ME302 Static DS1

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ME-302: Fluid Mechanics

Deepak Singhal

Asst. ProfessorDept. of Mech. Engg.

KIIT University Bhubaneswar Orissa

E-mail: [email protected]

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ME-304

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States of MatterMatter comes in a variety of states: solid, liquid, gas, and

plasma. The molecules of solid are locked in a rigid structure and can

only vibrate. (Add thermal energy and the vibrations increase.)

Some solids are crystalline, like table salt, in which the atoms

are arranged in a repeating pattern. Some solids are

amorphous, like glass, in which the atoms have no orderly

arrangement. Either way, a solid has definite volume and

shape.

A liquid is virtually incompressible and has definite volume but

no definite shape. (If you pour a liter of juice into severalglasses, the shape of the juice has changed but the total

volume hasnt.)

A gas is easily compressed. It has neither definite shape nor

definite volume. (If a container of CO2 is opened, it will diffuse


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Fluids

The term fluid refers to gases and liquids. Gases

and liquids have more in common with each other,

both have atoms/molecules that are free to move

around. They are not locked in place as they are in

a solid. The hotter the fluid, the faster its moleculesmove on average, and the more space the fluid will

occupy (if its container allows for expansion.) Also,

unlike solids, fluids can flow.


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Definition of a Fluid a fluid, such as water or air, deforms continuously

when acted on by shearing stresses of any magnitude.

Water

Oil

Air

Water

Oil

Air

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Distinction between solids and

fluidsAccording to our experience: A solid is hard and

not easily deformed. A fluid is soft and deforms

easilycontinuously when acted on by a shearing stress ofany magnitude

Solid may regain partly or fully its original shape whenthe tangential stress is removed

A fluid can never regain its original shape, once it hasbeen distorded by the shear stress

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Introduction Any characteristic of a system is called a property.

Familiar: pressure P, temperature T,volume V, and mass m.

Less familiar: viscosity, thermal conductivity, modulus of elasticity,

thermal expansion coefficient, vapor pressure, surface tension.n ens ve proper es are n epen en o e mass o e

system. Examples: temperature, pressure, and density.

Extensive properties are those whose value depends onthe size of the system. Examples: Total mass, total

volume, and total momentum. Extensive properties per unit mass are called specific

properties. Examples include specific volume v = V/mand specific total energye=E/m.

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Continuum

Atoms are widely spaced in the gasphase.

However, we can disregard theatomic nature of a substance.

View it as a continuous,

homogeneous matter with noholes, that is, a continuum.

This allows us to treat properties assmoothly varying quantities.

Continuum is valid as long as sizeof the system is large in

comparison to distance betweenmolecules.


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Density and Specific Gravity Density is defined as the mass per unit volume = m/V.

Density has units of kg/m3

Specific volume is defined as v = 1/= V/m.

For a gas, density depends on temperature and pressure.

Specific gravity, or relative density is defined as the ratioof the density of a substance to the density of some standardsubstance at a specified temperature (usually water at 4C),i.e., SG=/

H20. SG is a dimensionless quantity.

The specific weight is defined as the weight per unit volume, i.e.,s =gwhereg is the gravitationalacceleration. s has units of N/m

3.

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Vapor Pressure and Cavitation

Vapor Pressure Pv is defined asthe pressure exerted by its vaporin phase equilibrium with itsliquid at a given temperature

If P drops below Pv, liquid is

locally vaporized, creatingcavities of va or. Vapor cavities collapse when

local Prises above Pv. Collapse of cavities is a violent

process which can damagemachinery.

Cavitation is noisy, and cancause structural vibrations.


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Energy and Specific Heats Total energyEis comprised of numerous forms: thermal,

mechanical, kinetic, potential, electrical, magnetic,chemical, and nuclear.

Units of energy arejoule (J) or British thermal unit (BTU). Microscopic energy Internal energyu is for a non-flowing fluid and is due to molecular

activity. Enthalpy h=u+Pv is for a flowing fluid and includes flow energy

(Pv). Macroscopic energy

Kinetic energyke=V2/2 Potential energype=gz

In the absence of electrical, magnetic, chemical, andnuclear energy, the total energy is eflowing=h+V

2/2+gz.

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Viscosity

Viscosityis a propertythat represents theinternal resistance of afluid to motion.

The force a flowingu exerts on a o y

in the flow direction iscalled the drag force,

and the magnitude ofthis force depends, inpart, on viscosity.


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Viscosity

To obtain a relation for viscosity,consider a fluid layer between two

very large parallel plates separatedby a distance

Definition of shear stress is =F/A.

Using the no-slip condition,u 0 = 0 an u = , t e ve ocityprofile and gradient are u(y)= Vy/and du/dy=V/

Shear stress for Newtonian fluid: = du/dy

is the dynamic viscosityandhas units ofkg/ms, Pas, or poise.


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Newton's law of viscosity

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Types of fluid

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Effect of Temp on the fluid Viscosity

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Surface Tension

Liquid droplets behave like smallspherical balloons filled withliquid, and the surface of theliquid acts like a stretched elasticmembrane under tension.

The pulling force that causes thisis ue to t e attract ve orces

between molecules

called surface tension s.

Attractive force on surfacemolecule is not symmetric.

Repulsive forces from interiormolecules causes the liquid tominimize its surface area andattain a spherical shape.


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Surface TensionEver wonder why water beads up on a car, or how some

insects can walk on water, or how bubbles hold themselves

together? The answer is surface tension: Because of

cohesion between its molecules, a substance

tends to contract to the smallest area possible. Water on awaxed surface, for example, forms round beads because in this

shape, more weak bounds can be formed between molecules

than if they were arranged in one flat layer. Cohesive forces are

greater in mercury than in water, so it forms a more spherical

shape. Cohesive forces are weaker in alcohol than in water, so it

forms a more flattened shape.

mercury water alcohol


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Capillary Effect

Capillary effect is the riseor fall of a liquid in a small-diameter tube.

The curved free surface inthe tube is call the

meniscus.

because water is a wettingfluid.

Mercury meniscus curvesdown because mercury is a

nonwetting fluid. Force balance can describe

magnitude of capillary rise.


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Cohesion & Adhesion

The force of attraction between unlike charges in the atoms or molecules of

substances are responsible for cohesion and adhesion.

.

wonder why rain falls in drops rather than individual water molecules? Its because

water molecules cling together to form drops.

Adhesion is the clinging together of molecules/atoms of two different substances.

Adhesive tape gets its name from the adhesion between the tape and other objects.

Water molecules cling to many other materials besides clinging to themselves.

continued


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Cohesion & Adhesion (cont.)The meniscus in a graduated cylinder of water is due to the adhesion between

water molecules the sides of the tube. The adhesion is greater than the cohesion

between the water molecules.

The reverse is true about a column of mercury: Mercury atoms are attracted to

H2O Hg

eac o er more s rong y an ey are a rac e o e s es o e u e. s

causes a sort of reverse meniscus.


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Pressure Pressure is defined as a normal force exerted by a f luid

per unit area.

Units of pressure are N/m

2

, which is called a pascalPa .

Since the unit Pa is too small for pressuresencountered in practice, kilopascal(1 kPa = 103 Pa) andmegapascal(1 MPa = 106 Pa) are commonly used.

Other units include bar, atm, kgf/cm2, lbf/in2=psi.

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Pressure in a Fluid The pressure is just the weight of all the fluid above

you

Atmospheric pressure is just the weight of all the air

above on area on the surface of the earth In a swimming pool the pressure on your body surface

is just the weight of the water above you (plus the airpressure above the water)

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Absolute, gage, and vacuum

pressuresActual pressure at a give point is called the absolute

pressure.

Most pressure-measuring devices are calibrated to

,gage pressure, Pgage=Pabs - Patm.

Pressure below atmospheric pressure are called

vacuum pressure, Pvac=Patm - Pabs.

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Absolute, gage, and vacuum

pressures


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Pressure at a Point Pressure at any point in a fluid is the same in all

directions.

Pressure has a magnitude, but not a specific direction,

.

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Direction of fluid pressure on

boundaries

Furnace duct Pipe or tube

ea exc anger

Dam

Pressure is a Normal Force

(acts perpendicular to surfaces)

It is also called a Surface Force

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Variation of Pressure with Depth

In the presence of a gravitationalfield, pressure increases withdepth because more fluid rests ondeeper layers.

To obtain a relation for the

,consider rectangular element Force balance in z-direction gives

Dividing byx and rearranging gives2 1

0

0

z z F ma

P x P x g x z

= =

=

2 1 s P P g z z = = =


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Variation of Pressure with Depth

Pressure in a fluid at rest is independent of theshape of the container.

Pressure is the same at all points on a horizontalplane in a given fluid.


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Scuba Diving and Hydrostatic

Pressure

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Compressible fluid

Gases are compressible i.e. their density varies withtemperature and pressure =P M /RT

For small elevation changes (as in engineeringapplications, tanks, pipes etc) we can neglect the effectof elevation on pressure

In the general case start from:

g

dz

dP=

=

==

o

o

RT

zzMgPP

Tfor

)(exp

:constT

1212


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Pascals Law

Pressure applied to aconfined fluid increasesthe pressure throughout bythe same amount.

In picture, pistons are at

1 2 2 21 2

1 2 1 1

F F AP P

A F A= = =

Ratio A2/A1 is called idealmechanical advantage


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The Manometer

An elevation change ofzin a fluid at restcorresponds to P/g.

A device based on this iscalled a manometer.

A manometer consists of a-

1 2

2 atm

P P

P P gh

=

= +

more fluids such asmercury, water, alcohol, oroil.

Heavy f luids such as

mercury are used if largepressure differences areanticipated.


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Mutlifluid Manometer

For multi-fluid systems Pressure change across a f luid column

of height h is P =gh.

Pressure increases downward, anddecreases upward.

Two points at the same elevation in acontinuous fluid are at the samepressure.

Pressure can be determined by addingand subtractinggh terms.

2 1 1 2 2 3 3 1 gh gh gh P + + + =


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Measuring Pressure Drops

Manometers are well--suited to measure pressuredrops across valves, pipes,heat exchangers, etc.

Relation for pressure drop

P1-P2 is obtained bystart ng at po nt 1 anadding or subtractingghterms until we reach point2.

If f luid in pipe is a gas,2>>1 and P1-P2=gh


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The Barometer

Atmospheric pressure ismeasured by a device called abarometer; thus, atmosphericpressure is often referred to asthe barometric pressure.

PCcan be taken to be zero since

there is only Hg vapor above

C atm

atm

gh P

P gh

+ =

=

,relative to Patm.

Change in atmospheric pressuredue to elevation has manyeffects: Cooking, nose bleeds,

engine performance, aircraftperformance.


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Fluid Statics Fluid Statics deals with problems associated with

fluids at rest. In fluid statics, there is no relative motion between

adjacent fluid layers.,trying to deform it.

The only stress in fluid statics is normal stress Normal stress is due to pressure

Variation of pressure is due only to the weight of thefluid fluid statics is only relevant in presence ofgravity fields.

Applications: Floating or submerged bodies, waterdams and gates, liquid storage tanks, etc.

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Motivation?What are the pressure forces behind the Hoover Dam?

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Hydrostatic Forces on Plane

Surfaces On aplane surface, the

hydrostatic forces form asystem of parallel forces

For many applications,magnitude and location of

application, which is calledcenter o pressure, mustbe determined.

Atmospheric pressure Patmcan be neglected when it

acts on both sides of thesurface.


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First moment of an area and the

centroid

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econ momen o area

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Resultant Force

The magnitude ofFR acting on a plane surface of a

completely submerged plate in a homogenous fluidis equal to the product of the pressure PCat thecentroid of the surface and the areaA of the surface


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Center of Pressure

Line of action of resultant forceFR=PCA does not pass throughthe centroid of the surface. Ingeneral, it lies underneath wherethe pressure is higher.

Vertical location ofCenter of

Pressure is determined by

,xx C

p C

c

y yy A

= +

resultant force to the moment ofthe distributed pressure force.

$Ixx,C is tabulated for simplegeometries.


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Surfaces

FR on a curved surface is more involved since itrequires integration of the pressure forces that

change direction along the surface. Easiest approach: determine horizontal and

vertical components FHand FVseparately.


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Hydrostatic Forces on Curved

Surfaces Horizontal force component on curved surface:

FH=Fx. Line of action on vertical plane givesycoordinate of center of pressure on curved surface.

Vertical force component on curved surface:,the enclosed block W=gV. x coordinate of thecenter of pressure is a combination of line ofaction on horizontal plane (centroid of area) andline of action through volume (centroid of

volume). Magnitude of force FR=(FH

2+FV2)1/2

Angle of force is = tan-1(FV/FH)

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Liquid subjected to Horizontal acceleration

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Vertical acceleration

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Buoyancy and Stability

Buoyancy is due to the fluid displaced by a body.

FB=fgV.

Archimedes principal : The buoyant force acting ona body immersed in a fluid is equal to the weight of

the fluid displaced by the body, and it acts upwardthrough the centroid of the displaced volume.

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Buoyancy and Stability

Buoyancy force FB is equalonly to the displacedvolumefgVdisplaced.

Three scenarios possible

1. bodyfluid: Sinking body


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Stability of Immersed Bodies

Rotational stability of immersed bodies depends uponrelative location ofcenter of gravity G and center ofbuoyancy B. G below B: stable

G above B: unstable

G coincides with B: neutrally stable.


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Stability of Floating Bodies

If body is bottom heavy (Glower than B), it is alwaysstable.

Floating bodies can be

stable when G is higher

location of centerbuoyancy and creation ofrestoring moment.

Measure of stability is themetacentric height GM. IfGM>1, ship is stable.


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The Golden Crown of Hiero II, King of Syracuse

Archimedes, 287-212 B.C.

Hiero, 306-215 B.C.

Hiero learned of a rumor where the

goldsmith replaced some of the gold.asked Archimedes to determinewhether the crown was pure gold.

Archimedes had to develop anondestructive testing method


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The Golden Crown of Hiero II, King of Syracuse

The weight of the crown andnugget are the same in air: Wc =cVc = Wn =nVn.

If the crown is pure gold,c=nwhich means that the volumesmust be the same, V

c

=Vn

. ,

B=H2OV. If the scale becomes unbalanced,

this implies that the Vc Vn, whichin turn means that thec n

Goldsmith was shown to be afraud!

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ME302 Static DS1