Ch1 Introduction fundamentals of fluid mechanics

8/15/2019 Ch1 Introduction fundamentals of fluid mechanics

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INTRODUCTION


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2

Types of Forces on Materials

The internal resistance of a material to theapplied force is called Stress.

Stress


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3

What is a Fluid?

Any substance that is continuously deformedwhen subjected to shear stress is called a

fluid.

Mechanics is the branch of physics that isconcerned with the analysis of the action of forces

on matter.

Fluid Mechanics is the analysis of action of

forces on fluids.


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Branches of Fluid Mechanics

Fluid Statics: The analysis of action of forces

on fluids at rest e.. Water stored in a tan!

Fluid "ynamics: The analysis of action of

forces on mo#in fluids e.. water flowin in

a ri#er or a pipe flow


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• !or a solid" application of a shear stress ca#ses a deformationwhich" if modest" is not permanent and solid re$ains ori$inal

position.

Attached

plates Solid

Characteristics of fl#ids


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• !or a fl#id" contin#o#s deformation ta%es place with an infiniten#mber of layers slidin$ o&er each other. Deformationcontin#es #ntil the force is remo&ed.

• ' fl#id is defined as a s#bstance that deforms contin#o#slywhen acted #pon by a shearin$ stress of any ma$nit#de

Fluid

Characteristics of fl#ids


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(

Solid

Fluid

$Stress is proportional to

strain%

$Stress is proportional to

strain rate%

Shear force on a Solid & Fluid

F

A

τ α = µ

F V

A hτ µ = µ


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)

'ewton(s )aw of *iscosity


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*iscosity

• !or elastic solids shearin$ strain is proportional to theshearin$ stress

• !or fl#ids shearin$ stress is proportional to the rate ofshearing strain

• !or Newtonian fl#ids shearin$ stress is linearlyproportional to the rate of shearin$ strain

• The st#dy of non-Newtonian fl#ids is called rheolo$y

• *iscosity is &ery sensiti&e to temperat#re


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+,ample -.: The &elocity distrib#tion for the flow of a Newtonian fl#id between two wide"parallel plates is $i&en by the e+#ation

where V is the mean &elocity. The fl#id has a &iscosity of ,., lb-sft2. /hen V 0 2 fts

and h 0 ,.2 in. determine1 a the shearin$ stress actin$ on the bottom wall" and b the

shearin$ stress actin$ on a plane parallel to the walls and passin$ thro#$h the centerline

midplane

2

31

2

V yu

h

= − ÷


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+,ample -.: The &elocity distrib#tion for the flow of a Newtonian fl#id between two wide"parallel plates is $i&en by the e+#ation

where V is the mean &elocity. The fl#id has a &iscosity of ,.,r lb-sft2. /hen V 0 2 fts

and h 0 ,.2 in. determine1 a the shearin$ stress actin$ on the bottom wall" and b the

shearin$ stress actin$ on a plane parallel to the walls and passin$ thro#$h the centerline

midplane

Solution.

Shearin$ stress

*elocity distrib#tion

a 'lon$ the bottom wall" y 0 4h shearin$ stress

b 'lon$ the midplane" y 0 , shearin$ stress

2

31

2

V yu

h

= − ÷

du

dyτ µ =

0

du

dy =

2

3du Vy

dy h

= −

3du V

dy h=

2

bot wall 14.4 lb/ftτ =

midplane 0τ =


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/roblem -.0: ' 524in.4diameter circ#lar plate is placed o&er a fi6ed bottom plate with a,.54in. $ap between the two plates filled with $lycerin.

Determine the tor+#e re+#ired to rotate

the circ#lar plate slowly at 2 rpm.

'ss#me that the &elocity distrib#tion inthe $ap is linear and that the shear

stress on the ed$e of the rotatin$ plate

is ne$li$ible.


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is ne$li$ible.

Sol#tion

Tor+#e d#e to shearin$ stress on plate1


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is ne$li$ible.

Sol#tion


where Th#s

and

dT r dAτ =

2dA rdr π =

2dT r rdr τ π =

2

02

R

T r dr π τ = ∫


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is ne$li$ible.

Sol#tion



Searin$ stress

2

02

R

T r dr π τ = ∫


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is ne$li$ible.

Sol#tion



Searin$ stress

2

02

R

T r dr π τ = ∫

du V r dy

ω δ δ

= =

du r

dy

ω τ µ µ

δ = =


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is ne$li$ible.

Sol#tion


Searin$ stress

Tor+#e

2

02

R

T r dr π τ = ∫

du r dy

ω τ µ µ δ = =

43

0

2 2

4

R RT r dr

πµω πµω

δ δ = =∫


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is ne$li$ible.

Sol#tion

Tor+#e

( )

4

4 2

lb s rev rad 1 min 62 0.0313 2 2

2 ft min rev 60 s 12 ft0.02 ft lb

0.14 ft 412

RT

π π πµω

δ

× ÷ ÷ ÷ ÷ ÷ = = = ×

÷


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Dimensions and Units

• !l#id characteristics are described +#alitati&ely in terms ofbasic dimensions1 len$th" L" time" T " and mass" M .

• 'll theoretically deri&ed e+#ations are dimensionallyhomogeneous.

• !or a +#antitati&e description #nits are re+#ired

• Two system of units will be #sed1

– International System SI7 m" s" %$" 8

– 9ritish :ra&itational 9: System7 ft" s" lb" ;! or ;R

• /hen sol&in$ problem #se consistent system of #nits" don


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• !rom a microscopic point of &iew a fl#id is not a contin#o#sand homo$eneo#s s#bstance.

• /e ta%e the en$ineerin$ macroscopic &iew s#ch that we can

e6amine a s#fficiently lar$e >particle< of fl#id to allow the

concept of &elocity and density >at a point


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• ?ress#re 0 normal force area. The press#re at a point is1

• In the absence of shear forces fl#id at rest or in #niformmotion press#re at a point is independent of direction

?roperties and characteristics of fl#ids

=

→ A

F p

A 0lim


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• ?erfect $as law

– In this co#rse all $ases obey the perfect $as law


RT por mRT pV ρ==


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• Compressibility1 all fl#ids are compressible" especially $ases.

@ost li+#ids can be re$arded as incompressible for most

p#rposes.

• The b#l% mod#l#s of elasticity" E v , is a property which is #sed

to acco#nt for compressi&e effects1

•

Speed of so#nd is the &elocity at which small dist#rbancespropa$ate in a fl#id. !or ideal $ases speed of so#nd1

v

p E ρ

ρ

∂= ∂

c kRT =


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• Vapor pressure is a press#re e6erted by a &apo#r on the fl#idwhen they are in e+#ilibri#m in a closed &essel

• *apor press#re is a f#nction of temperat#re

• ' li+#id boils when the press#re is red#ced to &apor press#re

• /hen the li+#id press#re is dropped below the &apor press#red#e to flow phenomena" we call the process cavitation

• Cavitation is the formation and s#bse+#ent collapse of &aporb#bbles in a flowin$ fl#id

*apor ?ress#re


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• Ai+#id" bein$ #nable to e6pand freely" will form an interface with asecond li+#id or $as

• This s#rface phenomenon is d#e to #nbalanced cohesi&e forcesactin$ on the li+#id molec#le on the fl#id s#rface

• The intensity of molec#lar attraction per #nit len$th alon$ any line inthe s#rface is called the surface tension coefficient, (Nm!

• The &al#e of s#rface tension decreases as temperat#re increases

• If the interface is c#r&ed" then there is a press#re difference acrossthe interface" the press#re bein$ hi$her on the conca&e side drop offl#id" b#bble

• Capillary action in small t#bes" which in&ol&es a li+#id4$as solidinterface" is also ca#sed by s#rface tension

S#rface Tension


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Bffect of capillary action in small t#bes. a Rise of col#mn for a li+#id that wets the t#be.

b !ree4body dia$ram for calc#latin$ col#mn hei$ht. c Depression of col#mn for anonwettin$ li+#id.

The hei$ht h is $o&erned by the &al#e of the s#rface tension" " t#be radi#s" " " specific

wei$ht of the li+#id"" and the an$le of contact"#


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Bffect of capillary action in small t#bes. a Rise of col#mn for a li+#id that wets the t#be.

b !ree4body dia$ram for calc#latin$ col#mn hei$ht. c Depression of col#mn for anonwettin$ li+#id.

The hei$ht h is $o&erned by the &al#e of the s#rface tension" " t#be radi#s" " " specific

wei$ht of the li+#id"" and the an$le of contact"#

2 2 cos

2 cos

R h R

h R

γπ π σ θ

σ θ

γ

=

=


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+,ample -.1: ?ress#res are sometime determined by meas#rin$ the hei$ht of a col#mn ofli+#id in a &ertical t#be. /hat diameter of clean $lass t#bin$ is re+#ired so that the rise of

water at 2,;C in a t#be d#e to capillary action as opposed to press#re in the t#be is less

than 5., mm


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+,ample -.1: ?ress#res are sometime determined by meas#rin$ the hei$ht of a col#mn ofli+#id in a &ertical t#be. /hat diameter of clean $lass t#bin$ is re+#ired so that the rise of

water at 2,;C in a t#be d#e to capillary action as opposed to press#re in the t#be is less

than 5., mm

Solution

!or water at 2,;C from Table B.2" 0 ,.,(2) Nm and 0 .() %Nm3. Since ,;

it follows that for h 0 5., mm"

'nd the minim#m re+#ired t#be diameter" $" is

2 cosh

R

σ θ

γ =

2 cos R

h

σ θ

γ =

( ) ( )

( ) ( )

( )

3 3 3

2 0.02! "/m 10.014# m

#.!# 10 "/m 1.0 mm 10 m/mm R

−= =

×

2 0.02#! m 2#.! mm D R= = =


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+,ample: ' solid cylindrical needle of diameter d " len$th L" and density ρn may float in

li+#id of s#rface tension σ . Ne$lect b#oyancy and ass#me a contact an$le of ,E. Deri&e a

form#la for the ma6im#m diameter d ma, able to float in the li+#id. Calc#late d ma, for a steel

needle S: 0 (.) in water at 2,EC.


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+'" 3F )+4T56+


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Dimensions 'ssociated with

Common ?hysical F#antities

bac%


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Density of water as a f#nction of temperat#re

bac%


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9eha&ior of a fl#id placed

between two parallel plates

bac%


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Ainear &ariation of shearin$ stress with rate

of shearin$ strain for common fl#id

bac%


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*ariation of shearin$ stress with rate of shearin$ strain for

se&eral types of fl#ids" incl#din$ common non4Newtonian fl#ids.

bac%


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Dynamic absol#te &iscosity

of some common fl#ids as a

f#nction of temperat#re

bac%


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!orces actin$ on one4half of a li+#id drop

?ress#re drop across the s#rface of the droplet

bac%


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!orces actin$ on one4half of a li+#id drop

?ress#re drop across the s#rface of the droplet

bac%

22

2i e

R p R

p p p R

π σ π

σ

= ∆

∆ = − =


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Ch1 Introduction fundamentals of fluid mechanics

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