1
White,Chapter 2
潘锦珊, 第一章
Chapter 2
Pressure Distribution
in a fluid
(Fluid Statics Basic)
2
A
Fp
A ?
??
??
lim
0
Definition:
Unit,2mNPa ? (SI)
2cmkg atm
PSI
mmHg
Omm H2 (Pound per Squire Inch)
? Vertical to the surface and point into it.
?At any point,pressure is independent of orientation.
Properties of Pressure
Pressure
3
At any point in a static fluid,pressure
is independent of orientation.
Verification:
??
?
?
?
???
?
zgpp
pp
nz
nx
?
2
1
0? ??? zWhen pppp nzx ???
np
y?
x?
z
x
zp
xp ?
)21( zxbgG ??? ?
(up)
?? s ins in ybpybp nx ???
?s in
ybpP
nn
??
ybpP xx ??
Forces on left and up surface:
:x?
4
Fluid Mechanics
AerodynamicsFluid at rest
Fluid Statics
Fluid Dynamics
Fluid in motion
5
? Pressure is the only surface force.
? Pressure distribution relates to body force only.
? Dams（ 水坝）
? Buoyancy related instrument（ 利用浮力的装置）
? Fluid power system（ 液压驱动系统 ）
? Connected vessel （ 连通器）
? ……
Applications:
§ 2.1 Fluid @ rest
6
Consider a cube in a static fluid
Pressure at the center is p;
Body forces are kZjYiXR ???? ???
dx
dy
dz rightpleftp
p
?
?
?
?
?
?
?
??
?
?
??
2
2
dx
x
p
pp
dx
x
p
pp
l e f t
l e f t
§ 2.2 Equilibrium of a Fluid Element
7
dx
dy
dz Xd x d y d z?
Z d x d y d z?
Yd x d y d z?
dx
dy
dz 2
dx
x
pp
?
??
2
dx
x
pp
?
?? p
Pressure:
Body force:
8
022 ???
?
??
?
?
?
????
?
??
?
?
?
?? X d x d y d zd y d zdx
x
ppd y d zdx
x
pp ?
In x direction:
Force on
left surface
Force on
right surface
Body force in
x direction
Xxp ????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Z
z
p
Y
y
p
?
?
Euler Equilibrium Equations
(Euler 1775)
)( Z d zY d yXd xdp ??? ?
9
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Z
z
p
Y
y
p
X
x
p
?
?
?
Pressure increase in the
direction of body force,
Surfaces in fluid with same pressure,vertical
to body force everywhere,in gravity field it is
a horizontal plane.
Equipressure surface（ 等压面）
Rp ????
)( kZjYiXkzpjypixp ?????? ??????????? ?
10
)( Z d zY d yXd xdp ??? ?
x
z
z
h
z0
p0
Z d zdp ?? )( gZ ??gdz???
1 Basic rule:
Cgzp ??? ? General solution
2 Boundary
condition,00,ppzz ?? 00 gzpC ???
)( 00 zzgpp ??? ?
hpghpp ?? ???? 00
§ 2.3 Pressure Distribution under Gravity
11
ghphpp ?? ???? 00
Pressure at
free surface
Pressure due to
weight on top
Any point with the same depth h under free surface has
the same pressure.
equipressure surface（ 等压面）
Free surface is an equipressure surface
12
p0
Water manometer
（水柱压力计）
p
h
Pressure
source
Connected
water tube
Application?
Absolute pressure （绝对压力）
Relative pressure（ 相对压力）
Gauge pressure（ 表压力）
Vacuum degree（ 真空度）
Pressure measurement
h (mmH2O)
h+p0 (mmH2O) pA 绝压
pG 表压
13
P2.7
Homework:
14
?? ???
AA
AhAppP dd 0 ?
?sinyh ?
dAhppdAdP )( 0 ????
?? ????
AA
AyApAyApP ds inds in 00 ????
A
y d A
yA
x d A
x AcAc ?? ??,Centroid（ 形心，重心）
AhApAyApP cc ??? ???? 00 s in
hc
C
h
A
α
p0
Find total force P
§ 2.4 Hydrostatic Force on Plane
Surface
15
AhpP c )( 0 ??? hc:depth of centroid
The force on a submerged
plane equals the pressure
at the plate center times
the plate area,
independent of the shape
of the plate or the angle ?.
Center of pressure
Is the center of pressure at centroid?
hc
C
hd
D
h
A
α
p0
16
Ay
Iyy
C
C
CD ??
? ??
A
y d Aayp )s in( 0 ?
???
A
c dAyAyp
2
0 s in ??
AyI
A
?? 2 AyII CC 2??
hc
C
hd
D
h
A
α
p0
?? ??
AA
y d Ahpy d p )( 0 ?
Moment to x axis
17
Example
The gate is 5m wide,is hinged
at point B,and rest against
a smooth wall at point A.
Find
(a) The force on the gate
exerted by seawater
pressure,
(b) The horizontal force Px
exerted by the wall at
point A
m8
m6
?
A
B
mN 21 0 0 5 0??
m15
c
xP
18
AhF c??
105121 0 0 5 0 ????
N1003.6 6??
(a) Centroid,3m above B
Solution:
3
12
1 bLI
c ?AL
ILL
c
c
cd ??
6?? xd PFL(b)
?B
Px
A
c
cp
F
NP x 106 0 5 9.4 6??
m8
m6
?
A
B
mN 21 0 0 5 0??
m15
c
xP
19
Hoover Dam Channel
1,Select a dA and find the three forces on it
2,Integration
§ 2.5 Hydrostatic Forces on Curved Surfaces
20
Conclusion:
x
1,Horizontal forces,
xcx Ahpp )( 0 ???
xO
z
AAx
2,Vertical forces,
VApp zz ??? 0
O
z
V
21
Example,Find the forces acting on the hemi-
spherical covers.
R
F
O x
y
H
45o
?
Solution:
2
2
2 g H RAHF
xx ??? ???
sy gVF ???
FFF yX 22 ?? FF
x
yta n 1???
22
2
245s in RRA
x ?? ???
??? 45c o s32 23 HRRV s ??
22
§ 2.6.1 Uniform Linear Acceleration （ 恒加速度直线运动）
aX= -a
g
x
Gravity
Body force
Inertia Force
§ 2.6 Fluid in rigid body motion
23
a?
z
?
g?
xa?? x
xa?
gdzdxa x ?? ???
Boundary condition:
0,0 ppzx ???
)(0 gzxapp x ???? ?
0pp ?
At free surface（ 自由液面）
xgaz x??
Equipressure surface（ 等压面）
Cxgaz x ???
Euler Equilibrium Equations
)( Z d zXd xdp ?? ?
24
A cup of coffee is 7cm deep at rest.
1,Whether it will spill out while ax=7m/s2?
2,Gage pressure of point A?
Example:
cmDz 14.2ta n2 ??? ?
g
a x??tanSolution:
AA ghP ??
? ?31 0 1 0 mkg??
Pa9 0 6
)0 2 1 4.007.0(8.91 0 1 0
?
????
?
A
z?
cm6
cm7
cm3
g
27 sma x ?
xa?
It will not spill out !
25
§ 2.6.2 Rigid body rotation （ 整体旋转）
g
ω2r
z
f
O
ω2y
ω
x
y
R
O
f
ω2xr y
x
θ
gZyYxX ????,,22 ??
Body force:
)( Z d zY d yXd xdp ??? ?
)( 22 g d zy d yx d x ??? ???
Equipressure surface,dp=0
022 ??? g d zy d yx d x ??
Cgzyx ??? )(21 222?
26
Cyxgz ??? )(21 222? Crgz ?? 2221 ?
Parabola dish （ 抛物面）
Free surface:
0,0 zzr s ??
0
22
2
1 zr
gz s ?? ?
g
ω2r
z
f
O
z0
How to find z0?
旋转抛物面体的体积是同底面
积和高的圆柱体积的一半。
27
P2.64
P2.97 (selective)
P2.147
P2.152
Homework:
White,Chapter 2
潘锦珊, 第一章
Chapter 2
Pressure Distribution
in a fluid
(Fluid Statics Basic)
2
A
Fp
A ?
??
??
lim
0
Definition:
Unit,2mNPa ? (SI)
2cmkg atm
PSI
mmHg
Omm H2 (Pound per Squire Inch)
? Vertical to the surface and point into it.
?At any point,pressure is independent of orientation.
Properties of Pressure
Pressure
3
At any point in a static fluid,pressure
is independent of orientation.
Verification:
??
?
?
?
???
?
zgpp
pp
nz
nx
?
2
1
0? ??? zWhen pppp nzx ???
np
y?
x?
z
x
zp
xp ?
)21( zxbgG ??? ?
(up)
?? s ins in ybpybp nx ???
?s in
ybpP
nn
??
ybpP xx ??
Forces on left and up surface:
:x?
4
Fluid Mechanics
AerodynamicsFluid at rest
Fluid Statics
Fluid Dynamics
Fluid in motion
5
? Pressure is the only surface force.
? Pressure distribution relates to body force only.
? Dams（ 水坝）
? Buoyancy related instrument（ 利用浮力的装置）
? Fluid power system（ 液压驱动系统 ）
? Connected vessel （ 连通器）
? ……
Applications:
§ 2.1 Fluid @ rest
6
Consider a cube in a static fluid
Pressure at the center is p;
Body forces are kZjYiXR ???? ???
dx
dy
dz rightpleftp
p
?
?
?
?
?
?
?
??
?
?
??
2
2
dx
x
p
pp
dx
x
p
pp
l e f t
l e f t
§ 2.2 Equilibrium of a Fluid Element
7
dx
dy
dz Xd x d y d z?
Z d x d y d z?
Yd x d y d z?
dx
dy
dz 2
dx
x
pp
?
??
2
dx
x
pp
?
?? p
Pressure:
Body force:
8
022 ???
?
??
?
?
?
????
?
??
?
?
?
?? X d x d y d zd y d zdx
x
ppd y d zdx
x
pp ?
In x direction:
Force on
left surface
Force on
right surface
Body force in
x direction
Xxp ????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Z
z
p
Y
y
p
?
?
Euler Equilibrium Equations
(Euler 1775)
)( Z d zY d yXd xdp ??? ?
9
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Z
z
p
Y
y
p
X
x
p
?
?
?
Pressure increase in the
direction of body force,
Surfaces in fluid with same pressure,vertical
to body force everywhere,in gravity field it is
a horizontal plane.
Equipressure surface（ 等压面）
Rp ????
)( kZjYiXkzpjypixp ?????? ??????????? ?
10
)( Z d zY d yXd xdp ??? ?
x
z
z
h
z0
p0
Z d zdp ?? )( gZ ??gdz???
1 Basic rule:
Cgzp ??? ? General solution
2 Boundary
condition,00,ppzz ?? 00 gzpC ???
)( 00 zzgpp ??? ?
hpghpp ?? ???? 00
§ 2.3 Pressure Distribution under Gravity
11
ghphpp ?? ???? 00
Pressure at
free surface
Pressure due to
weight on top
Any point with the same depth h under free surface has
the same pressure.
equipressure surface（ 等压面）
Free surface is an equipressure surface
12
p0
Water manometer
（水柱压力计）
p
h
Pressure
source
Connected
water tube
Application?
Absolute pressure （绝对压力）
Relative pressure（ 相对压力）
Gauge pressure（ 表压力）
Vacuum degree（ 真空度）
Pressure measurement
h (mmH2O)
h+p0 (mmH2O) pA 绝压
pG 表压
13
P2.7
Homework:
14
?? ???
AA
AhAppP dd 0 ?
?sinyh ?
dAhppdAdP )( 0 ????
?? ????
AA
AyApAyApP ds inds in 00 ????
A
y d A
yA
x d A
x AcAc ?? ??,Centroid（ 形心，重心）
AhApAyApP cc ??? ???? 00 s in
hc
C
h
A
α
p0
Find total force P
§ 2.4 Hydrostatic Force on Plane
Surface
15
AhpP c )( 0 ??? hc:depth of centroid
The force on a submerged
plane equals the pressure
at the plate center times
the plate area,
independent of the shape
of the plate or the angle ?.
Center of pressure
Is the center of pressure at centroid?
hc
C
hd
D
h
A
α
p0
16
Ay
Iyy
C
C
CD ??
? ??
A
y d Aayp )s in( 0 ?
???
A
c dAyAyp
2
0 s in ??
AyI
A
?? 2 AyII CC 2??
hc
C
hd
D
h
A
α
p0
?? ??
AA
y d Ahpy d p )( 0 ?
Moment to x axis
17
Example
The gate is 5m wide,is hinged
at point B,and rest against
a smooth wall at point A.
Find
(a) The force on the gate
exerted by seawater
pressure,
(b) The horizontal force Px
exerted by the wall at
point A
m8
m6
?
A
B
mN 21 0 0 5 0??
m15
c
xP
18
AhF c??
105121 0 0 5 0 ????
N1003.6 6??
(a) Centroid,3m above B
Solution:
3
12
1 bLI
c ?AL
ILL
c
c
cd ??
6?? xd PFL(b)
?B
Px
A
c
cp
F
NP x 106 0 5 9.4 6??
m8
m6
?
A
B
mN 21 0 0 5 0??
m15
c
xP
19
Hoover Dam Channel
1,Select a dA and find the three forces on it
2,Integration
§ 2.5 Hydrostatic Forces on Curved Surfaces
20
Conclusion:
x
1,Horizontal forces,
xcx Ahpp )( 0 ???
xO
z
AAx
2,Vertical forces,
VApp zz ??? 0
O
z
V
21
Example,Find the forces acting on the hemi-
spherical covers.
R
F
O x
y
H
45o
?
Solution:
2
2
2 g H RAHF
xx ??? ???
sy gVF ???
FFF yX 22 ?? FF
x
yta n 1???
22
2
245s in RRA
x ?? ???
??? 45c o s32 23 HRRV s ??
22
§ 2.6.1 Uniform Linear Acceleration （ 恒加速度直线运动）
aX= -a
g
x
Gravity
Body force
Inertia Force
§ 2.6 Fluid in rigid body motion
23
a?
z
?
g?
xa?? x
xa?
gdzdxa x ?? ???
Boundary condition:
0,0 ppzx ???
)(0 gzxapp x ???? ?
0pp ?
At free surface（ 自由液面）
xgaz x??
Equipressure surface（ 等压面）
Cxgaz x ???
Euler Equilibrium Equations
)( Z d zXd xdp ?? ?
24
A cup of coffee is 7cm deep at rest.
1,Whether it will spill out while ax=7m/s2?
2,Gage pressure of point A?
Example:
cmDz 14.2ta n2 ??? ?
g
a x??tanSolution:
AA ghP ??
? ?31 0 1 0 mkg??
Pa9 0 6
)0 2 1 4.007.0(8.91 0 1 0
?
????
?
A
z?
cm6
cm7
cm3
g
27 sma x ?
xa?
It will not spill out !
25
§ 2.6.2 Rigid body rotation （ 整体旋转）
g
ω2r
z
f
O
ω2y
ω
x
y
R
O
f
ω2xr y
x
θ
gZyYxX ????,,22 ??
Body force:
)( Z d zY d yXd xdp ??? ?
)( 22 g d zy d yx d x ??? ???
Equipressure surface,dp=0
022 ??? g d zy d yx d x ??
Cgzyx ??? )(21 222?
26
Cyxgz ??? )(21 222? Crgz ?? 2221 ?
Parabola dish （ 抛物面）
Free surface:
0,0 zzr s ??
0
22
2
1 zr
gz s ?? ?
g
ω2r
z
f
O
z0
How to find z0?
旋转抛物面体的体积是同底面
积和高的圆柱体积的一半。
27
P2.64
P2.97 (selective)
P2.147
P2.152
Homework:
