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Chapter 3

Fluid Statics


،ةلولحم نيرامتو حرش تاركذم


هاندأ داوملا نم ديدعلل ةقباس تاناحتما


نيروكذملا نيعقوملا ىلع

اناجم ةحاتم

ةمهم تارارق ذختت لا


امدنع



اقهرم نوكت


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Fluid statics:



By definition, a fluid must deform continuously when a shear stress of any
magnitude is applied. The absence of relative motion (and thus, angular
deformation) implies the absence of shear stresses. Therefore, fluids either at rest
or in "ri
gid
-
body" motion are able to sustain only normal stresses.



We may apply Newton’s seco
nd law of motion

to

evaluate t
he reaction of the
particle to the applied forces.


3
-
1
THE BASIC EQUATION OF FLUID STATICS:

Our primary objective is to obtain an equation
that will enable us to determine the
pressure field within a static fluid. To do this, we apply Newton's second law to a
differential fluid element of mass







, with sides
dx
,
dy
, and
dz
, as shown
in Fig. 3.1. The fluid element is stationary relati
ve to the stationary rectangular
coordinate system shown.


From our previous discussion, recall that, two general types of forces may be
applied to a fluid: body forces and surface forces. The only body force that must be
considered in most engineering pro
blems is due to gravity. In some situations body
forces due to electric or magnetic fields might be present; they will not be
considered in this text.

For a differential fluid element, the body force,








, is











































In a static fluid no shear stresses can be present. Thus the only surface force is
the pressure force.

Pressure is a field quantity,










: the pressure varies
with position within the
fluid, The net pressure force that results from this variation
can be evaluated by summing the forces that act on the six faces of the fluid
element.

نأ كعسو يف هنأ ملعأ


اميظع

لاجر نوكت
.


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Fig. 3.1 Differential fluid element and pressure forces in the y direction.


Let the pressure at the center,
O
, of the element be
p
. To determine the pressure at
each of the six faces of the element, we use a Taylor series expansion of the
pressure about the point
O
. The pressure at the left face of the differential element is



















(



)








(Terms of higher order are omitted because they will vanish in the subsequent
limiting
process
)
.

The pressure on the right face of the differential element is






















P
ressure forces on the other faces of the element are obtained in the same way.

Combining all such forces gives the net surface force acting on the element. Thus









(






)







̂


(






)








̂











(






)







̂


(






)








̂













(






)





(

̂
)

(






)





(


̂
)

Collecting and canceling terms, we obtain









(




̂




̂




̂
)



























(



̂




̂




̂
)



































انءاطخأ ىسنن اننأ

ةعرسب


ادحأ نلأ

.اهب انركذي لا



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The term in parentheses is called the gradient of the pressure or simply the pressure
gradient and may be written grad
p

or



. In rectangular coordinates









(

̂




̂




̂


)

(

̂




̂




̂


)

The gradient can be viewed as a vector operator; taking the gradient of a scalar field
gives a vector field. Using the gradient designation, Eq. 3.1a can be written as






































































Physically the gradient of pressure is the negative of the surface force per unit
volume
due to pressure. We note that the level of
pressure is not important in
evaluating the ne
t pressure force. Instead, what matters is the rate at which pressure
changes occur with distance, the
pressure gradient
. We shall find this term very
useful throughout our study of fluid mechanics.


We combine the formulations for surface and body forces
that we have
developed to obtain the total force acting on a fluid element. Thus


















































or on a per unit volume basis


































































For a fluid particle, Newton's second law
gives















.
For a
static fluid,





. Thus













Substituting for







from Eq. 3.2, we obtain


































































Let us review briefly our derivation of this equation. The physical significance
of each term is
























































{
















}

{













}





نأ نيرخلآا نم بلطت لا
،كوبحي
مهلعجا

كوبحي
.




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This is a vector equation, which means that if consists of three
component equations
that must be satisfied individually. The components are






























































}







































































































Restrictions:

(1) Static

fluid
.


(2)

Gravity is the only body force
.


(3)
The z axis is vertical
.

This equation is the basic pressure
-
height relation of fluid statics. It is subject to

the
restrictions noted. Therefore it must be applied only where these restrictions are

reasonable for the physical situation. To determine the pressure distribution in a
static

fluid, Eq. 3.6 may be integrated and appropriate boundary conditions applied.

it is important

to note that pressure values must be stated with respect to a reference
level. If the

reference level is a vacuum, pressures are termed
absolute
,
as shown in
Fig. 3.
2.


Most pressure gages read a pressure
difference



the difference between the
measured pressure and the ambient level (usually atmospheric pressure). Pressure
l
evels measured with respect to atmospheric pressure are termed
gage

pressures.
Thus












Absolute pressures must be used in all calculations with the
ideal

gas
. or other

equations of state.

ل
لأسأ تخلأا هميق كردت يك
تاوخأ ةيدل سيل

اصخش
.





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3
-
2
PRESSURE VARIATION IN A STATIC FLUID































































Pressure variation in a compressible fluid can

be evaluated by integrating
Eq.

3.6. Before this can be done, density must be ex
pressed as a function of one
of
the other variables in the equation.




The density of gases generally depends on pressure and temperature. The ideal
gas equation of state






























































Where

R is the gas constant (see Appendix A) and
T

the absolute temperature,
accurately models die behavior of most gases under engineering conditions.




For an incompressible fluid,



. Then for constant
gravity,














































With h measured positive downward, the
n



































































Equation 3.7 indicates that the pressure difference between two points in a static
fluid can be determined by measuring the elevation difference between the two
points. Devices used for this purpose are called
manometers
.



Atmospheric pressure may be obtained from a
barometer
, in which the height of
a

mercury column is measured. The measured height may be converted to
engineering

units using Eq. 3.7 and the data for specific gravity of mercury
given
in Appendix

A.

corrections must be applied to the measured
level and the effects
of surface tension must be considered.

ةايحلا هميق كردت يكل
نع لأسأ
نم ساسحأ
توملا شارف ىلع
.






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A simple U
-
tube manometer is shown in Fig. 3.3
. Since the right leg is open to
the atmosphere, measurements of
h
1

and
h
2

will allow the determination of the
gage

pressure at A. Using the notation of Fig. 3.3, and applying Eq. 3.7 between
A and

B

and between B and C, gives

























an
d
























Ad
ding the two equations




















Since






, then











For many liquids, density is only a weak function of temperature. At modest
pressures, liquids may be considered incompressible. However, at high pressures,
compressibility effects in liquids can be important. Pressure and density changes
in liquids are related by the bulk compressibility modulus, or modulus of
elasticity,





























































If the bulk modulus is
assumed constant, then density is only a function of
pressure

(the fluid is barofropic) and Eq. 3.8 provides the additional density relation
needed

to integrate the basic "pressure
-
height relation. Bulk modulus data for some
common liquids are given in App
endix A.



Manometers

Manometers are simple and inexpensive
devices used frequently for pressure
measurements,
the

following rules are
useful:

1.

Any two points at the same elevation in a
continuou
s length of the same liquid are
at the same pressure.

2.

Pressure increases as one goes down a liquid column (remember the pressure
change on diving into a swimming pool).

نك
مئاد ةطقلاك


ادعتسمو

ابغار ا
كرحلاو زفقلل
ة
تقو يأ يف
.







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,
















































































































































































(















































)




























































































Basic equations:


























Assumptions:
(1) Static fluid


(2) Incompressible

Then






































For



















Beginning at point
A

and applying the equation between successive points around the
manometer gives

Multiplying all equations by (
-
1) and adding:

Substituting





























































ف ءاوس سانلا
إ
ن

اونيابت نحملا تءاج
.









Solution:

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,






























































































(


)














[


(


)

]







































[


(


)

]










[








]










[








]





Solution:

Basic equations:


























Assumptions: (1) Static fluid


(2) Incompressible

To eliminate
H,
note that the
volume
of manometer liquid must remain
constant.

Substituting

This equation can be simplified by expressing the applied pressure differential as an
equivalent water column of height ∆


,

Noting that
:













For Meriam red oil,






(Table A.l). Thus, the sensitivity is






ب
نلأ ءامظع سانلا ضع
راغص مهب نيطيحملا
.









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3
-
3
THE STANDARD ATMOSPHERE

The sea level

conditions of the U.S. Standard Atmosphere are summarized in Table
3.1.

Property

Symbol

SI

English

Temperature

T

288 K

59 F

Pressure

p

101.3 kPa (abs)

14.696 psia

Density



1.225 kg/m
3

0.002377 slug/ft
3

Specific
weight



ـــ

0.07651 lbf/ft
3

Viscosity












kg/m . sec

(Pa . sec)










lbf . sec/ft
2


The temperature profile of the U.S. Standard Atmosphere is shown in Fig, 3.4.

Additional property values are tabulated as functions of elevation in
Appendix A.




إ
هناسلب هرس ىشفأ ءرملا اذ
،
قمحأ وهف ةريغ هيلع ملاو
.









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,





































[










]














[










]


(



)





[












]





(



)














(



)



























































































































(



)





(






)

















{








































}

Basic equations:


















Assumptions: (1) Static fluid


(2) Air behaves as an ideal gas


(3) Temperature varies linearly with altitude

Substituting into the basic pressure
-
height relation yields

But temperature varies l
inearly with elevation, so















Integrating from



in Denver to
p
at Vail, we obtain

Evaluating gives


































The percent cha
nge in density is given by






















































كيددل نوكي نأ مهملا نم
هلجأ نم ظقيتست ام
.








Solution:

OR

and

Thus

and

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(

















)












Solution:




سوءر يناملأا
لاومأ

نيسلفملا
.








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ب
تاركذم
عقوملا
ةلاسرب

SMS

ينورتكللاا ديربلاب وأ

داوم
ماع
ة
:

Circuits
,
English 123,
Numerical Methods,
Dynamics
,

Strength
,
S
tatics

C++,

Java,

MATLAB
,

Data Structures
,

Algorithms,

Discrete Math,

Digital Logic,

Concepts
:
رتويبمك داوم

Mechanical Design


I/II
,

Structural Analysis I/II
,
Management
,
CAD
,

Fluid Mechanics,

System Dynamics

:
ميمصت داوم


نابعش ةدامح

260

4444

9


hs.com
-
eng
info@







ةلولحم لئاسمو حرش


اناجم

نيعقوملاب

com
hs.
-
eng
net
hs.
-
eng
,





































































































































































Solution:

Pressure at level passing point c is same if calculated by both sides


























يف اريثك تأطخأ دقل
يتايح
،
ححصأسو
أطخلا
باوصلا ىلع ريسلاب
.







By sum left and right

sides


October

2010

رانيد
نا

ةيده

لـك ىلع هيبنتلا دـنع
أـطخ

ب
تاركذم
عقوملا
ةلاسرب

SMS

ينورتكللاا ديربلاب وأ

داوم
ماع
ة
:

Circuits
,
English 123,
Numerical Methods,
Dynamics
,

Strength
,
S
tatics

C++,

Java,

MATLAB
,

Data Structures
,

Algorithms,

Discrete Math,

Digital Logic,

Concepts
:
رتويبمك داوم

Mechanical Design


I/II
,

Structural Analysis I/II
,
Management
,
CAD
,

Fluid Mechanics,

System Dynamics

:
ميمصت داوم


نابعش ةدامح

260

4444

9


hs.com
-
eng
info@







ةلولحم لئاسمو حرش


اناجم

نيعقوملاب

com
hs.
-
eng
net
hs.
-
eng
,








































































































































































Solution:

Pressure at level passing point c is same if calculated by both sides



حيحص لامع لمعت نلأ
ا

لا


ت
هيلع ركش
،

لمعت نأ نم ريخ
لامع

هيلع بقاعت لا ائطاخ
.








By sum both equations

October

2010

رانيد
نا

ةيده

لـك ىلع هيبنتلا دـنع
أـطخ

ب
تاركذم
عقوملا
ةلاسرب

SMS

ينورتكللاا ديربلاب وأ

داوم
ماع
ة
:

Circuits
,
English 123,
Numerical Methods,
Dynamics
,

Strength
,
S
tatics

C++,

Java,

MATLAB
,

Data Structures
,

Algorithms,

Discrete Math,

Digital Logic,

Concepts
:
رتويبمك داوم

Mechanical Design


I/II
,

Structural Analysis I/II
,
Management
,
CAD
,

Fluid Mechanics,

System Dynamics

:
ميمصت داوم


نابعش ةدامح

260

4444

9


hs.com
-
eng
info@







ةلولحم لئاسمو حرش


اناجم

نيعقوملاب

com
hs.
-
eng
net
hs.
-
eng
,








































































































(




















)









































Solution:

Starting from left, we can formulate the following four equations:



نأ ةيمولعمبو ةعبرلاا تلاداعملا عمجب

†††

㴠0











كريغ ءاطخأ نم يساقت نأ
نم لضفأ
تنأ اهبكترت نأ
.








p
1

p
2

p
3

p
4

p
5

0

October

2010

رانيد
نا

ةيده

لـك ىلع هيبنتلا دـنع
أـطخ

ب
تاركذم
عقوملا
ةلاسرب

SMS

ينورتكللاا ديربلاب وأ

داوم
ماع
ة
:

Circuits
,
English 123,
Numerical Methods,
Dynamics
,

Strength
,
S
tatics

C++,

Java,

MATLAB
,

Data Structures
,

Algorithms,

Discrete Math,

Digital Logic,

Concepts
:
رتويبمك داوم

Mechanical Design


I/II
,

Structural Analysis I/II
,
Management
,
CAD
,

Fluid Mechanics,

System Dynamics

:
ميمصت داوم


نابعش ةدامح

260

4444

9


hs.com
-
eng
info@







ةلولحم لئاسمو حرش


اناجم

نيعقوملاب

com
hs.
-
eng
net
hs.
-
eng
,









































































































































































































































Solution:

Starting from right at a, we can formulate the following three equations:

نأ ةيمولعمب و تلاداعملا عمجب















،
(









)



نم كل ريخ ائيش فرعت لائل
ةفرعم نم

ةقيقحلا فصن
.








p
1

p
2

p
3

October

2010

رانيد
نا

ةيده

لـك ىلع هيبنتلا دـنع
أـطخ

ب
تاركذم
عقوملا
ةلاسرب

SMS

ينورتكللاا ديربلاب وأ

داوم
ماع
ة
:

Circuits
,
English 123,
Numerical Methods,
Dynamics
,

Strength
,
S
tatics

C++,

Java,

MATLAB
,

Data Structures
,

Algorithms,

Discrete Math,

Digital Logic,

Concepts
:
رتويبمك داوم

Mechanical Design


I/II
,

Structural Analysis I/II
,
Management
,
CAD
,

Fluid Mechanics,

System Dynamics

:
ميمصت داوم


نابعش ةدامح

260

4444

9


hs.com
-
eng
info@







ةلولحم لئاسمو حرش


اناجم

نيعقوملاب

com
hs.
-
eng
net
hs.
-
eng
,


















































































(

















)






















































































Solution:

Starting from left, we can formulate the following three equations:

تلاداعملا عمجب


نورخلآا كمهفيس ابلاغ
كسفن نم رثكأ
،
مهلأسأ
.








p
1

p
2

p
b

p
a

October

2010

رانيد
نا

ةيده

لـك ىلع هيبنتلا دـنع
أـطخ

ب
تاركذم
عقوملا
ةلاسرب

SMS

ينورتكللاا ديربلاب وأ

داوم
ماع
ة
:

Circuits
,
English 123,
Numerical Methods,
Dynamics
,

Strength
,
S
tatics

C++,

Java,

MATLAB
,

Data Structures
,

Algorithms,

Discrete Math,

Digital Logic,

Concepts
:
رتويبمك داوم

Mechanical Design


I/II
,

Structural Analysis I/II
,
Management
,
CAD
,

Fluid Mechanics,

System Dynamics

:
ميمصت داوم


نابعش ةدامح

260

4444

9


hs.com
-
eng
info@







ةلولحم لئاسمو حرش


اناجم

نيعقوملاب

com
hs.
-
eng
net
hs.
-
eng
,



















































(









)


















(













)
















































































Solution:

Volume decreased in tube D=volume increased in tube d






فعض هعضوم ريغ يف ءايحلا
.









H

October

2010

رانيد
نا

ةيده

لـك ىلع هيبنتلا دـنع
أـطخ

ب
تاركذم
عقوملا
ةلاسرب

SMS

ينورتكللاا ديربلاب وأ

داوم
ماع
ة
:

Circuits
,
English 123,
Numerical Methods,
Dynamics
,

Strength
,
S
tatics

C++,

Java,

MATLAB
,

Data Structures
,

Algorithms,

Discrete Math,

Digital Logic,

Concepts
:
رتويبمك داوم

Mechanical Design


I/II
,

Structural Analysis I/II
,
Management
,
CAD
,

Fluid Mechanics,

System Dynamics

:
ميمصت داوم


نابعش ةدامح

260

4444

9


hs.com
-
eng
info@







ةلولحم لئاسمو حرش


اناجم

نيعقوملاب

com
hs.
-
eng
net
hs.
-
eng
,






















































































































































































Solution:

Pressure distribution

is as in figure:





دقحلا نطاب نم ريخ باتعلا رهاظ
.








+

1.0

1.5

0

p
1

p
2

F
1

F
2

9.81

24.53

R