Chap 2 Fluid Mechanics

8/10/2019 Chap 2 Fluid Mechanics

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Chapter 2 Fl

Assistant Professor of

FUNDAMENTALS OF

FLUID MECHANICS

id Statics

echanical Engineering


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MAIN TOPICS

Pressure at a Point

Basic Equation for Pressure Fiel

Standard Atmosphere

Measurement of Pressure

ManometryMechanical and Electronic Press

H drostatic Force on a Plane Su

Pressure Prism

Hydrostatic Force on a Curved Suoyancy, oa ng, an a

Rigid-Body Motion

re Measuring Devices

face

urface

2


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Pressure at a Point

Pressure ?

Indicating the normal for

point acting on a given pla

n eres .

How t e pressure at a po nt

the plane passing through the

e per unit area at a given

e within the fluid mass of

ar es w t t e or entat on o

oint ?

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Pressure at a Point

Consider the free-body di

in which there are no sheforces acting on the wedge a

agram within a fluid mass.

ring stress, the only externale due topressure andweight

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The equations of motion (Newton

directions are, respectively,

whereps, py,andpzare the avera

are the fluid s ecific wei ht and

accelerations. Note that a pres

appropriate area to obtain the foo ows rom e geome ry a

second law, F= ma) in the y and z

e pressures on the faces, and

ensit res ectivel and a a the

sure must be multiplied by an

rce generated by the pressure. It


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nce we are rea y n eres e n

take the limit as x, yand z

the angle, and it follows that

independent of the directi

.

w a s appen ng a a po n , we

pproach zero (while maintaining

, ,

n as long as there are no


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Pressure at a Point

Independent of direction

law named in honor of Blaise Pascal (1623-1662).

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Basic Equation for

How does the pressure in a

shearing stresses vary from p

To answer this question c

element of fluid removed f

w u

due to the pressure, and a b

o the element

*Other possible types of body forces, such

Pressure Field

fluid in which there are no

int to point?

onsider a small rectangular

rom some arbitrary position

.

dy force equal to the weight

as those due to magnetic fields, will not be


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we et t e pressure at t

designated as p, then the av

TheResultantsurfaceforceinthey

Similarl forthexandzdirectionsth

e center o t e e ement e

rage pressure on the various

directionis

resultantsurfaceforcesare


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Theresultantsurfaceforcecanbeex r

Thus,theresultantsurfaceforceper

ssedinvectorformas

nitvolumecanbeexpressedas


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,

negativesignindicatesthattheforced

,

Equation

mo onshearing sEq.2.2

uetotheweightisdownward

2.2 is the general equation of

r a u n w c ere are notresses


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,

pointinafluiddependsonlyonthesp

Forafluidatresti.e.a=0 andEq.2.2reduces

These equations show that the presswe move from point to point in a ho

xy plane) the pressure does not cha

Eqs. can be written as the ordinary dif

equation

ecificweightofthefluidatthatpoint.

o

re does not depend onx or y.Thus, asizontal plane (any plane parallel to the

ge. Since p depends only on z, above

ferential


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This is Fundamental equation for

determine how ressure chan es with

This equation indicates thatthe pre

rest

fluids at rest and can be used to

elevation.

sure gradient in the vertical direction


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Incompressiblefluid

Ingeneral,afluidwithconstantdensi

Since the specific weight is equal

acceleration of gravity (=g), changeor g. or mos eng neer ng app ca

our main concern is with the possible

For liquids the variation in density

vertical distances, so that the assump

dealing with liquids is a good one.

yiscalledanincompressiblefluid

to the product of fluid density and

in are caused either by a change inns e var a on n g s neg g e, so

variation in the fluid density.

is usually negligible, even over large

tion of constant specific weight when


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where h is t

fluid measur

This type of pressure distribution

.

pressure varies linearly with dept

depth to hold up the fluid above

Inthiscaseh

is

called

the

ressurehea

columnoffluidofspecificweightrequir

e distance, z2z1 which is the depth of

d downward from the location of p2.

is commonly called a hydrostatic

. The pressure must increase with

it.

andisinter retedasthehei hto a

dtogiveapressuredifference


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The pressure p at any depth

given by p =h + po

The pressure in a homogeneous,

depends on the depth of the fluid

relative to some reference plane,

and it is not influenced by t

size or shape of the tank or

held.

below the free surface is

e

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Fluid pressure in contai

The pressure in a homogeneous,incompressible fluid at rest depends

on the depth of the fluid relative to

,

influenced by the size or shape of ttank or container in which the fluid

held.

ners of arbitrary shape

es

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Transmission of Fl

The required equality of pre

throughout a system is importhydraulic jacks, lifts, and pres

controls on aircraft and other t

F = pA F = pA = F21 1 2 2

The transmission of fluid pressu

throughout a stationary fluid is

hydraulic devices are based.

id Pressure

sure at equal elevations

nt for the operation ofes, as well as hydraulic

pe of heavy machinery.

A2 F1A1

e

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e un amen a ea e n

demonstrated in previous figure.closed system filled with a liquid, s

the pressure throughout the system

force F1 to a second piston where

pressurep acting on the faces of bof elevation changes is usually negli

Device), it follows that

The piston area A2 can be made m

large mechanical advantage can b

app ie at t e sma er piston can e

larger piston. The applied force c

some type of mechanical device, scompressed air acting directly on th

hydraulic lifts commonly found in se

suc ev ces an sys ems s

piston located at one end of auch as oil, can be used to change

, and thus transmit an applied

he resulting force is F2. Since the

th pistons is the same (the effectible for this type of hydraulic

uch larger than A1 and therefore a

developed; that is, a small force

use to eve op a arge orce at t e

uld be created manually through

ch as a hydraulic jack, or throughsurface of the liquid, as is done in

vice stations.


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an ar mosp

Variationinpressureintheearths

ea y t e pressure vs. a t tu e

conditions (temperature, refer

Theinformationisnotavailable

The standard atmosphere h

that can be used in themissiles, and spacecraft, and

performance under standard co

re

atmosphere?

over t e spec c

nce pressure) for

.

s been determined

design of aircraft,in comparing their

ditions.

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an ar mosp

Standard atmosphere was fi

The currently accepted Stan

report published in 1962

The so-called U.S. standar

representationofmiddl

mean conditions of the

re

st developed in the 1920s.

ard atmosphere is based on a

and updated in 1976.

atmosphere is an idealized

-latitude, year-around

arths atmosphere.

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Standard Atmosph

For example, in the troposp

the temperature variation is ofthe form

T = Taz

where Ta is the temperature asea level (z=0) and is the

lapse rate (the rate of cha

of temperature with elevatio

P=pressur

re

ere,

ge

n).

variationthroughoutthetroposphere


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Measurement of Press

Absolute pressure: measure

Gage pressure: measured wi

toasasuctionorvacuumpre

ure: Absolute and Ga e

with respect to vacuum.

h respect to atmospheric pressure

sure

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Barometers A exam

Mercury Barometer is used

pressure:Patmh +Pvapor

Pva or 0.000023 lb / in2 6

specific weight of mer

The height of a mercury column

converted to atmosphere pressur

P atm=h+Pva or

le of one-type of manometer.

o measure atmosphere

8oF

ury

is

by

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Water vs. Mercury


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use ofliquid column in ver

Pressure measuring device

called manometers. Th

example of one type of mother configuration possible,

application.

Piezometer Tube.U-Tube manometer.

Inclined-Tube manome

ical or inclined tubes

based on this technique are

mercury barometer is an

nometer, but there are manydepending on the particular

er.45


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ezome er u e

P = P0+ h PA= 1A : gage pressure 0=

1 :the specific weight of the liq

container

h1: measured from the meniscussurface to point(1)

Only suitable if the pressu

greater than atmospheric pr

the required height of the c

fluid in the container must

a gas.

h1

id in the

t the upper

re in the container is

ssure, and the

olumn is reasonable. The

be a liquid rather than46


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Blood pressure measurements

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mp e - u e an

The fluid in the manometer

A(1)(2)(3)Open

PA +1 h 1 - 2h 2 = 0

>> PA =2h 2 -1 h 1

If i e A contains a as

then1h 10

>> PA = 2h 2

ome er

s called the gage fluid.

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The contribution of gas colu

negligible since the weight of th

AmajoradvantageoftheUtuthegagefluidcanbedifferent

whichthepressureistobedet

mns in manometers is usually

e gas is so small

emanometerliesinthefactthatromthefluidinthecontainerin

rmined


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Differential U-Tube

A(1)(2)(3)(4)(5)

PA1h12h2 3h3

PA PB2h23h31

Manometer

B

B

h1

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pressures, or pressures that ar

To overcome some of these p

of pressuremeasuring instru

os o ese ma e use o

acts on an elastic structure, t

pressure

changing rapidly with time

roblems numerous other types

ments have been developed.

e ea a w en a pressure

he structure will deform, and

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Bourdon Pressure

Bourdon tube pressure gage u

curved tube to measure res

As thepressurewithin the tub

,

translated into the motion of

age

es a hollow, elastic, and

sure

e increases the tube tends to

,

pointer on dial

Coiled spring

Connected to

the pressure

source 57


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Aneroid Barometer

The Aneroid barometer is us

pressure.

The Aneroid barometer co

e emen s w c s evacua e

element is near absolute zer

As the external atmospheric

deflects, and this motionmovement of an attached dia

d for measuring atmospheric

tains a hallow, closed, elastic

so a e pressure ns e e

pressure changes, the element

can be translated into thel

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Hydrostatic Forces on a Plane


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Hydrostatic For

When a surface is submerg

the fluid.

The determination of these

of storage tanks, ships,structures

ces on a Plane

d in a fluid, forces develop on

orces is important in the design

dams, and other hydraulic


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due to the hydrostatic pressu

To determine completely

Specifying the direc

pec y ng e ne

e distribution on the surface

the resultant force acting

.

ion of the force.

ac on o e orce.

62

Determination of the res ltant force acting on the


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esu an orce

pressure acting at the centroid

total area

whereyc is the y coordinate of thhc is the vertical distance

centroid of the area

of the area multiplied by the

e centroid of the area A.rom the fluid surface to the

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The resultant force does not pas

always below it

The coordinate of the resulsummation of moments around

thedistributedpressureforce

But

through the centroid but is

ant force can be determined bhe x axis


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The integral in the numerator is the

inertia) Ix, with respect to an axis f

con a n ng e sur ace an e ree s

Using

econd moment of the area (moment of

rmed by the intersection of the plane

r ace x ax s . us, we can wr e

arallel axis theorem


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Resultant force does not pass t

horizontal surfaces is alwaysbel

rough the centroid but for non

w it, since


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Assignm

Study and submit a (5~6

to ics of cha 2 before Mi

.

Article # 2.11 Buo anc

nt#2

ages) report on following

Terms

loatation and Stabilit

Chap 2 Fluid Mechanics

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