Fluid Mechanics_1_ Fluid Propeties

8/9/2019 Fluid Mechanics_1_ Fluid Propeties

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FLUIDMECHANICS

Prof. C.S.P. Ojha

Dept. of Civil Engineering

IIT OO!EE


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FLUID

A "#i$ is a substance that deforms continuously under theapplication of a shear stress, irrespective of the magnitudeof the applied shear stress. If a uid is at rest no shear force can exist.

Attain equilibrium deformation under the application of shear force, nomatter how small.

Types of uids: iquids

!ases

"i#erentiated in the basis of e#ect ofcohesive force between the molecules ofuid

Fl#i$ Me%hani%& deals with the study of uid either in

rest$uid statics% or uid in motion $ uid dynamics%.

&luid mechanics 'nds its application in breathing, blood ow,swimming, windmills, pumps, pipes, channels, rivers, ships,automobiles, airplanes etc.


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CONTINUUM

All uids are conglomeration of separate molecules which are widely

spaced in gases and closely spaced in liquids.

The distance between the molecules in both cases, however, is very

large as compared to their molecular diameter and the molecules move

freely relative to each other.

Thus a uid property such as mass density does not have speci'c

meaning since the no of molecules occupying a given space changes

continuously.

Therefore the interest centers on the average conditions of the uid

properties$density% and ow properties$velocity etc.%.

In uid mechanics it is assumed that the uid have continuous

distribution of matter with no empty space i.e. the uid is a continuum.

The uid properties are considered to be continuous function of space

and time.


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DIMENSIONS AND UNITS

Sl. No. '#antit( Unit S()*ol

) ength meter m

* +ass ilogram -g

Time interval /econd s

0 Temperature elvin

1 2lectric 3urrent Ampere A

4 uminous intensity 3andela cd5 Amount of substance mole mol

6ames and symbols of basic units of primary dimensions

Sl.No.

'#antit( Unit S()*ol

E+#ivalent%o)*ination& of

other #nit&

) &orce 6ewton 6 -g m7s*

* 8ressure andstress

8ascal 8 67m*

9or- and 2nergy oule 67m

0 8ower 9att 9 7s or 6m7s

1 &re uenc hert;


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FLUID POPETIES

"ensity

8ressure

>iscosity/urface tension

>apor 8ressure

?ul- +odulus of elasticity


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DENSIT,

+ass density$@%: mass per unit volume. Its unit is g7m

9eight density$%: weight per unit volume also called unit weight orspeci'c weight. Its unit is 67m

B @g where, g B C.D)m7s*$acceleration due to gravity%

At atmospheric pressure and *EF3@w B )EEE -g7m @air B ).*E1 -g7m

@w B CDEE 67m @air B)).D 67m

The density of gases is highly variable and increases nearlyproportionally to the pressure level but for liquids the density is

nearly constant.

&or example the density of water would increase by only )G if thepressure is increased by a factor of **E. for this reason most of theliquids are treated as incompressible.


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/peci'c volume $>s%: Heciprocal of mass

density i.e. volume per unit mass$m

7-g%.

/peci'c gravity$!%: It is the ratio ofmass$weight% density of liquid to the mass$weight% densityof water at 0o3. &or gases, however, thereference$standard% density may that of air or hydrogen.


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PESSUE

8ressure intensityB force7unit area $67m* called 8ascal, 8a%

)atmospheric pressure B )E).*1 -8a

+olecules of any uid are in a continuous state of collision.

2very part of the uid as well as the solid surface, the uid is in contactwith, experiences a force exerted upon it by the surrounding uid.

This force varies from location to location in magnitude and7or direction.

8ressure is the stress$compression% at a point in uid and is theconsequences of normal force acting on a plane in the uid or a planesurface the uid is in contact with.

&or ideal gas


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-ISCOSIT,

>iscosity may be de'ned as the property of uid due to which theuid is able to resist the shear $or the movement of uid%

In a way viscosity of uid is a quantitative measure of theresistance to ow that the owing uid o#ers.

The resistance of any uids to its movement depends upon thecohesion and the rate of transfer of molecular momentum.

I liquids cohesion is the prime cause of viscosity whereas in gasesthe rate of transfer of molecular momentum is the prime cause.

In liquids the viscosity increases with increase in temperature

however in gases the viscosity decreases with increase intemperature.

The viscosity of uid increases only marginally with pressure.


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3onsider a uid element abcdwith its upper face dcmobbing at a speed ofrelative to the lower surface ab $'gure a%

The uid element is therefore subected to a shear stress , due to whichshear strain angle $ at any time % continuously increases as long as the

shear stress exists. 3ommon uids such as air and water exhibit a linear relation between stress

and shear strain rate.

&rom 'gure

&or in'nitesimal change B JJJJJJJJJ


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Therefore the applied shear stress is proportional to the velocity gradient

Fr JJJJJJJJJ..$*%

"imensions of is +7T . "ynamic viscosity has therefore a unit of -g7ms or

6s7m*

2quation$*% is called the Neton/& la of vi&%o&it( and the uid that followthis law are called Netonian "#i$.

9here is called the coeKcient of dynamic

viscosity or simply viscosity

The 'gure shows the velocity pro'leof a moving uid near a solidboundary.

At the boundary the velocity of uidrelative to the boundary is ;ero. This iscalled the no &lip %on$itionthat wouldexist for all viscous uid ow pasta solidsurface.

The shear stress at any point isproportional to the slope of the velocity


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inematic viscosity: it is the ratio of dynamic viscosity and mass densityJJJJJ..J.$%

The dimension of is

*

7T and the unit is m

*

7s At *Eo3

0)12&3 04g2)&3

Air ).1 L )E=1 ).DL )E=1

water )E4 )E=

Air ).1 L )E=1 ).DL )E=1

water )E4 )E=

A uid having no viscosity $i.e. BE % is called non viscous or inviscidor ideal uid. 6o real uid is an inviscid.

&luid mechanics deals with 6ewtonian uid. &luid that do notfollow 6ewtonMs law are called non=6ewtonian uid

n

9here, B 3onsistency indexn B &low behavior

index

JJJJJJJJ.$0%

&or 6ewtonian uid B) and nB)

2quation0 can also be written as

JJJJJJJ..$1%

9here is called apparent viscosity


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Types of 6on 6ewtonian uid

). Dilatent 0&hear thi%4ening3 "#i$ 5

2xample:3oncentrated solution of sugar in water.Hesistance increases with increasing applied shear stress$nN)%*. P&e#$opla&ti% 0&hear thinning3"#i$5

2xample: !elatin, blood, mil-, liquid cement.Hesistance decreases with increasing applied shear stress$nO)%.

If the thinning e#ect is very strong, the uid is called plastic. /ome

plastic uid require a 'nite stress before they begin to ow. 6ingha) pla&ti%2i$eal pla&ti%5

2xample: clay suspension, drilling mud, toothpaste2xhibit linear relationship between applied shear stress and rateof deformation.

9here, is the viscosity of plastic uid.

0. I$eal "#i$


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SUFACE TENSION 03 If one imagines a line drawn on a surface of a liquid, then surface

tension may be de'ned as the magnitude of tensile force acting across

the perpendicular to a unit length of straight elements of the line. Alternatively the surface tension can be regarded as the surface

energy $ in 6m% per unit area $in m*%.

/urface tension has dimension +7T* or unit 67m

The value of surface tension forwater=air interface at *Eo3 is E.E567m

+ost organic liquid has surfacetension E.E* to E.E 67m

+ercury has surface tension about

E.1) 67m &or water the surface tension

decreases by adding some organic solutesli-e soap

detergent etc.

/alinity in water increases its surfacetension


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A liquid always forms an interface with another liquid or gasunless the liquid is in a container and is 'lled completely.

+olecules inside the liquid being surrounded by similar moleculesare subected to intermolecular force of attraction of equalmagnitude in all direction.

The molecules on the interface are surrounded by molecules ofother liquid or gas over the surface resulting imbalanced forcealong the interface to create a hypothetical membrane.

The unbalanced tensile force,acting on the plane of surface istermed as surface tension thatopposes any increase of surfacearea of a liquid surface.

The e#ect of surface tension is to

reduce the surface of a body ofliquid to a minimum. "ue to this the drop of liquid tend

to attain spherical shape so as tominimi;e the surface area.


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The surface tension of a liquid drop creates a higher pressurewithin the drop as compared to that of the surrounding.

3onsider the free body diagram of &pheri%al $rop of radius

The force developed around the surface due to surface tension B

It has to be balanced by the pressure di#erence between internalpressure pi and external pressure paacting over the circular area

/o,

Fr,

For hollo %ir%#lar *#**le ofradius H the surface tension force

would be *Las there is additional surface in sideof the bubble on which surfacetension also exists,/o,

Fr,


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Li+#i$ 7et

3onsider a section of et $radius H% of unitlength i.e. the ring between two planes normal

to the et axis and unit length apart.If half of the body were ta-en as free body:The tension at top and bottom are and&or high pressure the center of pressure canbe ta-en as the center of et and then B B

And the hori;ontal component of force actsthrough the pressure center of the proectedarea B *pH

Therefore, P *Fr,


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Another common phenomenon related to surface tension is the rise or fall of aliquid in a capillary tube $tube of very small diameter%.Con&i$er a %apillar( t#*e in&erte$ in ater

Capillarit(

&rom free body diagram 'gure 3

>ertical force due to surface tension B9eight of uid in capillary tube B?oth should be equal $why Q%

B

Fr,

When an interface between two fluids such as water and air or mercury and air meets a solid

surface the interface forms an angle w.r.t. the solid surface. This angle is called angle of contact

and depends on liquid, surrounding fluid and solid surface.

&or non wetting liquids li-emercury the angle of contactNCEo$mercury in contact with

clean glass )Eo

%

for contact with water andclean glass o


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-APOU PESSUE

The molecules of any liquid are at a state of continuous motion.

/ome of the molecules in the surface layer of the liquid would have

suKcient energy to enable them to overcome the cohesive attractionof the surrounding molecules and escape in space above the layer.

/ome of these molecules would return bac- to the layer but other

molecules of the liquid would ta-e their place. If the space above the liquid surface is con'ned then an equilibrium

will reach when the number of molecules of liquid in the space abovethe liquid surface is constant and the gas above the surface issaturated with vapors.

These vapors exerts a partial pressure on the liquid surface which is

-nown as vapour pressure. Vapour pressure increases with increase in temperature.

When the pressure above the liquid equals the vapour pressure ofliquid ,the liquid starts boiling.


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3avitation In a owing liquid, under certain condition the local pressure may

reduce less than the vapour pressure of the liquid and the liquid

starts boiling causing formation of bubbles. These vapour bubbles are carried by the liquid in to high pressure

;ones resulting in sudden implosive collapse.

If this collapse occurs in contact with a solid surface gets damageddue to the very large force with which the liquid hits the surfaceleading to the formation of cavities.

This process is called cavitation

3avitation can also occur when the dissolved air or gas is releaseddue to the reduced solubility as the pressure is reduced.

3avitation a#ects the performance of hydraulic machinery.


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6UL! MODULUS OFELASTICIT,

The compressibility of a liquid is expressed by its bul- modulusof elasticity de'ned as the ratio of change in pressure dp andthe resulting volumetric strain, d where is the volume of liquid.

B = B The dimension of is same as that of pressure +7T* and its unit is 67m* .

The Rve sign appears because with increase in pressure the volumedecreases.

/ince the decrease in volume of a given mass results in anincrease in densit, one can also write

B B

The for common liquids are large. &or water B *.E L )EC

67m*

indicating that a large pressure change is required for smallchange in volume.


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Fluid Mechanics_1_ Fluid Propeties