fluid statics.pdf

Hydrostatic Pressuredistributioninastaticfluidanditseffectsonsolidsurfacesandon

floatingandsubmergedbodies.

M.Bahrami ENSC283 Spring2009 1FluidStatics

Fluidatrest

hydrostaticcondition:whenafluidvelocityiszero,thepressurevariationisdueonlytotheweightofthefluid.

Thereisnopressurechangeinthehorizontaldirection. Thereis apressurechangeintheverticaldirectionproportionaltothe

density,gravity,anddepthchange.

2M.Bahrami ENSC283 Spring2009

Inthelimitwhenthewedgeshrinkstoapoint,

FluidStatics

Pressureforces(pressuregradient) Assumethepressurevaryarbitrarilyinafluid,p=p(x,y,z,t).

Thepressuregradientisasurfaceforce thatactsonthesidesoftheelement. Note that the pressure gradient (not pressure) causes a net force that must be Notethatthepressuregradient(notpressure)causesanetforcethatmustbe

balancedbygravityoracceleration.


Equilibrium

Thepressuregradientmustbebalancedbygravityforce,orweightoftheelement,forafluidatrest.

Thegravityforceisabodyforce,actingontheentiremassoftheelement.Magneticforceisanotherexampleofbodyforce.


Gagepressureandvacuum

Theactualpressureatagivenpositioniscalledtheabsolutepressure,anditismeasuredrelativetoabsolutevacuum.

Pgage

P

Patm

Pabs

Ab l t

Pvac

atm Absolute(vacuum)=0


Hydrostaticpressuredistribution

Forafluidatrest,pressuregradientmustbebalancedbythegravityforce

Recall:pisperpendiculareverywheretosurfaceofconstantpressurep. Inourcustomarycoordinatezisupwardandthegravityvectoris:

where g = 9 807 m/s2 The pressure gradient vector becomes:whereg=9.807m/s2.Thepressuregradientvectorbecomes:


Hydrostaticpressuredistribution

Pressureinacontinuouslydistributeduniformstaticfluidvariesonlywithverticaldistanceandisindependentoftheshapeofthecontainer.

Thepressureisthesameatallpointsonagivenhorizontalplaneinafluid.

Forliquids,whichareincompressible,wehave:

M.Bahrami ENSC283 Spring2009 7

Thequantity,p isalengthcalledthepressureheadofthefluid.

FluidStatics

Themercurybarometer

Patm =761mmHg

Mercury has an extremely small vapor pressure at room temperature Mercuryhasanextremelysmallvapor pressureatroomtemperature(almostvacuum),thusp1 =0.Onecanwrite:


Hydrostaticpressureingases Gasesarecompressible,usingtheidealgasequationofstate,p=RT:

Forsmallvariationsinelevation,isothermalatmospherecanbeassumed:

Ingeneral(forhigheraltitudes)theatmospherictemperaturedropsofflinearlywithz

TT0 BzwhereT0 isthesealeveltemperature(inKelvin)andB=0.00650K/m.

Note that the P is nearly zero (vacuum condition) at z = 30 km


NotethatthePatm isnearlyzero(vacuumcondition)atz =30km.

FluidStatics

Manometry Astaticcolumnofoneormultiplefluidscanbeusedtomeasurepressure

differencebetween2points.Suchadeviceiscalledmanometer.

/ / Adding/subtractingz asmovingdown/upinafluidcolumn.

JumpingacrossUtubes:anytwopointsatthesameelevationina


continuousmassofthesamestaticfluidwillbeatthesamepressure.

FluidStatics

Hydrostaticforcesonsurfaces Consideraplanepanelofarbitraryshapecompletelysubmergedinaliquid.

Thetotalhydrostaticforceononesideoftheplaneisgivenby:


Hydrostaticforcesonsurfaces Afterintegrationandsimplifications,wefind:

Theforceononesideofanyplanesubmergedsurfaceinauniformfluidequalsthepressureattheplatecentroidtimestheplatearea,independentoftheshapeoftheplateorangle.p p p g

Theresultantforceactsnotthroughthecentroidbutbelowittowardthehighpressureside.Itslineofactionpassesthroughthecentreofpressureg p p g pCPoftheplate(xCP,yCP).


Hydrostaticforcesonsurfaces Centroidalmomentsofinertiaforvariouscrosssections.

Note:forsymmetricalplates,Ixy =0andthusxCP =0.Asaresult,thecenter


yofpressureliesdirectlybelowthecentroidontheyaxis.

Hydrostaticforces:curvedsurfaces Theeasiestwaytocalculatethepressureforcesonacurvedsurfaceisto

computethehorizontalandverticalforcesseparately.

Thehorizontalforceequalstheforceontheplaneareaformedbytheprojectionofthecurvedsurfaceontoaverticalplanenormaltothe

tcomponent.

Theverticalcomponentequalstotheweightoftheentirecolumnoffluid,bothliquidandatmosphericabovethecurvedsurface.


FV=W2 +W1 +WairFluidStatics

fluid statics.pdf

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