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