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Pressure and Fluid Statics
By:
Dr. Muhammad Farhan
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Pressure
Pressureis defned as a normal forceexerted by a uid per unit area.
Units o pressure are N/m2 !hich is
called a pascal "Pa#. Since the unit Pa is too small orpressures encountered in practicekilopascal "$ %Pa & $'(Pa# and
megapascal"$ MPa & $')
Pa# arecommonly used. *ther units include bar atm, kgf/cm2,
lbf/in2=psi.
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+,solute -a-e and acuumpressures
+ctual pressure at a -ie point is calledthe absolute pressure.
Most pressuremeasurin- deices arecali,rated to read 0ero in theatmosphere and thereore indicategage pressure P-a-e&Pa,s Patm.
Pressure ,elo! atmospheric pressureare called vacuum pressure Pac&Patm
Pa,s.
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+,solute -a-e and acuumpressures
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Pressure at a Point
Pressure at any point in a 1uid is thesame in all directions.
Pressure has a ma-nitude ,ut not aspecifc direction and thus it is ascalar uantity.
i i i h
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3ariation o Pressure !ithDepth
4n the presence o a-raitational feld pressureincreases !ith depth ,ecausemore 1uid rests on deeperlayers.
5o o,tain a relation or theariation o pressure !ithdepth consider rectan-ularelement
Force ,alance inzdirection -ies
Diidin- ,y xand rearran-in-
-ies
2 1
0
0
z zF ma
P x P x g x z
= =
=
2 1 sP P P g z z = = =
3 i i P i h
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3ariation o Pressure !ithDepth
Pressure in a 1uid at rest is independent othe shape o the container.
Pressure is the same at all points on ahori0ontal plane in a -ien 1uid.
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Scu,a Diin- and 6ydrostaticPressure
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Scu,a Diin- and 6ydrostaticPressure
( ),2 3 2
,2 ,2
1998 9.81 100
3.28
1298.5 2.95
101.325
2.95 1 3.95
gage
abs gage atm
kg m mP gz ft
m s ft
atmkPa atm
kPa
P P P atm atm atm
= =
= =
= + = + =
1 1 2 2
1 2
2 1
3.954
1
PV PV
V P atm
V P atm
=
= =
Pressure on dierat $'' t7
Dan-er oemer-ency ascent7
100 ft
1
2
Boyles law
If you hold your breath on ascent, your lung
volume would increase by a factor of 4, which
would result in embolism andor death!
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Pascal8s 9a!
1 2 2 2
1 2
1 2 1 1
F F F AP P
A A F A= = =
Pressure applied to aconfned 1uid increasesthe pressure throu-hout,y the same amount.
4n picture pistons are at
same hei-ht:
atio +2/+$is calledideal mechanicaladvantage
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5he Manometer
1 2
2 atm
P P
P P gh
=
= +
+n eleation chan-e ozin a 1uid at restcorresponds to P/g.
+ deice ,ased on thisis called a manometer.
+ manometer consistso a Utu,e containin-one or more 1uids suchas mercury !ateralcohol or oil.
6eay 1uids such as
mercury are used ilar-e pressuredi;erences areanticipated.
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Mutli1uid Manometer
For multi1uid systems Pressure chan-e across a 1uid
column o hei-ht his P = gh.
Pressure increases do!n!ardand decreases up!ard.
5!o points at the same eleationin a continuous 1uid are at thesame pressure.
Pressure can ,e determined ,y
addin- and su,tractin- gh
terms.
2 1 1 2 2 3 3 1P gh gh gh P + + + =
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Measurin- Pressure Drops
Manometers are !ellsuited to measurepressure drops acrossales pipes heate
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5he Barometer
C atm
atm
P gh P
P gh
+ =
=
+tmospheric pressure ismeasured ,y a deicecalled a barometer= thusatmospheric pressure isoten reerred to as thebarometric pressure.
P#can ,e ta%en to ,e 0erosince there is only 6-apor a,oe point > and itis ery lo! relatie to Patm.
>han-e in atmosphericpressure due to eleation
has many e;ects: >oo%in-nose ,leeds en-ineperormance aircratperormance.
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Fluid Statics
Fluid Staticsdeals !ith pro,lems associated!ith 1uids at rest.
4n 1uid statics there is no relatie motion,et!een ad?acent 1uid layers.
5hereore there is no shear stress in the 1uidtryin- to deorm it. 5he only stress in 1uid statics is normal stress
Normal stress is due to pressure 3ariation o pressure is due only to the !ei-ht o
the 1uid @ 1uid statics is only releant in presenceo -raity felds.
+pplications: Floatin- or su,mer-ed ,odies!ater dams and -ates liuid stora-e tan%setc.
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6ooer Dam
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6ooer Dam
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6ooer Dam
Ahapter
"Bernoullieuation#.
6 d t ti F Pl
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6ydrostatic Forces on PlaneSuraces
*n aplanesuracethe hydrostatic orcesorm a system oparallel orces
For many applications
ma-nitude andlocation o application!hich is called centerof pressure must ,edetermined.
+tmospheric pressurePatmcan ,e ne-lected!hen it acts on ,othsides o the surace.
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esultant Force
5he ma-nitude o $%actin- on a plane
surace o a completely su,mer-ed plate ina homo-enous 1uid is eual to the producto the pressure P#at the centroid o the
surace and the area&o the surace
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>enter o Pressure
,xx C
p C
c
Iy y
y A= +
9ine o action o resultantorce $%=P#& does not passthrou-h the centroid o thesurace. 4n -eneral it liesunderneath !here thepressure is hi-her.
3ertical location o Centerof Pressureis determined,y euation the moment othe resultant orce to themoment o the distri,utedpressure orce.
E4ur ed
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6ydrostatic Forces on >uredSuraces
$%on a cured surace is more inoled since
it reuires inte-ration o the pressure orces
that chan-e direction alon- the surace. Aasiest approach: determine hori0ontal and
ertical components $'and $(separately.
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6ydrostatic Forces on >uredSuraces
6ori0ontal orce component on cured surace:$'=$x. 9ine o action on ertical plane -iesy
coordinate o center o pressure on curedsurace.
3ertical orce component on cured surace:$(=$y)* !here *is the !ei-ht o the liuidin the enclosed ,loc% *=g(. xcoordinate othe center o pressure is a com,ination o lineo action on hori0ontal plane "centroid o area#
and line o action throu-h olume "centroid oolume#. Ma-nitude o orce $%=+$'2)$(2/2
+n-le o orce is = tan!+$(/$'
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Buoyancy and Sta,ility
Buoyancy is due to the 1uiddisplaced ,y a ,ody. $-=fg(.
Archimedes principal: 5he,uoyant orce actin- on a ,odyimmersed in a 1uid is eual to the!ei-ht o the 1uid displaced ,y the
,ody and it acts up!ard throu-h thecentroid o the displaced olume.
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Buoyancy and Sta,ility
Buoyancy orce $-iseual only to thedisplaced olume
fg(displaced.
5hree scenariospossi,le
1. bodyuid: Floatin- ,ody
2. body=uid: Neutrally
,uoyant
3. body"uidSin%in- ,ody
A
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A
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A
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A
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A
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A
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A
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Sta,ility o 4mmersedBodies
otational sta,ility o immersed ,odies dependsupon relatie location o center of gravity0and
center of buoyancy-. 0,elo! -: sta,le 0a,oe -: unsta,le
0coincides !ith -: neutrally sta,le.
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Sta,ility o Floatin- Bodies
4 ,ody is ,ottom heay"0lo!er than -# it isal!ays sta,le.
Floatin- ,odies can ,esta,le !hen 0is hi-her
than -due to shit inlocation o center,uoyancy and creationo restorin- moment.
Measure o sta,ility isthe metacentric hei-ht1. 4 01G$ ship issta,le.
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i-idBody Motion
5here are special cases !here a ,ody o 1uid can under-ori-id,ody motion: linear acceleration and rotation o acylindrical container.
4n these cases no shear is deeloped.
Ne!tonCs 2nd la! o motion can ,e used to derie an
equation of motion or a 1uid that acts as a ri-id ,ody
4n >artesian coordinates:
P gk a + =
( ), ,x y xP P P
a a g ax y z
= = = +
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9inear +cceleration
>ontainer is moin- on a strai-htpath
5otal di;erential o P
Pressure di;erence ,et!een 2points
Find the rise ,y selectin- 2 pointson ree surace P2& P$
0, 0
, 0,
x y z
x
a a a
P P Pa g
x y z
= =
= = =
xdP a dx gdz =
( ) ( )2 1 2 1 2 1xP P a x x g z z =
( )2 1 2 1x
s s s
az z z x x
g = =
otation in a >ylindrical
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otation in a >ylindrical>ontainer
2
2
, 0
, 0,
r za r a a
P P Pr g
r z
= = =
= = =
>ontainer is rotatin- a,out the 0a
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Aro!n o 6iero 44 Hin- o
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5he olden >ro!n o 6iero 44 Hin- oSyracuse
+rchimedes 2IJ2$2 B.>. 6iero (')2$ B.>.
6iero learned o a rumor !herethe -oldsmith replaced some o
the -old in his cro!n !ith siler.6iero as%ed +rchimedes todetermine !hether the cro!n!as pure -old.
+rchimedes had to deelop anondestructie testin- method
5he olden >ro!n o 6iero 44 Hin- o
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5he olden >ro!n o 6iero 44 Hin- oSyracuse
5he !ei-ht o the cro!n andnu--et are the same in air:*c= c(c= *n= n(n.
4 the cro!n is pure -oldc&n!hich means that theolumes must ,e the same(c=(n.
4n !ater the ,uoyancy orceis -='2(.
4 the scale ,ecomesun,alanced this implies thatthe 3cK 3n !hich in turnmeans that the c K n
oldsmith !as sho!n to ,e araudL
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6ydrostatic Bodyat 5estin-
hat is the ,est !ay to
measure ,ody at7 6ydrostatic Bodyat 5estin-
usin- +rchimedes PrincipleL
Process
Measure ,ody !ei-ht*=body(
et in tan% e