FLUID MECHANICS

Ih. Moha,mmeil H S Zangana

Email: moha,mmeil"zaagana@koyauniversity.org

"Lectures in Elementary Fluid Dynamics", By, J.M. McDonough, 2009.

Importance of n'luiils

Fluids are involved in our transportation systems in manyways;. They have an effect on our recreation (e.9., basketballs

and footballs are inflated with air).

. Entertainment (the sound from the speakers of a TVwould not reach our ears in the absence of air).

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Importance of Fluitl,s

Importance of fluids can be classified in two maincategories:

re i) Physical and natural science.

re ii) Technology.

Clearly, the second of these is often of more interest to anengineering student

Definition of a fluiil

A fluid is a substance that deforms continuously whensubjected to a shear stress, no matter how small thatshear stress may be.

A shear force is the force component tangent to asurface, and this force divided by the area of thesurface is the average shear stress over the area. Shearstress at a point is the limiting value of shear force toarea as the area is reduced to the point.

Definition of a fluiil

Figure l.l Deformation resulting from application of constant shear foroe.

ln Figure 1.1 a substance is placed between two closely spaced parallel platesso large that conditions at their edges may be neglected. The lower plate isfixed, and a force F is applied to the upper plate, which exerts a shear stressF/A on any substance between the plates, A is the area of the upper plate, ifthe force F causes the upper plate to move with a steady (no zero) velocity, nomatter how small the magnitude of F, then the substance between the twoplates is a fluid.

Definition of afluiil

c The fluid in immediate contact with a solidboundary has the same velocity as the boundaryGD.

x The fluid in the area abcd flows to the newposition ab'c'd.

ffi Each fluid particle moving parallel to tl,re plateand the velobity u varying from zero at thestationary plate to U at the upper plate.

Definition of a fluid

In Equation l.I, F is directly proportional to A and to U and is inverselyproportional to thickness t:

(1 .1)

(1.2)

Then: (1.3)

Where:-El = shear stress (N/m2)

H = Viscosity of the Fluid (N.s/m2)

It = the rate of deformation (s-1)

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Newtonian antl non- Newtonian Fluitl

Equation 1".3 is Newton's law of viscosity, based on that;

fluids are classified as Newtonian or non-Newtonian. In

Newtonian fluid there is a linear relation between the

magnitude of applied shear stress and the resulting rate ofdeformation ( IIt is constant), however in non-Newtonian fluidthere is a nonlinear relation between the magnitude of applied

shear stress and the resulting rate of deformation (See Figure

L.2). Gases and most common liquids tend to be Newtonian

fluids, while thick, long chained hydrocarbons may be non-

Newtonian.

n'igure 1.2

An Ideal plastic has a definite yield stress and a constant linerrelation of E to ffi . e thixotropic substance, such as printer'sink, has a viscosi-ty that is dependent upon the immediatelyprior angular deformation of the substance and has a tendency

to solidiSz when at rest.

For analysis pu{poses only the assumption can be made that

the fluid is not viscous. With zero viscosity the shear stress is

always zero,regardless of the motion of the fluids, If the is

also considered to be incompressible, it is then called an I deal

fluid.

Importance of n'luitls

fluid dynamics is one of the most important of all areas ofphysics. Life as we know it would not exist without fluids,and without the behavior that fluids exhibit ,e.9.:

. Motion of air keeps us comfortable in a warm room.

. air provides the oxygen we need to sustain life.

. most of our (liquid) body fluids are water based

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Importance of I'luids

Fluids are involved in our transportation systems in manyways;. They have an effect on our recreation (e.9., basketballsand footballs are inflated with air).

. Entertainment (the sound from the speakers of a TVwould not reach our ears in the absence of air),

Importance of nluitls

lmportance of fluids can be classified in two maincategories:

m, i) Physical and natural science.

&l ii) Technology.

Clearly, the second of these is often of more interest to anengineering student

Units anil Dimensions

Mass, Length and Time are commonly used as primary units,

other units being derived from them. Their dimensions are

written as M, L and T respectively" Sometimes force is used as

a primary unit and its dimension is written as F.

The dimensions of velocity, which is a rate of increase ofdistance with time, may be written as LT-I, and those ofacceleration, the rate of increase of velocity. are LT-2. An area

has dimensions L2 and a volume has the dimensions L3.

The volume of a body does not completely define the amountof material which it contains, and therefore it is usual to definea third basic quantity, the amount of matter in the body, that isits mass M. Thus the density of the material, its mass per unitvolume, has the dimensions ML3. However, in the BritishEngineering System force F is used as the third fundamentaland mass then becomes a derived dimension.

B

'etrole..r (r1r|^<sr

Students of most areas of engineering soon discover that the

data used are expressed in a great variety of different units, so

that quantities must be converted into a common system before

proceeding with calculations. Most of the physical properties

determined in the laboratory will originally have been

expressed in the cgs system, whereas the dimensions of the

full-scale plant, its throughput, design, and operating

characteristics appear either in some form of general

engineering units or in special units which have their origin inthe history of the particular industry.

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Familiarity with the various systems of units and an ability toconvert from one to another are therefore essential, as it willfrequently be necessary to access literature in which the SI

system has not been used.

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SYSTEMS OE UNITS

Although in scientific work mass is taken as the thirdfundamental quantity and in engineering force is sometimes

used as mentioned above, the fundamental quantities L, M, F,

T may be used interchangeably. A summary of the various

systems of units, and the quantities associated with them, isgiven in Table 1.1.

Tab1e 1.1

(h*$ {P * ti*

fls tffi *rFh lMdts** *fl9*r* ts* t$1iM sed Md *Mi&&1 iy* &ce W#Ss$f w (e 16* ldB.| kb t\*Fhn&tlms {s4*{@ a'tu *5edq r*su sd:tqBm t*&F$ cd*B tk! W'WI&Eeld

urlr*ii c.rrlxm't h{tutt,&$*rr x relBu*l4ks'{' I :T }, i r;*5dri6'ttrr.vr*stF. S-iir"lgrryft&"(' $)nr&NN ..*!n@d*&rd C \tI ll.)I'rr' r,&,ltrk&6d

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The centimeter- gra,m- seconil (cgs) system

In this system the basic units are of length L, mass M, and time T with the nomenclature:

&

YS

Length:

Mass:

Time:

Dimension L:

Dimension M:

Dimension T:

Unit I centimetre

Unit i gram

Unit I second

(1 cm)

(l e)(1S)

The unit of force is that force which will give a mass of I g an acceleration of I cm/s2 and is

&;

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known as the dyne:

Force:

Energy:

Power:

Dimension F = MLT-2:

Dimensions ML2T-2

Dimensions ML2T-3

Unit 1 dyre

Unit I erg

Unit 1 ergls

(1 dyn)

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S.usterue

It temati on aI d' Uni tes (SDThese systems are in essence modifications ofthe cgs system but employ larger units.

The basic dimensions are again of L, M, and T.

ffi Length:

14 Mass:

m Time:

Dimension L:

Dimension M:

Dimension T:

Unit 1 metre

Unit 1 kilogram

ljnit I second

(1m)(1 ke)

(1 s)Theunit of force, known as the Newton, is that force which will give an acceleration of I m/s2 to amass of one kilogram. Thus 1 N = I kg m/s2 with dimensions MLT-2, and one Newton equals105 dynes. The energy unit, theNewton-metre, is 10i ergs and is called the Joule;andlhepower unit, equal to one Joule per second, is known as the WatL

E

E

}i

Thus:

Force:

Energy:

Power:

Dimensions MLT'2:

Dimensions ML2T-2:

Dimensions ML2T-3:

Unit l Newton (IN) or l kgm/s2

Unit 1 Joule (1 J) or 1 kgm2/s2

Unit 1 Watt (l W) or 1 kg m2ls3

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The basic units in this system are:

Length: Dimension L: Unit 1 foot (1 ft)

Mass: Dimension M: Unit I Pound (l Ib)

Time: Dimension T: Unit I second (l s)

The unit offorce gives that which a mass of I Ib an aoceleration of I ff/s2 is known

as the poundal (pdl). The unit of energy (or work) is the foot-poundal, and the unit of power is

the foot poundal per second.

Thus:

Force Dimensions MLT-2 Unit I poundal (1 pdl)

Energy Dimensions ML2T-2 Unit I ft-poundal

Power Dimensions ML2T-3 Unit I foot-poundal/s

The British engineering system

In an alternative form of the fps system (Engineering

system) the units of length (ft) and time (s) are

unchanged, but the third fundamental is a unit offorce (F) instead of mass and is known as the pound

force (Ibfl. This is defined as the force which gives a

mass of 1 Ib an acceleration of 32J740 ft.1s2, the

"standard" value of the acceleration due to gravity.

t2

The British engineering system

It is therefore a fixed quantity and must not be confused withthe pound weight which is the force exerted by the earth'sgravitational field on a mass of one pound and which variesfrom place to place as g varies. It will be noted therefore thatthe pound force and the pound weight have the same valueonly when gis32.1740 ft21s. The unit of mass in this system isknown as the slug, and is the mass which is given anacceleration of I ff/s2 by a one pound force:

1 slug= 1 lbf ft-1s2

Ihe British engineerfug system

Misunderstanding often arises from the fact that the poundwhich is the unit of mass in the fps system has the same nameas the unit of force in the engineering system. To avoidconfusion the pound mass should be written as Ib or even lbmand the unit of force always as lbf. It will be noted that:

1 slug :32.1740Ib mass and 1 lbf : 32.1740 pdl

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h To summarise:The basic units are:

x Length Dimension L Unit 1 foot (1 ft)6 Force DimensionF Unit 1 pound-force (1 Ibl)si Time Dimension T Unit 1 second (1 s)

m The derived units are:

B Mass Dimensions FL-1T2 Unit6 Energy Dimensions FL Unit$i Power Dimensions FLT-I Unit

stug (= 32.1 740 pounds)

footpound-force (l ftlbflfoot-pound force/s (1 ft-1b17,

6,g: Note: t horsepower is defined as 550 ft-lbfls.

I{on-coherent system employing pounal mass antl pounil

force simultaneously

Two units which have never been popular in the last twosystems of units are the poundal (for force) and the slug (formass). As a result, many writers, particularly in America, useboth.the pound mass and pound force as basic units in thesame equAtion because theyare the units in common use. Thisis an essentially incoherent system and requires great care inits use.

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Derivetl units

h: The three fundamental units of the SI and of the cgs systemsare length (L), mass (M), and time (T). It has been shown thatforce can be regarded as having the dimensions of MLT-2, andthe dimensions of many other parameters may be worked outin terms of the basic MLT system (Table (1.2),

Qu{rl}ti{} Unit Shnerrsion* Llnits in kg, m, rNewt+n kg m/s?

.kg m?ls? 1*kg rnllss 1*kglrn *3 1- Ikg/rn s (* Is'

Etergy *r r+ork J<:ule M[,3T*;ML2"r".jML*1T.

1N m*l.llt J/$)l,#m:1

N xlmri

P+tt erPre*gureVisc:r.rsity.F:requency

W$:t

Pascal-second lVft.- lT?le*t 't't

CONVERSION OF'UNfTfi

Conversion of units from one system to another is simplycamied out if the quantities are expressed in terms of thefundamental units of mass, length, time, temperature. Typicalconversion factors for the British and metric systems are:

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;

Thank you

CONVERSION OX'IINfNi

Mffis

Leirgth

1fime

T*mpemt*rediff*renee

Force

, * * (*) s)ug* 453.6 s = *'4535 ks

1 tf * 30.48 cm * 0.3{J48 nr

'..-(#)h,'r* (*) *- i*) K (ordqs.K)

t pourd fur*e * I2.? pourrdal * 4.44 x Itri tty*re * 4.44 N

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