1

White: Chapter 2

潘锦珊： 第一章

Chapter 2

Pressure Distributionin a fluid

(Fluid Statics Basic)

2

A

Fp

A

lim

0Definition:

Unit:2mNPa (SI)

2cmkg atm

PSI

mmHg

OmmH2(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

0zWhen pppp nzx

np

y

x

z

xzp

xp

)2

1( zxbgG

(up)

sin

sin

ybpybp nx

sin

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

dxdy

dzrightpleftp p

2

2dx

x

ppp

dx

x

ppp

left

left

§ 2.2 Equilibrium of a Fluid Element

7dx

dy

dzXdxdydz

Zdxdydz

Ydxdydz

dxdy

dz 2

dx

x

pp

2

dx

x

pp

p

Pressure:

Body force:

8

022

Xdxdydzdydzdx

x

ppdydz

dx

x

pp

In x direction:

Force on left surface

Force on right surface

Body force in x direction

Xx

p

Zz

p

Yy

p

Euler Equilibrium Equations

(Euler 1775)

)( ZdzYdyXdxdp

9

Zz

p

Yy

p

Xx

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

)( kZjYiXkz

pj

y

pi

x

p

10

)( ZdzYdyXdxdp

x

z

z

hz0

p0

Zdzdp )( 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 dsindsin 00

A

ydAy

A

xdAx A

cA

c

, Centroid（形心，重心）

AhApAyApP cc 00 sin

hc

y cC

h

y

x

A

α

y

x

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

y cC

hd

y dD

h

y

x

A

α

y

x

p0

16

Ay

Iyy

C

CCD

A

ydAayp )sin( 0

A

c dAyAyp 20 sin

AyIA 2 AyII CC

2

hc

y cC

hd

y dD

h

y

x

A

α

y

xp0

AA

ydAhpydp )( 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

210050

m15

cxP

18

AhF c

1051210050

N1003.6 6

(a) Centroid: 3m above B

Solution:

3

12

1bLIc

AL

ILL

c

ccd

6 xd PFL(b)

B

PxA

cc p

F

L c

LNPx 106059.4 6

m8

m6

A

B

mN

210050

m15

cxP

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

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Example:

Find the forces acting on the hemi-spherical covers.

R

F

O x

y

H

45o

Solution:

2

2

2gHRAHF xx

sy gVF

FFF yX22

F

F

x

ytan 1

22

2

245sin RRAx

45cos3

2 23 HRRVs

22

§ 2.6.1 Uniform Linear Acceleration （恒加速度直线运动）

aX= -a

g

x

GravityBody force

Inertia Force

§ 2.6 Fluid in rigid body motion

23

a

z

g

xa

x

xagdzdxax

Boundary condition:

0,0 ppzx

)(0 gzxapp x

0pp At free surface（自由液面） xg

az x

Equipressure surface（等压面） Cxg

az x

Euler Equilibrium Equations

)( ZdzXdxdp

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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:

cmD

z 14.2tan2

g

axtanSolution:

AA ghP

31010 mkg

Pa906

)0214.007.0(8.91010

Az

cm6

cm7

cm3

g

27 smax

xaIt will not spill out !

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§ 2.6.2 Rigid body rotation （整体旋转）

g

ω2r

z

f

O

ω2y

ω

x

y

R

O

f

ω2xry

xθ

gZyYxX ,, 22

Body force:

)( ZdzYdyXdxdp

)( 22 gdzydyxdx

Equipressure surface: dp=0

022 gdzydyxdx

Cgzyx )(2

1 222

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Cyxg

z )(2

1 222 Crg

z 22

2

1

Parabola dish （抛物面）

Free surface: 0,0 zzr s

022

2

1zr

gzs 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。

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P2.64

P2.97 (selective)

P2.147

P2.152

Homework: