FMM Course Material

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FLUID MECHANICS AND MACHINERY

UNIT I

INTRODUCTION

Definition of Stress

Consider a small area A on the surface of a body (Fig. 1.1). The force acting on this area is FThis force can be resolved into two perpendiculr co!ponents

The component of force acting normal to the area called nor!l force and is denoted by Fn The component of force acting along the plane of area is called tn"entilforce and is denoted

by Ft

Fi" #$# Nor!l nd Tn"entil Forces on surfce

When they are expressed as force per unit area they are called as nor!l stressand tn"entilstress respectively$The tangential stress is also

called shear stress

The normal stress

(1.1)

And shear stress

Definition of Fluid

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A fluid is a substance that defor!s continuousl%in the face of tangential or shearstress irrespecti&e of t'e !"nitude of s'er stress.This continuous deformation under theapplication of shear stress constitutes a flo!.

"n this connection fluid can also be defined as the stte of !tter t't cnnot sustin n%s'er stress$

Example :Consider Fig 1.#

Fi" #$( S'er stress on fluid )od%

"f a shear stress is applied at any location in a fluid the element $11% !hich is initially at rest !ill moveto $##% then to $&&%. Further it moves to $''% and continues to move in a similar fashion.

"n other !ords the tn"entil stress in fluid )od% depends on &elocit% of defor!tion nd&nis'es s t'is &elocit% pproc'es *ero$ A "ood e+!ple ise!ton%s parallel plateexperimentw'ere dependence of s'er force on t'e &elocit% of defor!tion ws est)lis'ed$

Distinction ,etween Solid nd Fluid

Solid Fluid

ore Compact *tructure

Attractive Forces bet!een themolecules

are larger therefore more closely pac+ed

*olids can resist tangential stresses instatic condition

Whenever a solid is sub,ected to shearstress

a. "t undergoes a definite

-ess Compact *tructure

Attractive Forces bet!een the moleculesare smaller therefore more loosely

pac+ed

Fluids cannot resist tangential stresses instatic condition.

Whenever a fluid is sub,ected to shearstress

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deformationor brea+s

b. is proportional to shear stressupto some limiting condition

*olid may regain partly or fully its

original shape !hen the tangentialstress is removed

a. o fixed deformation

b. Continious deformation ta+esplaceuntil the shear stress is applied

A fluid can never regain its originalshape once it has been distorded by theshear stress

Fi" #$- Defor!tion of Solid ,od%

Concept of Continuu!

The concept of continuum is a +ind of ideali/ation of the continuous description of matter!here the properties of the matter are considered as continuous functions of spacevariables. Although any matter is composed of several molecules the concept ofcontinuum assumes a continuous distribution of mass !ithin the matter or system !ith no

empty space instead of the actual conglomeration of separate molecules. Fluid .roperties /

Characteristics of a continuous fluid !hich are independent of the motion of the fluid are calledbasic properties of the fluid. *ome of the basic properties are as discussed belo!.

.ropert% S%!)ol Definition Unit

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Densit%

The densityp of a fluid is its mass per unit volume . "f a fluid element enclosing apointP has a volume and mass m (Fig. 1.') then density ()at point P is !rittenas

0o!ever in a medium !here continuum model is valid one can !rite

(1.&)

Fi" #$0 A fluid ele!ent enclosin" point .

+g2m&

Specific1ei"'t

The specific wei"'tis the !eight of fluid per unit volume. The specific !eight isgiven

by = g(1.'

)

Where g is the gravitational acceleration. 3ust as !eight must be clearlydistinguished from mass so must the specific !eight be distinguished from density.

2m&

Specific2olu!e

v The specific &olu!e of a fluid is the volume occupied by unit mass of fluid.

Thus

m&2+g

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(1.4)

Specific3r&it% s

For li5uids it is the ratio of density of a li5uid at actual conditions to the density ofpure !ater at 1$1 +2m# and at '6C.

The specific gravity of a gas is the ratio of its density to that of either hydrogen or airat some specified temperature or pressure.

Howe&er4 t'ere is no "enerl stndrd5 so t'e conditions !ust )e stted w'ilereferrin" to t'e specific "r&it% of "s$

2iscosit% 6 7 /

7iscosity is a fluid property !hose effect is understood !hen the fluid is in motion.

"n a flo! of fluid !hen the fluid elements move !ith different velocities each element !ill feelsome resistance due to fluid friction !ithin the elements.

Therefore shear stresses can be identified bet!een the fluid elements !ith different velocities.

The relationship bet!een the shear stress and the velocity field !as given by *ir "saac e!ton.

Consider a flo! (Fig. 1.4) in !hich all fluid particles are moving in the same direction in such a !ay thatthe fluid layers move parallel !ith different velocities.

Fi" #$8 .rllel flow of fluidFi" #$9 Two d:cent l%ers of !o&in"

fluid$

The upper layer !hich is moving faster tries to dra! the lo!er slo!ly moving layer along !ith itby means of a force F along the direction of flo! on this layer. *imilarly the lo!er layer tries toretard the upper one according to e!ton%s third la! !ith an e5ual and opposite force F on it(Figure 1.8).

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*uch a fluid flo! !here xdirection velocities for example change !ith ycoordinate iscalled s'er flowof the fluid.

Thus the dragging effect of one layer on the other is experienced by a tangential force F on therespective layers. "f F acts over an area of contactA, then the shear stress is defined as

=F/A

2iscosit% 6 7 /

e!ton postulated that is proportional to the 5uantity u/ y!herey is the distance ofseparation of the t!o layers and u is the difference in their velocities.

"n the limiting case of u / ye5uals du2dy the velocity gradient at a point in a directionperpendicular to the direction of the motion of the layer.

According to e!ton and du2dy bears the relation

(1.9)

!here the constant of proportionality is +no!n as the coefficient of &iscosit% or simply viscosity !hich

is a property of the fluid and depends on its state. *ign of depends upon the sign of du2dy. For the profilesho!n in Fig. 1.4 du2dy is positive every!here and hence is positive. :oth the velocity and stress areconsidered positive in the positive direction of the coordinate parallel to them.;5uation

defining the viscosity of a fluid is +no!n as e!ton%s la! of viscosity. Common fluids vi/. !ater airmercury obey e!ton%s la! of viscosity and are +no!n ase!tonian fluids.


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Fi" #$< Mo&e!ent of fluid !olecules )etween two d:cent

!o&in" l%ers

olecules from layer aa in course of continous thermal agitation migrate into layer bb

omentum from the migrant molecules from layer aa is stored by molecules of layer bb by !ay ofcollision

Thus layer bb as a !hole is speeded up

olecules from the lo!er layer bb arrive at aa and tend to retard the layer aa

;very such migration of molecules causes forces of acceleration or deceleration to drag thelayers so as to oppose the differences in velocity bet!een the layers and produce the effect ofviscosity.

Cuses of 2iscosit% ; contd fro! pre&ious slide$$$

As the random molecular motion increases !ith a rise in temperature the viscosity also increasesaccordingly. ;xcept for very special cases (e.g. at very high pressure) the viscosity of bothli5uids and gases ceases to be a function of pressure.

For e!tonian fluids the coefficient of viscosity depends strongly on temperature but varies verylittle !ith pressure.

For li5uids molecular motion is less significant than the forces of cohesion thus&iscosit% ofli=uids decrese wit' increse in te!perture$

For gasesmolecular motion is more significant than the cohesive forces thus&iscosit% of "sesincrese wit' increse in te!perture$

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Fi" #$>/ C'n"e of 2iscosit% of 1ter nd Air under # t!

No;slip Condition of 2iscous Fluids

"t has been established through experimental observations that the relative velocity bet!een thesolid surface and the ad,acent fluid particles is /ero !henever a viscous fluid flo!s over a solidsurface. This is +no!n as noslip condition.

This behavior of noslip at the solid surface is not same as the !etting of surfaces by the fluids.For example mercury flo!ing in a stationary glass tube !ill not !et the surface but !ill have /erovelocity at the !all of the tube.

The !etting property results from surface tension !hereas the noslip condition is aconse5uence of fluid viscosity.

Idel Fluid

Consider a hypothetical fluid having a /ero viscosity ( ? $). *uch a fluid is called an idealfluid and the resulting motion is called as idel or in&iscidflow. In n idel flow4 t'ere is no

e+istence of s'er force )ecuse of &nis'in" &iscosit%$

All the fluids in relit% '&e &iscosit% ( @ $) and hence they are termed as real fluid and theirmotion is +no!n as viscous flo!.

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nder certain situations of very high velocity flo! of viscous fluids an accurate analysis of flo!field a!ay from a solid surface can be made from the ideal flo! theory.

Non;Newtonin Fluids

There are certain fluids !here the linear relationship bet!een the shear stress and the

deformation rate (velocity gradient in parallel flo!) as expressed by the is not valid. Forthese fluids the viscosity varies !ith rate of deformation.

Due to the deviation from e!ton%s la! of viscosity they are commonly termed as non;Newtonin fluids. Figure #.1 sho!s the class of fluid for !hich this relationship is nonlinear.

Fi"ure ($# S'er stress nd defor!tion rte reltions'ip of different fluids

The abscissa in Fig. #.1 represents the behaviour of ideal fluids since for the ideal fluids theresistance to shearing deformation rate is al!ays /ero and hence they exhibit /ero shear stressunder any condition of flo!.

The ordinate represents the ideal solid for there is no deformation rate under any loadingcondition.

The e!tonian fluids behave according to the la! that shear stress is linearly proportional to

velocity gradient or rate of shear strain . Thus for these fluids the plot of shearstress against velocity gradient is a straight line through the origin. The slope of the linedetermines the viscosity.

The none!tonian fluids are further classified as pseudoplastic dilatant and :ingham plastic.

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Co!pressi)ilit%

Compressibility of any substance is the measure of its change in volume under the action ofexternal forces.

The normal compressive stress on any fluid element at rest is +no!n as hydrostatic

pressure p and arises as a result of innumerable molecular collisions in the entire fluid.

The degree of compressibility of a substance is characteri/ed by the )ul? !odulus ofelsticit%; defined as

(#.&)

Where and p are the changes in the volume and pressure respectively and is the initialvolume. The negative sign (sign) is included to ma+e ; positive since increase in pressure!ould decrease the volume i.e for Bp@$ B $) in volume.

For a given mass of a substance the change in its volume and density satisfies the relation

m? $ ( ) ? $

(#.')

using

!e get

(#.4)

7alues of E for li5uids are very high as compared !ith those of gases (except at very highpressures). Therefore li5uids are usually termed as incompressible fluids though in fact nosubstance is theoretically incompressible !ith a value of E as .

For example the bul+ modulus of elasticity for !ater and air at atmospheric pressure areapproximately # x 1$8+2m #and 1$1 +2m #respectively. "t indicates that air is about #$$$$

times more compressible than !ater. 0ence !ater can be treated as incompressible.

For gases another characteristic parameter +no!n as compressibility K, is usually defined it isthe reciprocal of E

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(#.8)

D is often expressed in terms of specific volume .

For any gaseous substance a change in pressure is generally associated !ith a change in

volume and a change in temperature simultaneously. A functionl reltions'ip )etween t'epressure4 &olu!e nd te!perture t n% e=uili)riu! stte is ?nown s t'er!od%n!ic

e=ution of stte for t'e "s$

For an ideal gas the thermodynamic e5uation of state is given by

p ?RT (#.9)

!here T is the temperature in absolute thermodynamic or gas temperature scale (!hich are infact identical) and R is +no!n as the characteristic gas constant the value of !hich dependsupon a particular gas. 0o!ever this e5uation is also valid for the real gases !hich arethermodynamically far from their li5uid phase. For air the value of E is #9 32+g D.

D and ; generally depend on the nature of process

Surfce Tension of Li=uids

The phenomenon of surface tension arises due to the t!o +inds of intermolecular forces

(i) Co'esion /The force of attraction bet!een the molecules of a li5uid by virtue of !hich theyare bound to each other to remain as one assemblage of particles is +no!n as the force ofcohesion. This property enables the li5uid to resist tensile stress.

(ii) Ad'esion /The force of attraction bet!een unli+e molecules i.e. bet!een the molecules of

different li5uids or bet!een the molecules of a li5uid and those of a solid body !hen they are incontact !ith each other is +no!n as the force of adhesion. This force enables t!o differentli5uids to adhere to each other or a li5uid to adhere to a solid body or surface.

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Fi"ure ($- T'e inter!oleculr co'esi&e force field in )ul? of li=uid wit' free surfce

A and : experience e5ual force of cohesion in all directions C experiences a net force interior ofthe li5uid The net force is maximum for > since it is at surface

Wor+ is done on each molecule arriving at surface against the action of an in!ard force. Thus

mechanical !or+ is performed in creating a free surface or in increasing the area of the surface.Therefore a surface re5uires mechanical energy for its formation and the existence of a freesurface implies the presence of stored mechanical energy +no!n as free surface energy. Anysystem tries to attain the condition of stable e5uilibrium !ith its potential energy as minimum.Thus a 5uantity of li5uid !ill ad,ust its shape until its surface area and conse5uently its freesurface energy is a minimum.

The magnitude of surface tension is defined as the tensile force acting across imaginary shortand straight elemental line divided by the length of the line.

The dimensional formula is F/L or T#. "t is usually expressed in 2m in *" units.

*urface tension is a binary property of the li5uid and gas or t!o li5uids !hich are in contact !itheach other and defines the interface. "t decreases slightly !ith increasing temperature. Thesurface tension of !ater in contact !ith air at #$6C is about $.$9& 2m.

"t is due to surface tension that a curved li5uid interface in e5uilibrium results in a greaterpressure at the concave side of the surface than that at its convex side.

2pour pressure All li5uids have a tendency to evaporate !hen exposed to a gaseous atmosphere. The rate of

evaporation depends upon the molecular energy of the li5uid !hich in turn depends upon the typeof li5uid and its temperature. The vapour molecules exert a partial pressure in the space above

the li5uid +no!n as vapour pressure. "f the space above the li5uid is confined (Fig. #.4) and theli5uid is maintained at constant temperature after sufficient time the confined space above theli5uid !ill contain vapour molecules to the extent that some of them !ill be forced to enter theli5uid. ;ventually an e5uilibrium condition !ill evolve !hen the rate at !hich the number of vapourmolecules stri+ing bac+ the li5uid surface and condensing is ,ust e5ual to the rate at !hich theyleave from the surface. The space above the li5uid then becomes saturated !ith vapour. Thevapour pressure of a given li5uid is a function of temperature only and is e5ual to the saturationpressure for boiling corresponding to that temperature. 0ence the vapour pressure increases!ith the increase in temperature. Therefore the phenomenon of boiling of a li5uid is closelyrelated to the vapour pressure. "n fact !hen the vapour pressure of a li5uid becomes e5ual to thetotal pressure impressed on its surface the li5uid starts boiling. This concludes that boiling canbe achieved either by raising the temperature of the li5uid so that its vapour pressure iselevated to the ambient pressure or by lo!ering the pressure of the ambience (surrounding gas)

to the li5uid%s vapour pressure at the existing temperature.

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Fi"ure ($8 To nd fro !o&e!ent of li=uid !olecules fro! n interfce in confined spce s closed surroundin"

Recp

In t'is c'pter %ou '&e lernt t'e followin"

A fluid is a substance that deforms continuously !hen sub,ected to even an infinitesimalshear stress. *olids can resist tangential stress at static conditions undergoing a definitedeformation !hile a fluid can do it only at dynamic conditions undergoing a continuousdeformation as long as the shear stress is applied.

The concept of continuum assumes a continuous distribution of mass !ithin the matteror system !ith no empty space. "n the continuum approach properties of a system canbe expressed as continuous functions of space and time. A dimensionless parameter

+no!n as @nudsen nu!)er !here G is the mean free path and - is thecharacteristic length aptly describes the degree of departure from continuum. Theconcept of continuum usually holds good !hen Dn $.$1.

7iscosity is a property of a fluid by virtue of !hich it offers resistance to flo!. The shearstress at a point in a moving fluid is directly proportional to the rate of shear strain. For a

one dimensional flo! . The constant of proportionality H is +no!n as coefficientof viscosity or simply the viscosity. The relationship is +no!n as the e!ton%s la! ofviscosity and the fluids !hich obey this la! are +no!n as Newtonin fluids$

The relationship bet!een the shear stress and the rate of shear strain is +no!n as theconstitutive e5uation. The fluids !hose constitutive e5uations are not linear through

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origin (do not obey the e!ton%s la! of viscosity) are +no!n as non;Newtoninfluids. For a e!tonian fluid viscosity is a function of temperature only. With anincrease in temperature the viscosity of a li5uid decreases !hile that of a gasincreases. For none!tonian fluid the viscosity depends not only on temperature but

also on the deformation rate of the fluid. Dinematic viscosity v is defined as .

Compressibility of a substance is the measure of its change in volume or density underthe action of external forces. "t is usually characteri/ed by the )ul? !odulus ofelsticit%

A flo! is said to be incompressible !hen the change in its density due to the change inpressure brought about by the fluid motion is negligibly small. When the flo! velocity ise5ual to or less than $.&& times of the local acoustic speed the relative change indensity of the fluid due to flo! becomes e5ual to or less than 4 per cent respectively

and hence the flo! is considered to be incompressible

The force of attraction bet!een the molecules of a fluid is +no!n as cohesion !hile thatbet!een the molecules of a fluid and of a solid is +no!n as adhesion. The interplay ofthese t!o intermolecular forces explains the phenomena of surface tension and capillaryrise or depression. A free surface of the li5uid is al!ays under stretched conditionimplying the existence of a tensile force on the surface. The magnitude of this force perunit length of an imaginary line dra!n along the li5uid surface is +no!n as the surfacetension coefficient or simply the surfce tension.

One4 Two nd T'ree Di!ensionl Flows

Fluid flo! is t'ree;di!ensionlin nature.

This means that the flo! parameters li+e velocity pressure and so on vary in all the threecoordinate directions.

*ometimes simplification is made in the analysis of different fluid flo! problems by=

*electing the appropriate coordinate directions so that appreciable variation of the hydro dynamicparameters ta+e place in only t!o directions or even in only one.

One;di!ensionl flow

All the flo! parameters may be expressed as functions of time and one space coordinate only.

The single space coordinate is usually the distance measured along the centreline (notnecessarily straight) in !hich the fluid is flo!ing.

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E+!ple/the flo! in a pipe is considered onedimensional !hen variations of pressure andvelocity occur along the length of the pipe but any variation over the crosssection is assumednegligible.

"n reality flo! is never onedimensional because &iscosit%causes the velocity to decrease to/ero at the solid boundaries.

"f ho!ever thenon unifor!it% of t'e ctul flow is not too "ret valuable results may oftenbe obtained from a Ione di!ensionl nl%sisI.

The &er"e &luesof the flo! parameters at any given section (perpendicular to the flo!) areassumed to be applied to the entire flo! at that section.

Two;di!ensionl flow

All the flo! parameters are functions of time and t!o space coordinates (say x and y).

o variation in / direction.

The same streamline patterns are found in all planes perpendicular to / direction at any instant.

T'ree di!ensionl flow

The hydrodynamic parameters are functions of three space coordinates and time.

S%ste!

Definition

S%ste!/A 5uantity of matter in space !hich is analy/ed during a problem.

Surroundin"s/;verything external to the system.

S%ste! ,oundr%/A separation present bet!een system and surrounding.

Classification of the system boundary=

Eeal solid boundary

"maginary boundaryThe system boundary may be further classified as=

Fixed boundary or Control ass *ystem

oving boundary or Control 7olume *ystem The choice of boundary depends on the problem being analy/ed.

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Fi" $# S%ste! nd Surroundin"s

Clssifiction of S%ste!s

T%pes of S%ste!

Control Mss S%ste! 6Closed S%ste!7

1. "ts a system of fi+ed !ss!ith fi+ed identit%.#. This type of system is usually referred to as Bclosed s%ste!B.

&. There is no mass transfer across the system boundary.

'. ;nergy transfer may ta+e place into or out of the system.

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Clic? to pl% t'e De!onstrtion

Fi" $( A Control Mss S%ste! or Closed S%ste!

Control 2olu!e S%ste! 6Open S%ste!7

1. "ts a system of fi+ed &olu!e.#. This type of system is usually referred to as Bopen s%ste!or a Bcontrol &olu!eB

&. ass transfer can ta+e place across a control volume.

'. ;nergy transfer may also occur into or out of the system.

4. A control volume can be seen as a fixed region across !hich mass and energy transfers arestudied.

8. Control *urface "ts the boundary of a control volume across !hich the transfer of both mass andenergy ta+es place.

9. The mass of a control volume (open system) may or may not be Jxed.

. When the net influx of mass across the control surface e5uals /ero then the mass of the systemis fixed and viceversa.

K. The identity of mass in a control volume al!ays changes unli+e the case for a control masssystem (closed system).

1$. ost of the engineering devices in general represent an open system or control volume.

E+!ple/;

0eat exchanger Fluid enters and leaves the system continuously !ith the transfer of heat across

the system boundary. Lump A continuous flo! of fluid ta+es place through the system !ith a transfer of mechanical

energy from the surroundings to the system.

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Clic? to pl% t'e De!onstrtion

Fi" $- A Control 2olu!e S%ste! or Open S%ste!

Isolted S%ste!

1. "ts a system of fi+ed !ss!ith same identit% nd fi+ed ener"%.#. o interaction of mass or energy ta+es place bet!een the system and the surroundings.

&. "n more informal !ords an isolated system is li+e a closed shop amidst a busy mar+et.

Fi" $0 An Isolted S%ste!

Conser&tion of Mss ; T'e Continuit% E=ution

Lw of conser&tion of !ss

The la! states that mass can neither be created nor be destroyed. Conservation of mass is inherent to acontrol mass system (closed system).

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The mathematical expression for the above la! is stated as=

m/t = 0 !here m? mass of the system

For a control volume (Fig.K.4) the principle of conservation of mass is stated as

Eate at !hich mass enters ? Eate at !hich mass leaves the region M Eate of accumulation of mass in theregion

emonstration

Fi" $8 A Control 2olu!e in Flow Field

Continuit% E=ution ; Differentil For!

Deri&tion

1. The point at !hich the continuity e5uation has to be derived is enclosed by an elementary control

volume.

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#. The influx efflux and the rate of accumulation of mass is calculated across each surface !ithin

the control volume.

Fi" $9 A Control 2olu!e Approprite to Rectn"ulr Crtesin Coordinte S%ste!

Consider a rectangular parallelopiped in the above figure as the control volume in a rectangular cartesian

frame of coordinate axes.

et efflux of mass along x axis must be the excess outflo! over inflo! across faces normal to x

axis.

-et the fluid enter across one of such faces A:C> !ith a velocity u and a density O.The velocity

and density !ith !hich the fluid !ill leave the face ;FP0 !ill be and

respectively (neglecting the higher order terms in x).

Therefore the rate of mass entering the control volume through face A:C> ?u dy dz.

The rate of mass leaving the control volume through face ;FP0!ill be

(neglecting the higher order terms in dx)

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*imilarly influx and efflux ta+e place in all y and / directions also.

Eate of accumulation for a point in a flo! field

sing Eate of influx ? Eate of Accumulation M Eate of ;fflux

Transferring everything to right side

(K.#)

This is the E=ution of Continuit%for a compressible fluid in a rectangular cartesian coordinatesystem.

Conser&tion of Mo!entu!/ Mo!entu! T'eore!

"n e!tonian mechanics the conservation of momentum is defined by e!tonQs second la! of motion.

Newtons Second Lw of Motion

The rate of change of momentum of a body is proportional to the impressed action and ta+es

place in the direction of the impressed action.

"f a force acts on the body linear momentum is implied.

"f a tor5ue (moment) acts on the bodyangular momentum is implied.

Re%nolds Trnsport T'eore!

A study of fluid flo! by the ;ulerian approach re5uires a mathematical modeling for a control volumeeither in differential or in integral form. Therefore the physical statements of the principle of conservationof mass momentum and energy !ith reference to a control volume become necessary.

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This is done by invo+ing a theorem +no!n as the Eeynolds transport theorem !hich relates the controlvolume concept !ith that of a control mass system in terms of a general property of the system.

Stte!ent of Re%nolds Trnsport T'eore!

The theorem states that Ithe time rate of increase of property !ithin a control mass system is e5ual to

the time rate of increase of property !ithin the control volume plus the net rate of efflux of the property across the control surfaceR.

E=ution of Re%nolds Trnsport T'eore!

After deriving Eeynolds Transport Theorem according to the above statement !e get

(1$.K)

"n this e5uation

flo! property !hich is transported

S intensive value of the flo! property

Anl%sis Of Finite Control 2olu!es ; t'e ppliction of !o!entu! t'eore!

We%ll see the application of momentum theorem in some practical cases of inertial and noninertial controlvolumes.

Inertil Control 2olu!esApplications of momentum theorem for an inertial control volume are described !ith reference to three

distinct types of practical problems namely

Forces acting due to internal No!s through expanding or reducing pipe bends.

Forces on stationary and moving vanes due to impingement of fluid ,ets.

3et propulsion of ship and aircraft moving !ith uniform velocity.

Non;inertil Control 2olu!eA good example of noninertial control volume is a roc+et engine !hich !or+s on the principle of ,etpropulsion.

We shalll discuss each example seperately in the follo!ing slides.

D%n!ic Forces on .lne Surfces due to t'e I!pin"e!ent of Li=uid ets

Force on sttionr% surfce Consider a stationary Nat plate and a li5uid ,et of cross sectional areaRaR stri+ing !ith a velocity 7 at an angle to the plate as sho!n in Fig. 11.&a.

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Fi" ##$- I!pin"e!ent of li=uid ets on Sttionr% Flt .lte

To calculate the force re5uired to +eep the plate stationary a control volume A:C>;FA (Fig. 11.&a) ischosen so that the control surface >; coincides !ith the surface of the plate. The control volume issho!n separately as a free body in Fig. 11.&b. -et the volume flo! rate of the incoming ,et be and bedivided into 1and #gliding along the surface (Fig. 11.&a) !ith the same velocity 7 since the pressurethroughout is same as the atmospheric pressure the plate is considered to be frictionless and theinNuence of a gravity is neglected (i.e. the elevation bet!een sections C> and ;F is negligible).

Coordinate axes are chosen as !sand !nalong and perpendicular to the plate respectively. eglectingthe viscous forces. (the force along the plate to be /ero)the momentum conservation of the controlvolume A:C>;FA in terms of sand ncomponents can be !ritten from ;5.(1$.1d) as

(11.4a)

and

(11.4b)

!here Fsand Fnare the forces acting on the control volume along !sand !nrespectively

From continuity

? 1M # (11.8)

With the help of ;5s (11.4a) and (11.8) !e can !rite

(11.9a)

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(11.9b)

The net force acting on the control volume due to the change in momentum of the ,et by the plate isFnalong the direction I"nR and is given by the ;5. (11.9b) as

(11.9c)

0ence according to e!tonQs third la! the force acting on the plate is

(11.)

"f the crosssectional area of the ,et is RaR then the volume flo! rate stri+ing the plate can be !ritten as ? a7. ;5uation (11.) then becomes

(11.K)

Conser&tion of Ener"%

The principle of conservation of energy for a control mass system is described by the first la!of thermodynamics

0eat added to a control mass system the !or+ done W by the control mass system ?change in its internal energy ;

The internal energy depends only upon the initial and final states of the system. "t can be!ritten in the form of the e5uation as

(1&.1a)

;5uation (1&.1a) can be expressed on the time rate basis as

(1&.1b)

Where U and UW are the amount of heat added and !or+ done respectively during a timeinterval of Ut. To develop the analytical statement for the conservation of energy of a control

volume the ;5. (1$.1$) is used !ith ? ; (the internal energy) and S ? e (the internal energyper unit mass) along !ith the ;5. (1&.1b). This gives

(1&.#)

The ;5. (1&.#) is +no!n as the "enerl ener"% e=utionfor a control volume.

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The first term on the right hand side of the e5uation is the time rate of increase in the internalenergy !ithin a control volume and the second term is the net rate of energy efflux from thecontrol volume.

>ifferent forms of energy associated !ith moving fluid elements comprising a control volume

are

#$ .otentil ener"%The concept of potential energy in a fluid is essentially the same as that of a solid mass. Thepotential energy of a fluid element arises from its existence in a conservative body force field.This field may be a magnetic electrical etc. "n the absence of any of such external force fieldthe earthQs gravitational effect is the only cause of potential energy. "f a fluid mass m is storedin a reservoir and its C.P. is at a vertical distance / from an arbitrary hori/ontal datum planethen the potential energy is mg/ and the potential energy per unit mass is g/. The arbitrarydatum does not play a vital role since the difference in potential energy instead of its absolutevalue is encountered in different practical purposes.

($ @inetic Ener"%

"f a 5uantity of a fluid of mass m flo!s !ith a velocity 7 being the same throughout its massthen the total +inetic energy is m7#2# and the +inetic energy per unit mass is 7#2#. For a streamof real fluid the velocities at different points !ill not be the same. "f 7 is the local component ofvelocity along the direction of flo! for a fluid flo!ing through an open channel or closed conduitof crosssectional area A the total +inetic energy at any section is evaluated by summing upthe +inetic energy No!ing through differential areas as

The average velocity at a crosssection in a flo!ing stream is defined on the basis of volumetric

flo! rate as

The +inetic energy per unit mass of the fluid is usually expressed as !here is+no!n as the ?inetic ener"%correction fctor.Therefore !e can !rite

0ence

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(1&.&a)

For an incompressible flo!

(1&.&b)

-$ Inter!oleculr Ener"%The intermolecular energy of a substance comprises the potential energy and +inetic energy ofthe molecules. The potential energy arises from intermolecular forces. For an ideal gas thepotential energy is /ero and the intermolecular energy is therefore due to only the +ineticenergy of molecules. The +inetic energy of the molecules of a substance depends on itstemperature.

0$ Flow 1or?

Flo! !or+ is the !or+ done by a fluid to move against pressure.For a flo!ing stream a layer of fluid at any crosssection has to push the ad,acent neighboringlayer at its do!nstream in the direction of flo! to ma+e its !ay through and thus does !or+ onit. The amount of !or+ done can be calculated by considering a small amount of fluid mass

A11dx to cross the surface A: from left to right (Fig. 1&.1). The !or+ done by this mass offluid then becomes e5ual to p1A1dx and thus the flo! !or+ per unit mass can be expressed as

(!here p1is the pressure at section A: (Fig 1&.1)

Fi" #-$# 1or? done )% fluid to flow "inst pressure

Therefore the flo! !or+ done per unit mass by a fluid element entering the control volumeA:C>A (Fig. 1&.1) is p121*imilarly the flo! !or+ done per unit mass by a fluid elementleaving the control volume across the surface C> is p#21

"mportant "n introducing an amount of fluid inside the control volume the !or+ done against

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the frictional force at the !all can be sho!n to be small as compared to the !or+ done againstthe pressure force and hence it is not included in the flo! !or+.

Although Qflo! !or+Q is not an intrinsic form of energy it is sometimes referred to as QpressureenergyQ from a vie! point that by virtue of this energy a mass of fluid having a pressure p at

any location is capable of doing !or+ on its neighboring fluid mass to push its !ay through.

Sted% Flow Ener"% E=ution

The energy e5uation for a control volume is given by ;5. (1&.#). At steady state the Jrst term on the righthand side of the e5uation becomes /ero and it becomes

(1&.')

"n consideration of all the energy components including the flo! !or+ (or pressure energy) associated!ith a moving fluid element one can substitute QeQ in ;5. (1&.') as

and finally !e get

(1&.4)

The ;5. (1&.4) is +no!n as sted% flow ener"% e=ution

A Control 2olu!e Approc' for t'e Deri&tion of Eulers E=ution

;ulerQs e5uations of motion can also be derived by the use of the momentum theorem for a controlvolume.

Deri&tion"n a fixed x y / axes (the rectangular cartesian coordinate system) the parallelopiped !hich !as ta+enearlier as a control mass system is no! considered as a control volume (Fig. 1#.').

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Fi" #($0 A Control 2olu!e used for t'e deri&tion of Eulers E=ution

We can define the velocity vector and the body force per unit volume as

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The rate of x momentum influx to the control volume through the face A:C> is e5ual to Ou #dy d/. Therate of x momentum efflux from the control volume through the face ;FP0

e5uals

Therefore the rate of net efflux of x momentum from the control volume due to the faces perpendicular to

the x direction (faces A:C> and ;FP0) ? !here the elemental volume ? dx dy d/.

*imilarly

The rate of net efflux of x momentum due to the faces perpendicular to the y direction (face :CPF and

A>0;) ?

The rate of net efflux of x momentum due to the faces perpendicular to the / direction (faces >CP0 and

A:F;) ?

0ence the net rate of x momentum efflux from the control volume becomes

The time rate of increase in x momentum in the control volume can be !ritten as

(*ince by the definition of control volume is invariant !ith time)Applying the principle of momentum conservation to a control volume (;5. '.#b) !e get

(1#.11a)

The e5uations in other directions y and / can be obtained in a similar !ay by considering the ymomentum and / momentum Nuxes through the control volume as

(1#.11b)

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(1#.11c)

The typical form of ;ulerQs e5uations given by ;5s (1#.11a) (1#.11b) and (1#.11c) are +no!n as theconservative forms.

,ernoullis E=ution

Ener"% E=ution of n idel Flow lon" Stre!line

;ulerQs e5uation (the e5uation of motion of an inviscid fluid) along a stream line for a steady flo! !ithgravity as the only body force can be !ritten as

(1&.8)

Application of a force through a distance ds along the streamline !ould physically imply !or+ interaction.Therefore an e5uation for conservation of energy along a streamline can be obtained by integrating the;5. (1&.8) !ith respect to ds as

(1&.9)

Where C is a constant along a streamline. "n case of an incompressible flo! ;5. (1&.9) can be !ritten as

(1&.)

The ;5s (1&.9) and (1&.) are based on the assumption that no !or+ or heat interaction bet!een a fluidelement and the surrounding ta+es place. The first term of the ;5. (1&.) represents the flo! !or+ per unitmass the second term represents the +inetic energy per unit mass and the third term represents thepotential energy per unit mass. Therefore the sum of three terms in the left hand side of ;5. (1&.) can beconsidered as the total mechanical energy per unit mass !hich remains constant along a streamline for asteady inviscid and incompressible flo! of fluid. 0ence the ;5. (1&.) is also +no!n as Mec'niclener"% e=ution.

This e5uation !as developed first by >aniel :ernoulli in 19& and is therefore referred to as :ernoulliQse5uation. ;ach term in the ;5. (1&.) has the dimension of energy per unit mass. The e5uation can also

be expressed in terms of energy per unit !eight as

(1&.K)

"n a fluid flo! the energy per unit !eight is termed as head. Accordingly e5uation 1&.K can be interpretedas

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Lressure head M 7elocity head M Lotential head ?Total head (total energy per unit !eight).

,ernoullis E=ution wit' Hed Loss

The derivation of mechanical energy e5uation for a real fluid depends much on the information about thefrictional !or+ done by a moving fluid element and is excluded from the scope of the boo+. 0o!ever in

many practical situations problems related to real fluids can be analysed !ith the help of a modified formof :ernoulliQs e5uation as

(1&.1$)

!here hfrepresents the frictional !or+ done (the !or+ done against the fluid friction) per unit !eight of afluid element !hile moving from a station 1 to # along a streamline in the direction of flo!. The term h fisusually referred to as head loss bet!een 1 and # since it amounts to the loss in total mechanical energyper unit !eight bet!een points 1 and # on a streamline due to the effect of fluid friction or viscosity."tphysically signifies that the difference in the total mechanical energy bet!een stations 1 and # isdissipated into intermolecular or thermal energy and is expressed as loss of head h fin ;5. (1&.1$). The

term head loss is conventionally symboli/ed as h-instead of hfin dealing !ith practical problems. For aninviscid flo! h-? $ and the total mechanical energy is constant along a streamline.

UNIT II

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FLO1 THROU3H CIRCULAR CONDUITS

Lift nd Dr" for Flow .st C%linder wit'out Circultion

.ressure in t'e C%linder Surfce

Lressure becomes uniform at large distances from the cylinder ( !here the influence of doublet is small).

-et us imagine the pressure p$is +no!n as !ell as uniform velocity $.We can apply :ernoulli%s e5uation bet!een infinity and the points on the boundary of the cylinder.

eglecting the variation of potential energy bet!een the aforesaid point at infinity and any point on thesurface of the cylinder !e can !rite

(##.4)

!here the subscript b represents the surface on the cylinder.

*ince fluid cannot penetrate the solid boundary the velocity U)s'ould )e onl% in t'e trns&ersedirection or in other !ords only vVcomponent of velocity is present on the streamline ? $ .

Thus at

(##.8)

From e5s (##.4) and (##.8) !e obtain

(##.9)

Lift nd Dr"

Lift /force ctin" on t'e c%linder 6per unit len"t'7 in t'e direction nor!l to unifor! flow$

Dr"/ force ctin" on t'e c%linder 6per unit len"t'7 in t'e direction prllel to unifor! flow$

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Fi" (($0 Clcultion of Dr" in C%linder

The drag is calculated by integrating the force components arising out of pressure in the x direction onthe boundary. Eeferring to Fig.##.' the drag force can be !ritten as

infinitesimal length on the circumference

*ince

or,

(##.)

*imilarly the lift force may be calculated as

(##.K)

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The ;5s (##.) and (##.K) produce DG nd LG fter the integration is carried out.

0o!ever in relit%4 t'e c%linder will lw%s e+perience so!e dr" force$ T'is contrdiction)etween t'e in&iscid flow result nd t'e e+peri!ent is usull% ?nown s D Al!)ert prdo+$

:ernoulli%s e5uation can be used to calculate the pressure distribution on the cylinder surface

The pressure coefficient cpis therefore

(##.1$)

Lift nd Dr" for Flow A)out Rottin" C%linder

The pressure at large distances from the cylinder is uniform and given by p$.

>eploying :ernoulli%s e5uation bet!een the points at infinity and on the boundary of the cylinder

(#&.K)

0ence

(#&.1$)

From ;5s (#&.K) and (#&.1$) !e can !rite

(#&.11)

The lift may calculated as

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or,

(#&.1#)

The drag force !hich includes the multiplication by cosV (and integration over #X) is /ero.

Thus the inviscid flo! also demonstrates lift.

lift becomes a simple formula involving only the density of the medium free stream velocity and

circulation.

in t!o dimensional incompressible steady flo! about a boundary of any shape the lift is al!ays a

product of these three 5uantities. @utt; ou?ows?i t'eore!

Aerofoil T'eor%

Aerofoils are streamline shaped !ings !hich are used in airplanes and turbo machinery. These shapesare such that the drag force is a very small fraction of the lift. The follo!ing nomenclatures are used fordefining an aerofoil

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Fi" (-$0 Aerofoil Section

The c'ord(C) is the distance bet!een the leading edge and trailing edge.

The length of an aerofoil normal to the crosssection (i.e. normal to the plane of a paper) is

called the spnof a aerofoil.

The c!)er linerepresents the mean profile of the aerofoil. *ome important geometricalparameters for an aerofoil are the ratio of maximum thic+ness to chord (t2C) and the ratio ofmaximum camber to chord (h2C). When these ratios are small an aerofoil can be considered tobe thin. For the analysis of flo! a thin aerofoil is represented by its camber.

The theory of thic+ cambered aerofoils uses a complexvariable mapping !hich transforms the inviscidflo! across a rotating cylinder into the flo! about an aerofoil shape !ith circulation.

Flow Around T'in Aerofoil

Thin aerofoil theory is based upon the superposition of uniform flo! at infinity and a

continuous distribution of cloc+!ise free vortex on the camber line having circulationdensity per unit length .

The circulation density should be such that the resultant flo! is tangent to the

camber line at every point.

*ince the slope of the camber line is assumed to be small . The total

circulation around the profile is given by

(#&.1&)

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Fi" (-$8 Flow Around T'in Aerofoil

A vortical motion of strengt at != "evelo#s a velocity at te point p$icmay be e!#resse" as

The total induced velocity in the up!ard direction at point pdue to the entire vortex distributionalong the camber line is

(#&.1')

For a small camber (having small ) this expression is identically valid for the induced velocity

at point pdue to the vortex sheet of variable strength on the camber line. The resultant

velocity due to and v(x) must be tangential to the camber line so that the slope of acamber line may be expressed as

(#&.14)

From ;5s (#&.1') and (#&.14) !e can !rite

Consider an element ds on the camber line. Consider a small rectangle (dra!n !ith dotted line)around ds. The upper and lo!er sides of the rectangle are very close to each other and theseare parallel to the camber line. The other t!o sides are normal to the camber line. Thecirculation along the rectangle is measured in cloc+!ise direction as

Ynormal component of velocity at the camber line should be/eroZ

or

"f the mean velocity in the tangential direction at the camber line is given byit can be re!ritten as

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and

if v is very small becomes e5ual to . The difference in velocity across the

camber line brought about by the vortex sheet of variable strength causes pressuredifference and generates lift force.

3enertion of 2ortices Around 1in"

T'e lift round n erofoil is "enerted followin" @utt;ou?ows?i t'eore! . -ift is a

product of O and the circulation .

When the motion of a !ing starts from rest vortices are formed at the trailing edge.

At the start there is a velocity discontinuity at the trailing edge. This is eventual because near the

trailing edge the velocity at the bottom surface is higher than that at the top surface. Thisdiscrepancy in velocity culminates in the formation of vortices at the trailing edge.

Figure #&.8(a) depicts the formation of starting vortex by impulsively moving aerofoil. 0o!ever

the starting vortices induce a counter circulation as sho!n in Figure #&.8(b). The circulationaround a path (A:C>) enclosing the !ing and ,ust shed (starting) vortex must be /ero. 0ere !erefer to @el&ins t'eore!once again.

Fi" (-$9 2ortices 3enerted w'en n Aerofoil ust ,e"ins to Mo&e

"nitially the flo! starts !ith the /ero circulation around the closed path. Thereafter due to the

change in angle of attac+ or flo! velocity if a fresh starting vortex is shed the circulation aroundthe !ing !ill ad,ust itself so that a net /ero vorticity is set around the closed path.

Eeal !ings have finite span or finite aspect ratio (AE) G defined as

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(#&.18)

!here b is the span length Asis the plan form area as seen from the top..

For a !ing of finite span the end conditions affect both the lift and the drag. "n the leading edgeregion pressure at the bottom surface of a !ing is higher than that at the top surface. Thelongitudinal vortices are generated at the edges of finite !ing o!ing to pressure differencesbet!een the bottom surface directly facing the f lo! and the top surface.

Fi" (-$< 2ortices Around Finite 1in"

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Fi" (-$> 3enertion of Lon"itudinl 2ortices

H"en .oiseuille Flow

Consider full% de&eloped laminar flo! through a straight tube of circular crosssection as in Fig.

#8.&. Eotational symmetry is considered to ma+e the flo! t!odimensional axisymmetric.

-et us ta+e /axis as the axis of the tube along !hich all the fluid particles travel i.e.

Fig 26.3 - Hagen-Poiseuille flow through a pipe

o! from continuity e5uation !e obtain

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% For rotational symmetry &

!hich means

"nvo+ing in the

avier*to+es e5uations !e obtain

(in the /direction)

(#8.&)

For sted% flow the governing e5uation becomes

(#8.')

The boundary conditionsare 6i7At r ? $ is finite and 6ii7r ? E yields

;5uation (#8.') can be !ritten as

or

or

or

or

At r ?$ is finite !hich means A should be e5ual to /ero and at r ? E ? $ yields

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(#8.4)

T'is s'ows t't t'e +il &elocit% profile in full% de&eloped l!inr pipe flow is '&in"

pr)olic &rition lon" r$

At r ? $ as such

(#8.8a)

The average velocity in the channel

or

(#8.8b)

or(#8.8c)

o! the discharge () through a pipe is given by

(#8.9)

or YFrom ;5. #8.8bZ

or

(#8.)

.rllel Flow in Stri"'t C'nnel

Consider sted% flow )etween two infinitel% )rod prllel pltes as sho!n in Fig. #4.#.

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Flo! is independent of any variation in / direction hence / dependence is gotten rid of and ;5. (#4.11)becomes

FI3 (8$( .rllel flow in stri"'t c'nnel

(#4.1#)

The boundary conditions are at y ? b u ? $[ and y ? b u ?


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At y ? $ [ this yields

(#4.1'a)


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With the help of ;5s (#8.Kb) and (#8.Kc) it can be !ritten

(#8.1'b)

The s+in friction coefficient C fis called as Fanning%s friction factor . From comparison of ;5s (#8.1'a) and(#8.1'b) it appears

For fully developed turbulent flo! the analysis is much more complicated and !e generally

depend on experimental results. Friction factor for a !ide range of Eeynolds number (1$ 'to 1$)can be obtained from a loo+up chart . Friction factor for high Eeynolds number flo!s is also afunction of tube surface condition. 0o!ever in circular tube flo! is laminar for Ee \ #&$$ andturbulent regime starts !ith Ee ] '$$$.

The surface condition of the tube is another responsible parameter in determination of friction

factor.

Friction factor in the turbulent regime is determined for different degree of surface

roughness of the pipe !here is the dimensional roughness and $his usually thehydraulic diameter of the pipe .

Friction factors for different Eeynolds number and surfaceroughness have been determined

experimentally by various investigators and the comprehensive results are expressed through agraphical presentation !hich is +no!n as Mood% C'rt after L$F$ Mood%!ho compiled it.

The hydraulic diameter !hich is used as the characteristic length in determination of friction

factor instead of ordinary geometrical diameter is defined as

(#8.14)

!here A!is the flo! area and L!is the !etted perimeter .

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@inetic ener"% correction fctor , The +inetic energy associated !ith the fluid flo!ing !ith

its profile through elemental area and the total +inetic energy passing

through per unit time .

This can be related to the +inetic energy due to average velocity( ) through a correction

factor as

or

0ere for 0agenLoiseuille flo!

Flow )etween Two Concentric Rottin" C%linders

Another example !hich leads to an exact solution of avier*to+es e5uation is the flo!

bet!een two concentric rottin" c%linders$

Consider flo! in the annulus of t!o cylinders (Fig. #8.4) !here r1and r#are the radii of inner andouter cylinders respectively and the cylinders move !ith different rotationalspeeds #and (respectively

(

FIG 26. - Flow between two concentric rotating cylinders

From the physics of the problem !e +no! .

From the continuity ;5. and these t!o conditions !e obtain

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!hich means is not a function of . Assume % dimension to be large enough so that end effects can

be neglected and (any property) ? $.

This implies . With these simplifications and assuming that I s%!!etr%I holds

good avier*to+es e5uation reduces to

(#8.19)

and

(#8.1)

;5uation (#8.19) signifies that the centrifugal force is supplied by the radial pressure exerted bythe !all of the enclosure on the fluid. "n other !ords it describes the rdil pressuredistri)ution$From ;5. (#8.1) !e get

or(#8.1K)

For the a/imuthal component of velocity the boundary conditions are= at

at .

Application of these boundary conditions on ;5. (#8.1K) !ill produce

and

Finally the velocity distribution is given by

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(#8.#$)

1ll S'er Stress

With the profile +no!n !all shear can be evaluated as

o!

or

or

from Table #.1

61ll S'er Stress7

(#K.1a)

and the local s+in friction coefficient is

*ubstituting from (#K.1a) !e get

6S?in Friction Coefficient7

(#K.1b)

"n 1K41 -iepmann and >ha!an measured the shearing stress on a flat plate directly.

Their results sho!ed a stri+ing confirmation of ;5. (#K.1).

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Total frictional force per unit !idth for the plate of length - is

or

or

(#K.#)

and the average s+in friction coefficient is

(#K.&)

!here .

For a flat plate of length - in the stream!ise direction and !idth ! perpendicular to the flo! the>rag > !ould be

(#K.')

,oundr% L%er T'ic?ness

*ince it is customary to select the boundary layer thic+ness as

that point !here pproc'es $$

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From Table #.1 rec'es $ t G 8$ and !e can !rite

(#K.4)

0o!ever the aforesaid definition of boundary layer thic+ness is some!hat arbitrary a physically

more meaningful measure of boundary layer estimation is expressedthrough displce!ent thic+ness .

Fi"$ ($# 6Displce!ent t'ic?ness7 6)7 Mo!entu! t'ic?ness

Displce!ent t'ic?ness / "t is defined as the distance by !hich the external potential flo!is displaced out!ards due to the decrease in velocity in the boundary layer.

Therefore

(#K.8)

*ubstituting the values of and from ;5s (#.#1a) and (#.1K) into ;5.(#K.8) !e obtain

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This dictates the inviscid pressure distribution on the cylinder !hich is sho!n by a firm line in Fig.#K.#.

0ere

/ pressure in t'e free stre!

/ &elocit% in t'e free stre! nd

"is t'e locl pressure on t'e c%linder$

Fi"$ ($( Flow seprtion nd for!tion of w?e )e'ind circulr c%linder

Consider the forces in the flo! field.

"n the in&iscid re"ion4

1. Until the pressure force and the force due to stream!ise accelerationi.e. inertia forces are acting in the same direction (pressure"rdientbeing ne"ti&eJf&our)le)

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#. ,e%ond the pressure "rdient is positi&e or d&erse. >ue to theadverse pressure gradient the pressure force and the force due to acceleration!ill be opposing each other in the in viscid /one of this part.

*o long as no viscous effect is considered the situation does not cause any sensation."n the&iscid re"ion(near the solid boundary)

1. Up to 4 t'e viscous force opposes the combined pressure force and theforce due to acceleration. Fluid particles overcome this viscous resistance due tocontinuous con&ersion of pressure force into ?inetic ener"%$

#. :eyond !ithin the viscous /one the flo! structure becomes different."t is seen that the force due to acceleration is opposed by both the viscous forceand pressure force.

>epending upon the magnitude of adverse pressure gradient so!ew'ere round 4 t'e

fluid prticles4 in t'eboundr% l%er re seprted fro! t'e wll and driven in the upstream

direction. 0o!ever the far field external stream pushes bac+ these separated layers together!ith it and develops )rod pulstin" w?e )e'ind t'e c%linder.

T'e !t'e!ticl e+plntion of flow;seprtion / The point of separation may be defined as

the limit bet!een for!ard and reverse flo! in the layer very close to the !all i.e. at the point ofseparation

(#K.18)

This means that the shear stress at the !all . :ut at this point the adverse pressurecontinues to exist and at the do!nstream of this point the flo! acts in a reverse direction resultingin a bac+ flo!.

We can also explain flo! separation using the argument about the second derivative of

velocity &at the !all. From the dimensional form of the momentum at the !all !here & ' v ' !!e can !rite

(#K.19)

Consider the situation due to a f&our)le pressure "rdient !here !e have

1. . (From ;5. (#K.19))

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#. As we proceed towrds t'e free stre!4 t'e &elocit% upproc'es

s%!ptoticll%4so decreases at a continuously lesser rate in ydirection.

&. This means that remains less than /ero near the edge of the boundary layer.

'. The curvature of a velocity profile is al!ays negative as sho!n in (Fig. #K.&a)

Consider the case of d&erse pressure "rdient

1. At the boundary the curvature of the profile must be positive (since ).

#. ear the interface of boundary layer and free stream the previous argument

regarding and still holds good and the curvature is negative.

&. Thus !e observe that for an adverse pressure gradient there must exist a point for

!hich . This point is +no!n aspoint o( in()ectionof the velocity profile inthe boundary layer as sho!n in Fig. #K.&b

'. 0o!ever point of separation means at the !all.

4. at the !all since separation can only occur due to adverse pressure

gradient. :ut !e have already seen that at the edge of the boundary layer. "t is therefore clear that if t'ere is point of seprtion4 t'ere !ust e+ist point ofinflection in t'e &elocit% profile$

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Fi"$ ($- 2elocit% distri)ution wit'in )oundr% l%er

(a) Favourable pressure gradient

(b) adverse pressure gradient

1. -et us reconsider the flo! past a circular cylinder and continue ourdiscussion on t'e w?e)e'ind c%linder$The pressure distribution !hich !as sho!n by the firm line in Fig. #1.4 is

obtained from the potential flo! theory. 0o!ever. some!here near 6in e+peri!ents it

's )een o)ser&ed to )e t 7 $ t'e )oundr% l%er detc'es itself fro! t'e wll$#. ean!hile pressure in t'e w?e re!ins close to seprtion;point;pressuresince the

eddies (formed as a conse5uence of the retarded layers being carried together !ith the upperlayer through the action of shear) cannot convert rotational +inetic energy into pressure head. Theactual pressure distribution is sho!n by the dotted line in Fig. #K.&.

&. *ince the w?e *one pressure is less t'n t't of t'e forwrd st"ntion point (pressure atpoint A in Fig. #K.&) the cylinder experiences a drag force !hich is basically attributed to thepressure difference.

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T'e dr" force4 )rou"'t )out )% t'e pressure difference is ?nown s for# dragw'erest'e s'er stress t t'e wll "i&es rise to s$in friction drag$Penerally these t!o drag forcestogether are responsible for resultant drag on a body

Concept of Friction Fctor in pipe flow/

The friction factor in the case of a pipe flo! !as already mentioned in lecture #8.

We !ill elaborate further on friction factor or friction coefficient in this section.

*+in friction coefficient for a fully developed flo! through a closed duct is defined as

(&4.1)

!here * is the average velocity of flo! given by + andA are the volume flo! rate through theduct and the crosssectional area of the duct respectively.

From a force balance of a typical fluid element (Fig. &4.1) in course of its flo! through a duct of constantcrosssectional area !e can !rite

(&4.#)

FI3 -8$# Force ,lnce of fluid ele!ent in t'e course of flow t'rou"' duct

!here is the shear stress at the !all and is the pie/ometric pressure drop over a length

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of L . A and S are respectively the crosssectional area and !etted perimeter of the duct.*ubstituting the expression (&4.#) in ;5. (&4.1) !e have

(&4.&)

!here and is +no!n as the '%drulic di!eter .

"n case of a circular pipe $h?$ the diameter of the pipe. The coefficient #f defined by ;5s (&4.1) or (&4.&)is +no!n as Fnnin"s friction fctor .

To do a!ay !ith the factor 12' in the ;5. (&4.&) >arcy defined a friction factor f 6Drc%s friction

fctor7 as

(&4.')

Comparison of ;5s (&4.&) and (&4.') gives . ;5uation (&4.') can be !ritten for a pipe

flo! as

(&4.4)

;5uation (&4.4) is !ritten in a different fashion for its use in the solution of pipe flo! problems in

practice as

(&4.8a)

or in terms of head loss (energy loss per unit !eight)

(&4.8b)

!here hf represents the loss of head due to friction over the length - of the pipe.

;5uation (&4.8b) is fre5uently used in practice to determine hf

"n order to evaluate hf !e re5uire to +no! the value of f. The value of f can be determined from

oody%s Chart.

2rition of Friction Fctor

"n case of a laminar fully developed flo! through pipes the friction factor ( is found from theexact solution of the avier*to+es e5uation as discussed in lecture #8. "t is given by

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(&4.9)

"n the case of a turbulent flo! friction factor depends on both the Eeynolds number and the

roughness of pipe surface.

*ir Thomas ;. *tanton (1841K&1) first started conducting experiments on a number of pipes of

various diameters and materials and !ith various fluids.After!ards a Perman engineeri+uradse carried out experiments on flo!s through pipes in a very !ide range of Eeynoldsnumber.

A comprehensive documentation of the experimental and theoretical investigations on the la!s of

friction in pipe flo!s has been presented in the form of a diagram as sho!n in Fig. &4.# by L$F$

Mood%to sho! the variation of friction factor ( !ith the pertinent governing parameters namely

the Eeynolds number of flo! and the relative roughness of the pipe. This diagram is +no!n

as Mood%s C'rt !hich is employed till today as the best means for predicting the values of ( .

Fi"$ -8$( Friction Fctors for pipes 6dpted fro! Trns$ ASME4 9949


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The region !here (vs Eecurves (Fig. &4.#) become hori/ontal sho!ing that f is independentof Re is +no!n as the rou"' *one and the intermediate region bet!een the smooth and rough/one is +no!n as the trnsition *one.

The position and extent of all these /ones depend on the relative roughness of the pipe. "n the

smooth /one of flo! the laminar sublayer becomes thic+ and hence it covers appreciably the

irregular surface protrusions. Therefore all the curves for smooth flo! coincide.

With increasing Eeynolds number the thic+ness of sublayer decreases and hence the surface

bumps protrude through it. The higher is the roughness of the pipe the lo!er is the value of Reat

!hich the curve of ( vs Rebranches off from smooth pipe curve (Fig. &4.#).

"n the rough /one of flo! the flo! resistance is mainly due to the form drag of those protrusions.

The pressure drop in this region is approximately proportional to the s5uare of the average

velocity of flo!. Thus ( becomes independent of Rein this region.

"n practice there are three distinct classes of problems relating to flo! through a single pipe line asfollo!s=

1. The flo! rate and pipe diameter are given.


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Application of :ernoulli%s e5uation bet!een t!o pointsA and at the free surfaces in the t!o

reservoirs gives

(&4.1$)

!here is the loss of head in the course of flo! fromA to .

Therefore ;5. (&4.1$) states that under steady state the head causing flo! becomes e5ual

to the total loss of head due to the flo!.

Considering the possible hydrodynamic losses the total loss of head can be !ritten in terms

of its different components as

(&4.11)

-oss of heat atentry to the pipefrom reservoir A

Friction loss inpipe over itslength -

;xit loss to thereservoir :

!here * is the average velocity of flo! in the pipe.

The velocity * in the above e5uation is usually substituted in terms of flo! rate + since under steadystate the flo! rate remains constant throughout the pipe even if its diameter changes. Therefore

replacing 7 in ;5. (&4.11) as !e finally get

or(&4.1#)

!here

(&4.1&)

The term R is defined as the flow resistnce .

"n a situation !here f becomes independent of Re the flo! resistance expressed by ;g. (&4.1&)becomes simply a function of the pipe geometry. With the help of ;5. (&4.1$) ;5. (&4.1#) can be !rittenas

(&4.1')

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in ;5. (&4.1') is the head causing the flo! and is defined as the difference in flo! potentials

bet!een A and :.

This e5uation is comparable to the voltagecurrent relationship in a purely resistive electrical

circuit. "n a purely resistive electrical circuit !here is the voltage or electricalpotential difference across a resistor !hose resistance isR and the electrical current flo!ing

through it is-. The difference ho!ever is that !hile the voltage drop in an electrical circuit is linearly proportional

to current the difference in the flo! potential in a fluid circuit is proportional to the s5uare of theflo! rate.

Therefore the fluid flo! system as sho!n in Fig. &4.& and described by ;5. (&4.1') can be

expressed by an e5uivalent electrical net!or+ system as sho!n in Fig. &4.'.

Fi" -8$0 E=ui&lent electricl networ? s%ste! for si!ple pipe flow pro)le! s'own inFi"$-8$-

Flow T'rou"' ,rnc'ed .ipes

"n several practical situations flo! ta+es place under a given head through different pipes ,ointed together

either in series or in parallel or in a combination of both of them.

.ipes in Series

"f a pipeline is ,oined to one or more pipelines in continuation these are said to constitute pipes in

series. A typical example of pipes in series is sho!n in Fig. &8.1. 0ere three pipesA, and # are,oined in series.

Fi" -9$# .ipes in series

"n this case rate of flo! + remains same in each pipe. 0ence

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"f the total head available at *ec. 1 (at the inlet to pipe A) is !hich is greater than thetotal head at *ec. # (at the exit of pipe C) then the flo! ta+es place from 1 to # through thesystem of pipelines in series.

Application of :ernoulli%s e5uation bet!een *ecs.1 and # gives

!here is the loss of head due to the flo! from 1 to #. Eecogni/ing the minor and ma,or losses

associated !ith the flo! can be !ritten as

(&8.1)

Friction lossin pipe A

-oss due toenlargement atentry to pipe :

Friction lossin pipe :

-oss dueto abruptcontractionat entryto pipe C

Friction loss inpipe C

The subscriptsA, and # refer to the 5uantities in pipe A : and C respectively. Ccis the coefficient ofcontraction.

The flo! rate satisfies the e5uation

(&8.#)

7elocities 7A 7:and 7Cin ;5. (&8.1) are substituted from ;5. (&8.#) and !e get

(&8.&)

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(&8.')

;5uation (&8.') states that the total flo! resistance is e5ual to the sum of the different resistancecomponents. Therefore the above problem can be described by an e5uivalent electrical net!or+ systemas sho!n in Fig. &8.#.

Fi" -9$( E=ui&lent electricl networ? s%ste! for t'rou"' pipes in series

.ipes In .rllel

When t!o or more pipes are connected as sho!n in Fig. &8.& so that the flo! divides and

subse5uently comes together again the pipes are said to be in parallel.

"n this case (Fig. &8.&) e5uation of continuity gives

(&8.4)

!here + is the total flo! rate and and are the flo! rates through pipesA and respectively.

-oss of head bet!een the locations 1 and # can be expressed by applying :ernoulli%s e5uation

either through the path 1A# or 1:#.

Therefore !e can !rite

Fi" -9$- .ipes in .rllel

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and

;5uating the above t!o expressions !e get

(&8.8)

!here

;5uations (&8.4) and (&8.8) give

(&8.9)

!here(&8.)

The flo! system can be described by an e5uivalent electrical circuit as sho!n in Fig. &8.'.

Fi" -9$0 E=ui&lent electricl networ? s%ste! for flow t'rou"' pipes in prllel

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From the above discussion on flo! through branched pipes (pipes in series or in parallel or incombination of both) the follo!ing principles can be summari/ed=

1. The friction e5uation must be satisfied for each pipe.#. There can be only one value of head at any point.

&. Algebraic sum of the flo! rates at any ,unction must be /ero. i.e. the total mass flo! rate to!ardsthe ,unction must be e5ual to the total mass flo! rate a!ay from it.

'. Algebraic sum of the products of the flux (#) and the flo! resistance (the sense beingdetermined by the direction of flo!) must be /ero in any closed hydraulic circuit.

The principles & and ' can be !ritten analytically as

at a node (3unction) (&8.K)

in a loop (&8.1$)

While ;5. (&8.K) implies the principle of continuity in a hydraulic circuit ;5. (&8.1$) is referred toas pressure e5uation of the circuit.

.ipe Networ?/ Solution )% Hrd% Cross Met'od

The distribution of !ater supply in practice is often made through a pipe net!or+ comprising a

combination of pipes in series and parallel. The flo! distribution in a pipe net!or+ is determinedfrom ;5s(&8.K) and (&8.1$).

The solution of ;5s (&8.K) and (&8.1$) for the purpose is based on an iterative techni5ue !ith an

initial guess in +

The method !as proposed by 0ardyCross and is described belo!=

The flo! rates in each pipe are assumed so that the continuity (;5. &8.K) at each node is

satisfied. sually the flo! rate is assumed more for smaller values of resistance R andvice versa.

"f the assumed values of flo! rates are not correct the pressure e5uation ;5. (&8.1$) !ill

not be satisfied. The flo! rate is then altered based on the error in satisfying the ;5.(&8.1$).

-et $be the correct flo! in a path !hereas the assumed flo! be . The error d + in flo! is then

defined as

(&8.11)

-et (&8.1#a)

and (&8.1#b)

Then according to ;5. (&8.1$)

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in a loop (&8.1&a)

and in a loop (&8.1&b)

Where e is defined to be the error in pressure e5uation for a loop !ith the assumed values of flo! rate in

each path.From ;5s (&8.1&a) and (&8.1&b) !e have

or (&8.1')

Where dh (? h h% ) is the error in pressure e5uation for a path. Again from ;5. (&8.1#a) !e can !rite

or (&8.14)

*ubstituting the value of dh from ;5. (&8.14) in ;5. (&8.1') !e have

Considering the error d to be the same for all hydraulic paths in a loop !e can !rite

(&8.18)

The ;5. (&8.18) can be !ritten !ith the help of ;5s (&8.1#a) and (&8.1#b) as

(&8.19)

The error in flo! rate d+ is determined from ;5. (&8.19) and the flo! rate in each path of a loop is thenaltered according to ;5. (&8.11).

The 0ardyCross method can also be applied to a hydraulic circuit containing a pump or a turbine. Thepressure e5uation (;5. (&8.1$)) is only modified in consideration of a head source (pump) or a head sin+(turbine) as

(&8.1)

!here is the head delivered by a source in the circuit. Therefore the value of to be substitutedin ;5. (&8.1) !ill be positive for a pump and negative for a turbine.

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Losses In .ipe ,ends

:ends are provided in pipes to change the direction of flo! through it. An additional loss of head

apart from that due to fluid friction ta+es place in the course of flo! through pipe bend.

The fluid ta+es a curved path !hile flo!ing through a pipe bend as sho!n in Fig. &9.#.

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as the radius of curvature of the centre line of the bend. The factor K varies slightly !ith Eeynolds numberEe in the typical range of Ee encountered in practice but increases !ith surface roughness.

Losses In .ipe Fittin"s

An additional loss of head ta+es place in the course of flo! through pipe fittings li+evalves couplings and so on. "ngeneral more restricted the passage is greater is theloss of head.

For turbulent flo! the losses are proportional to the s5uare of the average flo! velocity

and are usually expressed by !here * is the average velocity of flo!. Thevalue of Kdepends on the exact shape of the flo! passages. Typical values of K are

Appro+i!te Loss Coefficients4 'for Co!!ercil .ipe Fittin"s (

T%pe nd position of fittin"s 2lues of '

3lo)e &l&e4wide open #3te &l&e4 wide open $(

3te &l&e4 t'ree;=urters open #$#8

3te &l&e4 'lf open 8$9

3te &l&e4 =urter open (0

.u!p foot &l&e #$8

Kel)ow6t'reded7 0.9

08Kel)ow6t'reded7 0.4Side outlet of T :unction #$>

*ince the eddies generated by fittings persist for some distance do!nstream the total

loss of head caused by t!o fittings close together is not necessarily the same as thesum of the losses !hicheach alone !ould cause.These losses are sometimes expressed in terms of an e5uivalent length of anunobstructed straight pipe in !hich an e5ual loss !ould occur for the same averageflo! velocity. That is

(&9.4)

!here represents the e5uivalent length !hich is usually expressed in terms of the

pipe diameter as given by ;5. (&9.4). T'us depends upon t'e frictionfctor f 4 nd t'erefore on t'e Re%nolds nu!)er nd rou"'ness of t'e pipe$

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UNIT III

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DIMENSIONAL ANALYSIS

T'e Appliction of D%n!ic Si!ilrit% ; T'e Di!ensionl Anl%sis

T'e concept/

A physical problem may be characterised by a group of dimensionless similarity parameters or variablesrather than by the original dimensional variables.

This gives a clue to the reduction in the number of parameters re5uiring separate consideration in anexperimental investigation.

For an e+!ple4 if the Eeynolds number Ee ? O7 >h2^ is considered as the independent variable incase of a flo! of fluid through a closed duct of hydraulic diameter > h then a change in Ee may be causedthrough a change in flo! velocity 7 only. Thus a range of Ee can be covered simply by the variation in 7!ithout varying other independent dimensional variables O>hand ^.

-n fact the variation in the Eeynolds number physically implies the variation in any of the dimensionalparameters defining it though the change in Ee may be obtained through the variation in anyoneparameter say the velocity 7.

A number of such di!ensionless pr!eters in relation to dynamic similarity are sho!n in Table 4.1.*ometimes it becomes diffcult to derive these parameters straight for!ard from an estimation of therepresentative order of magnitudes of the forces involved. An alternative !et'od of deter!inin" t'esedi!ensionless pr!eters )% !t'e!ticl tec'ni=ue is ?nown s di!ensionl nl%sis .

T'e Tec'ni=ue/

The re5uirement of dimensional homogeneity imposes conditions on the 5uantities involved in a physicalproblem and these restrictions placed in the form of an algebraic function by the re5uirement ofdimensional homogeneity play the central role in dimensional analysis.

There are t!o existing approaches[

one due to :uc+ingham +no!n as ,uc?in"'!s pi t'eore!

other due to Eayleigh +no!n as R%lei"'s Indicil !et'od

"n our next slides !e%ll see fe! examples of the dimensions of physical 5uantities.

Di!ensions of .'%sicl untities

All physical 5uantities are expressed by magnitudes and units.

For e(a#ple the velocity and acceleration of a fluid particle are m2s and 1$m2s#respectively. 0ere thedimensions of velocity and acceleration are ms 1and ms#respectively.

"n SI 6S%ste! Interntionl7 units4 t'e pri!r% p'%sicl =untities w'ic' re ssi"ned )sedi!ensions re t'e !ss4 len"t'4 ti!e4 te!perture4 current nd lu!inous intensit%$ Of t'ese4 t'e

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first four re used in fluid !ec'nics nd t'e% re s%!)oli*ed s M 6!ss74 L 6len"t'74 T 6ti!e74nd 6te!perture7$

Any physical 5uantity can be expressed in terms of these primary 5uantities by using the basicmathematical definition of the 5uantity.

The resulting expression is +no!n as the dimension of the 5uantity.

-et us ta+e some e+!ples/

#$ Di!ension of Stress

*hear stress is defined as force2area. Again force ? mass _ acceleration

>imensions of acceleration ? >imensions of velocity2>imension of time.

Dimension of area = (Length)2=L2

0ence dimension of shear stress

(1K.1)

($ Di!ension of 2iscosit%

Consider e!ton%s la! for the definition of viscosity as

or,

The dimension of velocity gradient du2dy can be !ritten as

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di!ension of duJd%G di!ension of uJdi!ension of % G 6L J T7JL G T ;#

The dimension of shear stress is given in ;5. (1K.1).

0ence dimension of

Di!ensions of 2rious .'%sicl untities in T)ulr For!t

Lhysical uantity >imension

ass

-ength -Time T

Temperature V

7elocity -T1

Angular velocity T1

Acceleration -T#

Angular Acceleration T#

Force Thrust Weight -T#

*tress Lressure -1T#

omentum -T1

Angular omentum -#T1

oment Tor5ue -#T#

Wor+ ;nergy -#T#

Lo!er -#T&

*tream Function -#T1

7orticity *hear Eate T1

7elocity Lotential -#T1

>ensity -&

Coefficient of >ynamic 7iscosity -1T1

Coefficient of Dinematic 7iscosity -#T1

*urface Tension T#

:ul+ odulus of ;lasticity -1T#

,uc?in"'!s .i T'eore!

Assume a physical phenomenon is described by ! nu!)er of independent &ri)les li?e +#4 +(4+-4 $$$4 +!

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The phenomenon may be expressed analytically by an implicit functional relationship of the controllingvariables as

(1K.#)

o! if n )e t'e nu!)er of fund!entl di!ensions li?e !ss4 len"t'4 ti!e4 te!perture etc .involved in these m variables then according to :uc+ingham%s p theorem

T'e p'eno!enon cn )e descri)ed in ter!s of 6! ; n7 independent di!ensionless "roups li?e#4( 4 $$$4 !;n4 !here p terms represent the dimensionless parameters and consist of differentcombinations of a number of dimensional variables out of the m independent variables defining theproblem.

Therefore. the analytical version of the phenomenon given by ;5. (1K.#) can be reduced to

(1K.&)

ccordin" to ,uc?in"'!s pi t'eore!

This physically implies that the p'eno!enon !hich is basically described by m independent

dimensional variables is ulti!tel% controlled )% 6!;n7 independent di!ensionlesspr!eters ?nown s ter!s$

Alternti&e Mt'e!ticl Description of 67 .i T'eore!

A physical problem described by m number of variables involving n number of fundamental dimensions (n m) leads to a system of n linear algebraic e5uations !ith m variables of the form

(((((((((((((((((((((((((((((((((((((((((((((((((((

(1K.')

or in a matrix form

(1K.4)

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!here

and

Deter!intion of ter!s

A group of n (n ? number of fundamental dimensions) variables out of m (m ? total number of

independent variables defining the problem) variables is first chosen to form a basis so that all ndimensions are represented . These n variables are referred to as repeating variables.

Then the p terms are formed by the product of these repeating variables raised to arbitrary

un+no!n integer exponents and anyone of the excluded (m n) variables.

For e(a#ple ifx1x#...xnare ta+en as the repeating variables. Then

(((((((((((((((((((((((((((((((((((

The sets of integer exponents #4 ($ $ $ nre diffe

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