statics part 1

7/29/2019 Statics Part 1

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FLUID STATICS

FLUIDS ARE AT REST OR MOVING IN SUCH A MANNER THAT THERE IS NO

RELATIVE MOTION BETWEEN ADJACENT PARTICLES

NO SHEAR STRESSES IN THE FLUID AT REST

PRESSURE IS THE ONLY FORCE DEVELOPED

PRESSURE AT A POINT

Pressure normal force per unit area at a given point acting on a given

plane within the fluid mass of interest

How the pressure at a point varies with the orientation of the plane passing

through the point ?


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sxPs

x

y

z

zxPy

yxPz

x

y

z

2

zyx

g

FORCES ON AN ARBITRARY WEDGE SHAPED ELEMENT OF FLUID

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 PASCALS LAW

s


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s

zsin

s

ycosSinsz Cossy

yy amF

yxy a2

zyxSinsxPzxP

yxy ayPP2

zz amF

z

xz

a2

zyx

2zyxgCossxPyxP

ga2

zPP zxz


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Taking the limit x, y, z approch zero while maintaining

zys PPP

IN THE ABSENCE OF SHEARING STRESSES, PRESSURE AT ANY POINT INA FLUID IS INDEPENDENT OF DIRECTION PASCALS LAW

PASCAL BLAISE 1623 1662 39 YEARS

1642 - invented the first mechanical adding machine

Pascal proved by experimentation in 1648 that the level of the mercury column in a

barometer is determined by an increase or decrease in the surrounding atmospheric

pressure rather than by a vacuum, as previously believed.

His methodology reflected his emphasis on empirical experimentation as opposed toanalytical, a priori methods, and he believed that human progress is perpetuated by

the accumulation of scientific discoveries resulting from such experimentation.


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y

z

x

zx2

y

y

PP

zx

2

y

y

PP

yx

2

z

z

PP

yx2

z

z

PP

zy

2

x

x

PP

zy2

x

x

PP

BASIC EQUATION FOR THE PRESSURE FIELD

How does the pressure in a fluid in which there are no shearing

stresses vary from point to point ?

j

i

K

z,y,xP


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T.O.H.......xx

)x(f

2

1x

x

)x(f)x(fxxf

2

2

2

Neglecting all higher order terms

zyx

y

PFy

T.O.H.....2

y

y

P

2

1

2

y

y

P

PP

2

2

2

2yy

2

y

y

PPP

2y

y

zx

2

y

y

PPzx

2

y

y

PPFy


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By Newtons Second Law,

m a = Surface forces (pressure) + Body forces (weight)

a

mF

azyxkzyxzyxP

akP

General equation of motion for a fluid no shearing stress


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PRESSURE VARIATION IN A FLUID AT REST

dz

dp

For liquids or gases at rest the pressure gradient in the vertical

direction at any point in a fluid depends only on the specific weight of

the fluid at that point

akp

PZ

0a;z

p;0

y

p;0

x

p


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INCOMPRESSIBLE FLUID

dz

dp

2

1

2

1

z

z

p

p

dzdp

)zz(pp 1221

This pressure distribution HYDROSTATIC DISTRIBUTION

hghPP 21

Hgofmm76081.910006.13

10325.101

g

PPh

3

21


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PAB

= h + Po

PRESSURE IN A HOMOGENOUS, INCOMPRESSIBLE FLUIDAT REST

depends on the depth of the fluid relative to somereference plane

not influenced by the size of shape of the tank in which

the fluid is held


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F1 = (A1/A2)F2

A1


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COMPRESSIBLE FLUID

dz

dp2z

1z

2p

1p

dzdp Gases are compressible

gTR

pg

dz

dp2z

1z

2p

1pT

dz

R

g

p

dp

2z

1z1

2

T

dz

R

g

p

pln zfT

For isothermal condition

o

12

1

2

TR

)zz(g

p

pln

o

12

1

2

TR

)zz(gexp

p

p


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STANDARD ATMOSPHERE

T = 150 C (288.15 K)

p = 101.33 kPa (abs)

= 1.225 kg/m3

= 12.014 N/m3

= 1.789 10-5 Pa.s

FOR TROPOSPHEREm/K0065.0

zTT a

Airplanes


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2z

1z1

2

T

dz

R

g

p

pln

zfT

m/K0065.0

zTT a

2z

1z

a1

2

zT

dz

R

g

p

pln

zTln1

R

g

p

pln a

1

2

Rg

a1

2 zTlnp

pln


16/22

Graphical Representation of Pressure

ABSOLUTE PRESSURES POSITIVE

GAGE PRESSURES POSITIVE & NEGATIVE

Pabs = Patm Pgage


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BAROMETER

vaporatm php

For mercury at 20o C,

papvapor 158585.0

hpatm

Evangelista Torricelli - 1644


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PIEZOMETER TUBE

Simple and accurate

Suitable only for pressure

greater than atmospheric

pressure

Pressure measured to bereasonably small height will

be large

Capillarity effects are negligible

for large bore tube ie.,diameters greater than 30 mm

atmA phP

0hh


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U-TUBE MANOMETER02211 hhpa

1122a hhp

If A contains gas,

22 hpa

Capillarity effects are

negligible for large bore

tube ie., diameters

greater than 30 mm

DIFFERENTIAL U TUBE MANOMETER


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DIFFERENTIAL U-TUBE MANOMETER

3322b11a hhphp

Capillarity effects cancel

INCLINED TUBE MANOMETER


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INCLINED TUBE MANOMETER

3322b11a hsinlphp

113322 sin hhlpp baIf A and B contain gas

sin22

ba ppl

Amplification issin1

sin

1sin


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Pressure Gage

Pressure gages

statics part 1

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