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Chap. 2
Fluid Statics
2. Fluid Statics
Fluid Statics : concerns the problems in which the fluid is
either at rest or moving in such a manner that there is no
relative motion between adjacent particles.
acts no shear stress, but only the pressure
The rigid body motion of a fluid is also involved in this
category because there is no deformation (i.e., no shear
strain).
Table of Hydraulics and Hydrostatics, from the 1728 “Cyclopaedia”
2.1 Pressure at a Point : Pascal’s Law
Pascal’s Law : The pressure at any point in a nonflowing fluid has a single value, independent of direction.
y
Mass
syy a2
zyxsinsxpzyxpF
z
MassWeight
szz azyxzyx
sxpyxpF
22
cos
(Newton’s second law, F = ma)
Since ; cossy ,sinsz
2
ypapp ysy
2)(
zpapp zsz
Since we are really interested in what is happening
at a point, we take the limit as x, y and z approach
zero (while maintaining the angle ), and it follows
that
or
sy pp sz pp
zys ppp
The angle was arbitrarily chosen so we can conclude that the
pressure at a point in a fluid at rest, or in motion, is
independent of direction as long as there are no shearing
stresses present. Pascal’s Law
Pascal’s Law
Physically, if , the fluid must flow,
not be in the rest.
zys ppp
Blaise Pascal (1623-1662)
2.2 Basic Equation for Pressure Field
Let’s consider a cubical fluid element at rest or in rigid body motion in which there is no shear stress.
Taylor series expansion
HOT2
z
2
1
2
zpp
22
z
2z
p
z
p
HOT2
z
2
1
2
zpp
22
z
2z
p
z
p
HOT2
x
2
1
2
xpp
22
x
2x
p
x
p
HOT2
x
2
1
2
xpp
22
x
2x
p
x
p Brook Taylor
(1685-1731)
onaccelerati of.compx
x
Mass
xxx a)y)(z)(x()y)(z(p)y)(z(pF
onaccelerati of.compy
y
MassWeight
zzz a)y)(z)(x()y)(z)(x()y)(x(p)y)(x(pF
X-component:
x
Mass
p
2
2
2
p
2
2
2
a)y)(z)(x(
)yz(HOT2
x
x
p
2
1
2
x
x
pp
2
x
x
p
2
1
2
x
x
pp
xx
z
Mass
Weight
pp
ayzx
yxzHOTz
z
pz
z
pp
z
z
pz
z
pp
zz
))()((
)()(22
1
222
1
2
2
2
22
2
2
z-component:
Combining and canceling terms where possible,
X-component: xa)y)(z)(x()y)(z)(x(.T.O.Hx
p
z-component:za)y)(z)(x()y)(z)(x(.T.O.H
z
p
That is,
X-component: xax
p
y-component:
z-component:
yay
p
zaz
p
In Vector Form,
ak
p
Sir Issac Newton
(1643-1727, aged 84 )A reputed descendant of Newton'sapple tree, found in the BotanicGardens in Cambridge
Newton's own copy of his Principia, with hand-written corrections for the second edition.
2.3 Pressure Variation in a Fluid at Rest
0kp
Since ,0a
Or,; 0
x
p
; 0
y
p
z
p
dz
dp
2.3.1 Incompressible Fluid
2
1
2
1
z
z
p
pdzdp
hzzpp 1221
hpp 21
that is,
or,
z
H
p
p0
Hpp 0
If letting p2= p0, the pressure at any depth H below the free surface is
Called “Hydrostatic Pressure”
Pressure Head
Hydraulic Jack
1
1
22 F
A
AF
Same Level
“Use Lecture Note”
2.3.2 Compressible Fluid
RT
gpg
dz
dp
RTp
2
1
2
1
z
z1
2p
p T
dz
R
g
p
pln
p
dp
Integrating gives
(assumed that g and R are constant)
If we assume that the temperature has a constant value T0 over
the range z1 to z2 (isothermal condition),
0
1212
RT
zzgexppp (in the isothermal layer)
Almost Isothermal
The temperature variation in the troposphere :
BzTT 0
z
0
p
p T
dz
R
g
p
dp
0
0
0z
00
z
000 T
BzTln
RB
gBzTln
RB
g
BzT
dz
R
g
p
pln
RBg
T
Bzpp
0
0 1
Then
Extremely rarified,
No longer continuum.
Atmosphere diagram showing the troposphere and other layers
Summer and Winter
An idealized view of three large circulation cells
2.4 Standard Atmosphere
2.5 Measurement of Pressure
vaporatm php
hpatm
Mercury barometer :
Since the vapor pressure can be neglected (for mercury,
pvapor, mercury=0.000023 lb/in2 (abs) at room temperature,
(for mercury, h=760 mm Hg =29.9 in. Hg)
If water is used instead of mercury,
h=10.36 m H2O = 34 ft H2O.
2.6 Manometry
Piezometer Tube
11atm11A hphp (patm =0 for gage pressure at point A)
U-Tube Manometer
2211A hhp
1122A hhp
that is,
11h
22A hp
If A contains a gas, is negligible.
21 pp
3Hg21oilair,gage hhhp
3322B11A hhphp
113322BA hhhpp
Or
32 pp
32 pp
22411A hphp
22511A hphp
22211B11A hhhphp
122BA hpp
That is,
2.6.3 Inclined-Tube Manometer
sinlhphp 2233B11A
112233BA hsinlhpp
sinlpp 22BA
or
If pipes A and B contain a gas,
2.7 Mechanical and Electronic Measuring Devices
Recommended