low thicklubricantsusedforfastmovingparts.
 Viscositydatahelptoforecasthow fluidactin particularconditionthathelpsinmachine
designs.
 Viscosityholdsliquidanywhere theyplaced. If fluidhasnoviscositytheycanflow forever
withoutanyinternal resistanceandtheycanflow outof container.
 For some application,viscositymustbe justintime togetdesirable properties.Forexample,if
viscosityof paintisverylow,itwill rundownwalls.Insimple way,if viscosityishigh,thenitis
toughto applypainton wall.
 So heatdependsonviscosity.
Formula of Viscosity:Consideredtwolayersof fluidwhichare ata distance dyapart. Let the velocityof
the lowerlayerof fluidis uand the upperadjacentlayerisu + duas showninthe figure.Due tothe
viscosityandrelative velocity,shearstressisinducedinbetweenthe twolayersof the fluid.
The top layerinducesshearforce onthe adjacentlowerandthe lowerlayer inducesshearforce onthe
adjacenttoplayer.The shear stresscausedinbetweenthe twolayersof fluidisproportional tothe rate
of change of velocitywithrespecttoy.The shearforce isdenotedbysymbol τ(tau).
Mathematically
Where τ = Shear stress
du/y= Rate of shearstrainor rate of seardeformationorvelocitygradient.
Here μ isthe constant of proportionalityandcalledascoefficientof dynamicviscosityoronlyViscosity.
From the above formulaof shearstress,the formulaof viscosityiscanbe writtenas

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