fluids lakshmikanth 19.05.12 - karnataka · they are called fluids. fluid is the name given to a...

Fluid Mechanics

Vikasana – Bridge Course 2012

FluidLiquids and gases can flow. Hence

they are called fluids.they are called fluids.

Fluid is the name given to aFluid is the name given to asubstance which begins to flow, whenexternal force is applied on it Thisexternal force is applied on it. Thisproperty distinguish fluids from solids.


Fluid Mechanics:-It is the branch of physics, which

deals with the properties of fluids underp pthe action of several forces.

Fluid Mechanics is classified into twobranchesbranches.

1. Fluid statics 2. Fluid dynamics


i) Fluid statics :- It deals with variousproperties of fluids at rest and thelaws that govern them.g

ii) Fluid dynamics : It deals with variousii) Fluid dynamics :- It deals with variousproperties of fluids in motion and thelaws that govern them.


Fluid Thrust The total normal force exerted by a

liquid at rest on any surface in contactq ywith it is called fluid thrust.


Pressure (Fluid Pressure/Hydro Static Pressure):(Fluid Pressure/Hydro Static Pressure):

Fluids have weight and they alsot th b d ll fexert pressure on the base and walls of

the container in which they arel denclosed.

Pressure at any point in a fluid is thefluid thrust per unit area around thatpoint.


Pressure = Thrustarea

1. Pressure is a Scalar quantity.2 SI unit of pressure is Nm-2 (pascal)2. SI unit of pressure is Nm 2 (pascal) 3. 1 Pa = 1 N/m2


APPLICATION’S OF THE CONCEPT OFPRESSUREPRESSURE :1. We can push a nail into a woodenplank by it’s pointed end rather than itshead. Because the area of the pointedend of the nail is much smaller than it’shead.

(As pressure, P = F )A


A

2. For the same reason, it is easier to cutvegetables with sharp knife rather thanvegetables with sharp knife rather thanblunk knife. The area over which theforce is applied plays a role in makingforce is applied, plays a role in makingthese tasks easier.

3 P t ’ l d i f l th3. Porter’s place a round piece of clothon their heads, when they have to carryh l d B d i thi thheavy loads. By doing this theyincrease the area of contact of the load

ith th i h dVikasana – Bridge Course 2012

with their head.

So the pressure on the head isreduced and they find it easier to carryreduced and they find it easier to carrythe load.

4 Th b d it id d4. The bags and suit cases are providedwith broad handles so that small

i t d th h d hilpressure is exerted on the hand whilecarrying them.


PRESSURE EXERTED BY A LIQUID:-Li id h llLiquids exerts pressure on the walls

of the container. Pressure exerted byth li id t th b tt f th t ithe liquid at the bottom of the containerdepends on the height of it’s column.

Pressure at any point at a depth ‘h’below the surface of a homogenousliquid of density ‘ρ’ is given by

p = ρ g h


p ρ gwhere ‘g’ isacceleration due to gravity.

p = ρ g hThus as depth below the surfaceThus as depth below the surface

increases the pressure also increases.


Activity :- To show that, liquid exertspressure let us do an activity Considerpressure, let us do an activity. Considera vessel containing a liquid. Make asmall hole near the bottom Now thesmall hole near the bottom. Now theliquid will flow out of it. If the flow liquidis stopped by pressing a finger againstis stopped by pressing a finger againstthe hole the finger will experience anout ward force due to the liquidout ward force due to the liquidcontained in the vessel.


The ratio of force to the area of thefinger in contact with it gives, pressurefinger in contact with it gives, pressureof the liquid. Therefore it is reasonableto conclude that the liquid exertsto conclude that the liquid exertspressure on the walls and base of thecontaining vessel.containing vessel.Ex:- Fountains of water comes out ofthe leaking joints or holes in pipesthe leaking joints or holes in pipessupplying water. It is due to thepressure exerted by water on the walls


pressure exerted by water on the wallsof the pipes.

Pressure exerted by a Gas :-G h ll fGases too exert pressure on the walls ofthe container.

Ex:-1) Air in the bicycle tube comes out) y

when it has a Puncture. Because airexerts pressure in all directions.p2) We can not inflate the balloon which

has holes for the same reason given


has holes, for the same reason givenabove.

ATMOSPHERIC PRESSURE:-1. We know that there is air all around

us. This envelop of air surrounding theearth is called atmosphere.

2. Earth’s atmosphere extends up to aheight of about 200 km above theheight of about 200 km above thesurface of the earth.


3. The pressure exerted by theatmosphere on anybody on the earth’satmosphere on anybody on the earth’ssurface is called the atmosphericpressure This is obviously due to thepressure. This is obviously due to theweight of the gases of the earth’satmosphereatmosphere.

4. Atmospheric pressure is maximum atth f f th th dthe surface of the earth and goes ondecreasing as we move up into the

th’ t hVikasana – Bridge Course 2012

earth’s atmosphere.

5. The device used to measureatmospheric pressure is known asatmospheric pressure is known asbarometer.

6 SI it f t h i i N 26. SI unit of atmospheric pressure is Nm-2

or Pascal (pa).7. Atmospheric pressure is also

measured in mm or cm of mercurycolumn.


8. Normal or standard atmosphericpressure is the pressure exerted by apressure is the pressure exerted by avertical column of mercury of height0 76 m at 00c at sea level0.76 m at 00c at sea level.

For normal pressure, we haveh= 0.76 mρ = 13.596103 kg/m3ρ gg= 9.806 m/s2


Since, p = ρ h g

Normal pressure,p = 0.76 x13.596x9.806

= 1 013105 Nm-2= 1.01310 Nm1 atmospheric pressure

1 t 1 013105 N 2= 1atm = 1.013105 Nm-2


PASCAL’S LAW :A P d hA Pressure exerted anywhere on aconfined fluid (enclosed) is transmitted

ll t ti f fl idequally to every portion of fluid.Applications of Pascal’s law :

(1) hydraulic lift(2) hydraulic brakes(2) hydraulic brakes(3) In the crushing of oil seeds and

t ti f il f thVikasana – Bridge Course 2012

extraction of oil from them.

HYDRAULIC LIFT :-S ll f li d th llSmall force applied on the smaller

piston will be appearing as a very largef th l i t A lt fforce on the large piston. As a result ofwhich a heavy load placed on the larger

i t i il lift d dpiston is easily lifted up wards.


Applied pressure P = F1

AA1

Transmitted pressure P = F2

A2

By Pascal’s Principle P = F1 = F2By Pascal s Principle P = F1 = F2

A1 A2

F A FF2 = A2 F1

A1

Vikasana – Bridge Course 2012Since A2 > A1 F2 > F1

Thus, by applying a small force onthe piston in X we can get a large forcethe piston in X, we can get a large forceon the piston in Y.

Thi i i l i d i d ti tThis principle is used in dentistchairs, car lift’s and Jacks.


ARCHIMEDES’S PRINCIPLE Wh b d i i d ti llWhen a body is immersed partially or

completely in a liquid, the body loosest f it’ i ht th l f i hta part of it’s weight; the loss of weight

of body is equal to weight of liquiddi l ddisplaced.

If Wa be the weight of a body in airand Wl when it is immersed in a liquid atrest, then

Vikasana – Bridge Course 2012Wa – We = weight of liquid displaced.

Activity : Immerse a “heavy stone” inwater Try to lift the stone immersed inwater .Try to lift the stone immersed inwater. You will feel it “lighter as long asit is in water Once the stone comes upit is in water. Once the stone comes upto the air, you will feel it to be heavier.This is because anybody inside a liquidThis is because anybody inside a liquidlooses some weight.

Buoyancy: The resultant up thrust actingBuoyancy: The resultant up thrust actingon a body immersed completely orpartially in a fluid is called


partially in a fluid is called

up ward thrust or buoyant force orsimply buoyancysimply buoyancy.

The resultant upward force actingth h th t f it fthrough the centre of gravity ofdisplaced liquid is called center ofbbuoyancy.


FLOATATION:When a body is completely immersed

in a liquid, two forces act on it.1. The weight W of the body, which

acts in down ward directionacts in down ward direction.2. The buoyant force Wl, which acts

i d di tiin upward direction.


There are three possibilities.Case (I) : If W > Wl, In this case the weight

of the body (downward force) is morethan the upward force (up thrust).Hence the body will sink to bottom ofthe liquid.


Case (II) : If W = Wl In this case, thedownward force is equal to the upwarddownward force is equal to the upwardforce. Hence the body will float with it’swhole volume is just immersed in thewhole volume is just immersed in theliquid.


Case (III) : If W < Wl In this case, thedownward force is less than the upwarddownward force is less than the upwardfore. Hence the body experiences a netupward thrust and it rises to theupward thrust, and it rises to thesurface. This is the condition forfloationfloation.


LAWS OF FLOATATION:1 Th i ht f th fl ti b d i1. The weight of the floating body is

equal to the weight of the liquiddi l ddisplaced.

2. The centre of gravity of the body andthe centre of buoyancy lie in the samevertical line.


FLUID DYNAMICSFluid dynamics is that branch of fluid

mechanics which deals with flow offluids.

The motion of real fluids isThe motion of real fluids iscomplicated and not yet completelyunder stood. For example, the motion ofunder stood. For example, the motion ofwater in a “falls” (Jog falls) is verypleasant to look at.


pleasant to look at.

The water falls from a great height andthen moves thumping and jumpingthen moves thumping and jumpingproducing a musical effect. It is verydifficult to analyse the motion of anydifficult to analyse the motion of anyparticle of water using the laws ofmechanicsmechanics.

So to describe the motion of a fluid,id “id l fl id”we consider an “ideal fluid”.


1. The description of motion is based onth f ll i tithe following assumptions:

2. The fluid is incompressible i.e., densityof fluid is a constant.

3. The fluid is non-viscous i.e., no,opposition between the different layersof fluid.

4. The flow of fluid is steady.


STREAM LINE FLOW :


Characteristics of stream line flow:1 A li i li l hi h h1.A stream line is a line along which the

particles of the fluid move.2. No two stream lines intersect each

other.3. A bundle of stream lines is called a

“tube of flow”.4. The shape of a stream line may be

straight or curved


straight or curved.

5. The velocity of the fluid at any point int li i t t t ll tia stream line is constant at all times.

6. The amount of fluid that enters a tubeat one end will at all times be equal tothe amount that leaves the tube at theother end.


Equation of Continuitya v = a va1 v1 = a2 v2


The equation of continuity says thatthe amount of liquid that enters one andthe amount of liquid that enters one andshould come out of the other end, asthe liquid can not stay inside the tubethe liquid can not stay inside the tube.For this to happen, the liquid has toflow fast where the area of crossflow fast where the area of crosssection is small and it has to flow slowwhere the area of cross section is largewhere the area of cross section is large.


i.e., the velocity of fluid is less if the areaf ti i d l it iof cross section is more and velocity is

more where the area of cross section islless.

Illustrations :-1) Velocity of liquid is high in the

“narrow” tube as compared to thepvelocity of the liquid in a wider tube.


2) Deep water running's slow, can beexplained from the equation ofexplained from the equation ofcontinuity i.e,

t tav = a constant.The area of cross section increaseswhere water is deep and hence velocitydecreases. That is the reason why thewater runs slow when the path is widerand runs fast when the path is

Vikasana – Bridge Course 2012narrower in case of a river .

3) The falling stream of ater becomes3) The falling stream of water becomesnarrower as the velocity of fallingstream of water increases so it’s area ofstream of water increases so it’s area ofcross section decreases

TURBULENT FLOWIf the velocity of fluid crosses aparticular limit then the flow is calledturbulent flow.


Example:The flow of water in a river in rainy

season, The water flowing out from adam.


ENERGY OF A FLUID IN STEADY FLOW:

A fluid in steady flow can have threeytypes of energy.

1.Potential Energy1.Potential Energy2) Kinetic Energy3) P3) Pressure energy


1.Potential energy of a fluid is the energypossessed by it by virtue of it’spossessed by it by virtue of it’sposition. If h be the height of a givenmass ‘m’ of a fluid above a level thenmass ‘m’ of a fluid above a level, thenit’s potential energy = m g hP t ti l it hPotential energy per unit mass = g h

2.Kinetic energy of a fluid is the energypossessed by virtue of its motion.

K.E per unit mass = ½ v2


p

3. pressure energy of a fluid is theenergy possessed by virtue of itsenergy possessed by virtue of itspressure. If P be the pressure of a fluidand V it’s volumeand V it’s volume,it’s pressure energy = PVpressure energy per unit mass = PV/m

= P/ρρTotal energy per unit mass of a fluid inmotion = g h + ½ v2 + p / ρ


motion = g h + ½ v + p / ρ

BERNOULLI’S THEOREM :Al h li fl f id lAlong the stream line flow of an ideal

fluid the sum of Potential energy,d Ki tipressure energy and Kinetic energy per

unit mass remains constant.g h + p/ ρ + ½ v2 = constant.

Bernoulli’s theorem is an outcomeof the principle of conservation ofenergy applied to a liquid in motion.


gy pp q

For a horizontal flow, h is constant.I f h i l flIn case of horizontal flow.

p/ ρ + ½ v2 = constantSignificance :

If the speed v of flow at any pointIf the speed v of flow at any pointincreases the pressure at that pointdecreases and vice versadecreases and vice versa.


APPLICATIONS OF BERNOULLI’S THEOREM :THEOREM :1. UPLIFT OF AN AIR CRAFT :

We know that, the pressure of anyfluid decreases with increase in speed.This fact is used in designing air craftwings.Air craft wings are so designed that

the air covers more distance above the

Vikasana – Bridge Course 2012wings than below the wings.

This leads to high velocity, lowt f i dpressure on upper part of wings and

low velocity, high pressure belowth i Thi i t h tthe wings. This gives a net push towings in upward direction, that lifts thei lair plane.

½ v2 + p / ρ = constant


2. SPRAYER OR ATOMISER


3. CURVED MOTION OF A SPINNINGBALL :

When a spinning ball is thrown, itp g ,deviates from it’s usual path in flight.

4. Blowing off of the roofs during strong4. Blowing off of the roofs during strongwinds.


5) If sheet of paper placed on a table isblown across it’s top surface the paperblown across it’s top surface the paperrises. This is because the jet of airblown over the top surface of the paperblown over the top surface of the papercauses a reduction in air pressureabove the paper A net upward forceabove the paper. A net upward forceacts on the paper which causes it toriserise.


SURFACE TENSIONThere are two types of molecular forces.1) Adhesive forces1) Adhesive forces2) Cohesive forces1 ADHESIVE FORCES It i th f f1.ADHESIVE FORCES :- It is the force of

attraction acting between the moleculesof different s bstancesof different substances.

Ex. The force of attraction between a


water molecule and an air molecule.

2) Water wets the surface of the glasscontainer The force attraction betweencontainer. The force attraction betweena water molecule and a glass molecule.

3)Sti ki f t P i t t3)Sticking of gum to a paper., Paint towall.

2.COHESIVE FORCE :- It is the force ofattraction between the molecules of thesame substance.


Example :1. The force of attraction between two

molecules of water.2.The force of attraction between two

molecules of sodium chloride (common(salt). Cohesive force is maximum insolids, lesser in liquids and least in, qgases.


Molecular range :It is the maximum distance upto

which a molecule can exert somemeasurable attraction on an anothermolecule. The order of molecular rangeis 10-9m in solids & liquids.


Surface Tension :A ll i f li id f fA small quantity of liquid free fromexternal forces always takes up aS h i l ShSpherical Shape.

For Example : Rain drops and tinydroplets of mercury on a sheet of glassare spherical. For a given volume asphere has the least surface area. Thusthe tendency of the liquid is to assume

Vikasana – Bridge Course 2012a shape for which the surface area is aminimum

The surface of a liquid thus acts like astretched membrane which alwaysstretched membrane, which alwaystries to contract and have the minimumsurface areasurface area.

Surface Tension is the property of theliquid by virtue of which the freeliquid by virtue of which the freesurface of liquid at rest tends to haveminimum area and as such it behavesminimum area and as such it behavesas if covered with a stretchedmembrane


membrane

It is measured by the ratio of totalforce on either side of the imaginaryforce on either side of the imaginaryline to length of the line.

SI it f f t i i N 1SI unit of surface tension is Nm-1


Experiments to illustrate surface tension:1 Wh i l ( i l i f1. When a wire loop (a circular ring of

wire) is dipped into a soap solution andt k t fil i f d thtaken, out a soap film is formed on theloop due to surface tension.

2. A small liquid drop takes Sphericalshape due to surface tension.

3. When a paint brush is dipped inwater, it’s bristles spread out. When the


, pbrush is taken out of water, the bristlescling together.

Angle of contact :- When a liquid is incontact with a solid the surface of thecontact with a solid, the surface of theliquid will be curved at the point ofcontactcontact.


The angle between tangent to liquidsurface at the point of contact and thesurface at the point of contact and thesolid surface inside the liquid is calledangle of contact It is denoted by ‘θ’angle of contact. It is denoted by ‘θ’1. If the angle of contact is acute, thenth li id t th fthose liquids wet the surface.2. If the angle contact is obtuse, thenthose liquids will not wet surface


3. For liquids which wet the surfaceθ < 900

4. For liquids which do not wet theqsurface, θ> 900

5 The angel of contact for pure water5. The angel of contact for pure waterand clean glass is zero. For mercuryand glass it is 1350and glass it is 135 .


CAPILLARITY :A t b f fi b i ll dA tube of very fine bore is called a

capillary tube.When a capillary tube is dipped

vertically into a liquid contained in abeaker the liquid rises, if θ < 900.

If θ > 900, there is depression in the, ptube.


The rise or fall in the level of a liquid ina capillary tube due to the property ofa capillary tube due to the property ofsurface tension is known as capillarity.


Expression for capillary rise of a liquid isgiven bygiven by

h = 2T / ρ g r

WhereT is S.T. of the liquidρ is density of the liquidρ is density of the liquidg is acceleration due to gravity

Vikasana – Bridge Course 2012r is radius of the capillary tube.

From this expression, it is clear that,capillary rise varies inversely as thecapillary rise varies inversely as theradius of the capillary tube. Thus thenarrower the capillary tube the highernarrower the capillary tube, the higheris the capillary rise.

APPLICATIONS OF SURFACE TENSION:1. Due to capillary action, water from the

stem of a tree reaches every branch of


ythe tree.

2. We can see insects like mosquitoesstanding safely on the surface of waterstanding safely on the surface of water.

3. A sheet of blotting paper contains fineIt b b i k t d t thpores. It absorbs ink or water due to the

forces of surface tension.4.Cotton dress is more comfortable in

summer. This is because sweat formedon the body in summer is sucked by thepores present in the cotton dress.


5. Swelling of wood in rainy season isdue to rise of moisture from air in thedue to rise of moisture from air, in thepores of the wood.

6 Pl hi f fi ld i ti l f6. Ploughing of fields is essential forpreserving moisture in the soil. By

l hi th fi ill i i th ilploughing, the fine capillaries in the soilare destroyed. Water from within the

il h ll t i d t ffsoil shall not rise and evaporate off.7. A towel soaks water on account of

Vikasana – Bridge Course 2012capillary action.

VISCOSITYVi i i h f fl idViscosity is the property of a fluid

(liquid or gas) by virtue of which ani t l f i ti l f i t linternal frictional force comes into playwhen the fluid is in motion and opposesth l ti ti f it’ diff tthe relative motion of it’s differentlayers. It is also called “fluid friction”.The opposing force is called theinternal frictional force or viscous force

Vikasana – Bridge Course 2012and this property of fluid is called

viscosity.

Thick liquids like castor oil, honey,diesel etc are more viscous Thindiesel etc., are more viscous. Thinliquids like water, alcohol, petrol, areless viscousless viscous.


Causes of viscosity: It is due to theinter molecular forces which areinter molecular forces which areeffective when the different layers of thefluid are moving with differentfluid are moving with differentvelocities. Thick liquids flow veryslowly where as thin liquids flow fastslowly where as thin liquids flow fast.


Viscous drag :- Due to viscous force,every fast moving liquid layer tends toevery fast moving liquid layer tends toaccelerate the adjoining slow movinglayer and every slow moving layerlayer and every slow moving layertends to retard the adjoining fastmoving layer of liquid As a result amoving layer of liquid. As a result, abackward dragging force called“viscous drag” comes into play which“viscous drag” comes into play whichaccount for viscosity of liquids.


Aero plane, boats and cars aredesigned such that they experience thedesigned such that they experience theleast viscous drag by the fluid throughwhich they movewhich they move.

Variation of viscosity with temperatureViscosity of liquids generally decreaseswith increase in temperature. This isbecause, as the temperature increases

Vikasana – Bridge Course 2012the force of cohesion decreases.

Viscosity of gases increases withincrease in temperature This isincrease in temperature. This isbecause viscous force of a gas is dueto the diffusion of gas molecules whichto the diffusion of gas molecules, whichincreases with the rise in temperature.


POISEUILLE’S FORMULA :-Volume of liquid flowing per second through aVolume of liquid flowing per second through a

horizontal capillary tube is

Where L = Length of the capillary tubeη = co efficient of viscosityη yP = Pressure differencer = radius of the tube.


This equation is true only for thet d fl f li id th hsteady flow of liquid through a

horizontal capillary tube.STOKES FORMULA :

When a body moves through a fluid ity gexperiences a retarding (draggingforce) force due to the viscosity of the) yfluid. If the body has an irregular shapethis viscous force cannot be calculated.


However, if the body is spherical inshape the viscous drag experienced byshape, the viscous drag experienced byit while moving in a fluid with aconstant velocity can be calculatedconstant velocity can be calculated.This calculation was first made bystokes The result obtained by him isstokes. The result obtained by him iscalled stokes formula.

Thi f l iThis formula isF =6 π η r v


WhereF = viscous force experienced by asphere of radius ‘r’

η = coefficient of viscosityv = velocity of the spherev velocity of the spherer = radius of the sphere

I tImportance :1) This law helps a man coming down

Vikasana – Bridge Course 2012with the help of parachute.

2) This law accounts for the formation ofcloudsclouds.

3) This law is used in the determinationf l t i h ith th h lof electronic charge with the help

of Millikan’s (oil drop) experiment.4) This law explains, why the speed of

rain drops is less than that of a bodyfalling freely with a constant velocityfrom the height of clouds.


STOKE’S LAW :-S k f d h b k d d iStoke found that backward dragging

force F acting on a small spherical bodyf di ‘ ’ i th h diof radius ‘r’, moving through a medium

of coefficient of viscosity with velocityi i bv is given by,

F = 6 π η r V


fluids lakshmikanth 19.05.12 - karnataka · they are called fluids. fluid is the name given to a...

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