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FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

IntroductionIntroductionvvSome Characteristics of FluidsSome Characteristics of FluidsvvAnalysis of Fluid BehaviorsAnalysis of Fluid BehaviorsvvFluid PropertiesFluid PropertiesvvViscosityViscosity

CSE Winter School 2012

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Characteristics of FluidsCharacteristics of Fluids

vvWhat’s a Fluid ?What’s a Fluid ?vvWhat’s difference between a solid and a fluid ?What’s difference between a solid and a fluid ?

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Fluid and Solid Fluid and Solid 2/32/3

vv Vague ideaVague ideaððFluid is soft and easily deformed. Fluid is soft and easily deformed. ððSolid is hard and not easily deformed.Solid is hard and not easily deformed.

vv Molecular structureMolecular structureððSolid has densely spaced molecules with large intermolecular Solid has densely spaced molecules with large intermolecular

cohesive force allowed to maintain its shapecohesive force allowed to maintain its shape

vv Fluids comprise the liquid and gas (or vapor) phase of the phyFluids comprise the liquid and gas (or vapor) phase of the physical formssical forms

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Fluid and Solid Fluid and Solid 3/33/3

ððLiquid has further apart spaced molecules, the intermolLiquid has further apart spaced molecules, the intermolecular forces are smaller than for solids, and the molececular forces are smaller than for solids, and the molecules have more freedom of movement. ules have more freedom of movement. At normal tempAt normal temperature and pressure, the spacing is on the order of 10erature and pressure, the spacing is on the order of 10--66

mm. The number of molecules per cubic millimeter is mm. The number of molecules per cubic millimeter is on the order of 10on the order of 1021 21 ..

ððGases have even greater molecular spacing and freedoGases have even greater molecular spacing and freedom of motion with negligible cohesive intermolecular fm of motion with negligible cohesive intermolecular forces and as a consequence are easily deformed.orces and as a consequence are easily deformed. At norAt normal temperature and pressure, the spacing is on the ordmal temperature and pressure, the spacing is on the order of 10er of 10--77mm. The number of molecules per cubic millimm. The number of molecules per cubic millimeter is on the order of 10meter is on the order of 1018 18 ..

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Definition of FluidDefinition of Fluid

vvA fluid is a substance that deforms continuously undeA fluid is a substance that deforms continuously under the application of a shear stress no matter how small r the application of a shear stress no matter how small the shear stress may be.the shear stress may be.

vvA shearing stress is created whenever a tangential forA shearing stress is created whenever a tangential force acts on a surface.ce acts on a surface.

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Fluid and Solid Fluid and Solid 1/31/3

vvWhen a constant shear force is applied:When a constant shear force is applied:ððSolid deforms or bendsSolid deforms or bendsððFluid continuously deforms.Fluid continuously deforms.

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ContinuumContinuum

vv Fluid : aggregation of moleculesFluid : aggregation of moleculesvv Most engineering problems are concerned Most engineering problems are concerned

with physical dimensions much larger than with physical dimensions much larger than the limiting volume, the limiting volume, ddV*, V*, so thatso that density is density is essentially a point function and fluid properties essentially a point function and fluid properties can be thought of as varying continuously in can be thought of as varying continuously in space such a fluid is called “continuum”.space such a fluid is called “continuum”.

vvddV* =V* =1010--99 mmmm33 = (1 = (1 μμm)m)3 3 for gasesfor gasesvvException : rarefied gas Exception : rarefied gas

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ContinuumContinuum

rr

ddV*V* ddVV

Vm

VV ddr

dd *lim®

=

ddmm

ddVV

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Analysis of Fluid Behaviors Analysis of Fluid Behaviors 1/21/2

vvAnalysis of any problem in fluid mechanics necessarilAnalysis of any problem in fluid mechanics necessarily includes statement of the basic laws governing the fy includes statement of the basic laws governing the fluid motion. The basic laws, which applicable to any fluid motion. The basic laws, which applicable to any fluid, are:luid, are:ððConservation of massConservation of massððNewton’s second law of motionNewton’s second law of motionððThe principle of angular momentumThe principle of angular momentumððThe first law of thermodynamicsThe first law of thermodynamicsððThe second law of thermodynamicsThe second law of thermodynamics

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Analysis of Fluid Behaviors Analysis of Fluid Behaviors 2/22/2

vvFluid statics : the fluid is at restFluid statics : the fluid is at restvFluid dynamics : the fluid is in motionv Fluid properties are closely related to fluid

behaviore.g.) gas : light and compressible

liquid : heavy and incompressibleThe flow of syrup is slower than that of

water

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DensityDensityvvThe density of a fluid, designated by the Greek symbThe density of a fluid, designated by the Greek symb

ol ol rr (rho), is (rho), is defined as its mass per unit volume.defined as its mass per unit volume.vvIIn SIn SI system,system, the units are kg/mthe units are kg/m33..vvThe value of density can vary widely between differeThe value of density can vary widely between differe

nt fluids, but for liquids, variations in pressure and tent fluids, but for liquids, variations in pressure and temperature generally have only a small effect on the vmperature generally have only a small effect on the value of density.alue of density.

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ViscosityViscosity

vvThe properties of density and specific weight are meaThe properties of density and specific weight are measures of the “heaviness” of a fluid.sures of the “heaviness” of a fluid.

vvIt is clear, however, that these properties are not sufficIt is clear, however, that these properties are not sufficient to uniquely characterize how fluids behave since ient to uniquely characterize how fluids behave since two fluids can have approximately the same value of two fluids can have approximately the same value of density but behave quite differently when flowing.density but behave quite differently when flowing.

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Viscosity DefinitionViscosity Definition

vv The constant of proportionality is The constant of proportionality is designated by the designated by the Greek symbol Greek symbol mm (mu) and is called the absolut(mu) and is called the absolute viscoe viscossity, dynamic viscosity, or ity, dynamic viscosity, or simply the viscosity of the fluid.simply the viscosity of the fluid.

vv The viscosity depends on the partiThe viscosity depends on the particular fluid, and for a particular flucular fluid, and for a particular fluidid,, the viscosity is alsothe viscosity is also dependendependent on temperature.t on temperature.

Newtonian fluids

dydumt =

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Newtonian and NonNewtonian and Non--Newtonian FluidNewtonian Fluid

vvFluids for which the shearing stress is linearly related Fluids for which the shearing stress is linearly related to the rate of shearing strain are designated as to the rate of shearing strain are designated as NewtoNewtonian fluidsnian fluids after I. Newton (1642after I. Newton (1642--1727).1727).

vvMost common fluids such as water, air, and gasoline Most common fluids such as water, air, and gasoline are Newtonian fluid under normal conditions.are Newtonian fluid under normal conditions.

vvFluids for which the shearing stress is not linearly relFluids for which the shearing stress is not linearly related to the rate of shearing strain are designated as ated to the rate of shearing strain are designated as nononn--Newtonian fluids.Newtonian fluids.

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NonNon--Newtonian Fluids Newtonian Fluids 11/2/2

vvShear thinning fluids: The viShear thinning fluids: The viscosity decreases with increascosity decreases with increasing shear rate sing shear rate –– the harder tthe harder the fluid is sheared, the less vhe fluid is sheared, the less viscous it becomes. Many colliscous it becomes. Many colloidal suspensions and polymoidal suspensions and polymer solutions are shear thinniner solutions are shear thinning. Latex paint is example.g. Latex paint is example.

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NonNon--Newtonian Fluids Newtonian Fluids 22/2/2

vvShear thickening fluids: The viscosity increases with iShear thickening fluids: The viscosity increases with increasing shear rate ncreasing shear rate –– the harder the fluid is sheared, tthe harder the fluid is sheared, the more viscous it becomes. Waterhe more viscous it becomes. Water--corn starch mixturcorn starch mixture watere water--sand mixture are examples.sand mixture are examples.

vvBingham plastic: neither a fluid nor a solid. Such matBingham plastic: neither a fluid nor a solid. Such material can withstand a finite shear stress without motioerial can withstand a finite shear stress without motion, but once the yield stress is exceeded it flows like a n, but once the yield stress is exceeded it flows like a fluid. Toothpaste and mayonnaise are common exampfluid. Toothpaste and mayonnaise are common examples.les.

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figun_01_p017a

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Viscosity vs. Temperature Viscosity vs. Temperature 1/31/3

vvFor fluids, the viscosity For fluids, the viscosity decreases with an increadecreases with an increase in temperature.se in temperature.

vvFor gases, an increase in For gases, an increase in temperature causes an intemperature causes an increase in viscosity.crease in viscosity.

ððWHY? molecular structWHY? molecular structureure..

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Viscosity vs. Temperature Viscosity vs. Temperature 2/32/3

vvThe liquid moleculesThe liquid molecules are closely spaced, with strong care closely spaced, with strong cohesive forces between molecules, and the resistance ohesive forces between molecules, and the resistance to relative motion between adjacent layers is related tto relative motion between adjacent layers is related to these intermolecular force. o these intermolecular force.

vvAs the temperature increases, these cohesive force are As the temperature increases, these cohesive force are reduced with a corresponding reduction in resistance treduced with a corresponding reduction in resistance to motion. Since viscosity is an index of this resistance, o motion. Since viscosity is an index of this resistance, it follows that it follows that viscosity is reduced by an increase in tviscosity is reduced by an increase in temperature.emperature.

vvThe Andrade’s equation μThe Andrade’s equation μ＝＝ DeDeB/TB/T

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Viscosity vs. Temperature Viscosity vs. Temperature 3/3 3/3

vvIn gases,In gases, the molecules are widely spaced and intermthe molecules are widely spaced and intermolecular force negligible. olecular force negligible.

vvThe resistance to relative motion mainly arises due to The resistance to relative motion mainly arises due to the exchange of momentum of gas molecules between the exchange of momentum of gas molecules between adjacent layers. adjacent layers.

vvAs the temperature increases, As the temperature increases, the random molecular the random molecular activity increases with a corresponding increase in vactivity increases with a corresponding increase in viscosity.iscosity.

vvThe Sutherland equation μThe Sutherland equation μ＝＝ CTCT3/23/2 / (T+S)/ (T+S)

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Example: Newtonian Fluid Shear StressExample: Newtonian Fluid Shear Stress

•• The velocity distribution for the flow of a Newtonian fluid betThe velocity distribution for the flow of a Newtonian fluid between two sides, parallel plates is given by the equationween two sides, parallel plates is given by the equation

úúû

ù

êêë

é÷øö

çèæ-=

2

hy1

2V3u

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Example Example SolutionSolution

where V is the mean velocity. The fluid has a where V is the mean velocity. The fluid has a viscosityviscosity of of 2N s/m2N s/m22. When V=. When V=0.60.6 mm/s and h=/s and h=5 mm5 mm. determine: (a) t. determine: (a) the shearing stress acting on the bottom wall, and (b) the he shearing stress acting on the bottom wall, and (b) the shearing stress acting on a plane parallel to the walls and shearing stress acting on a plane parallel to the walls and passing through the centerline (midplane).passing through the centerline (midplane).

03

/7203

3

02

22

2

=-==

=-==

-==

=

-=

ymidplane

hywallbottom

hVy

dydu

mNhVy

dydu

hVy

dydu

mmt

mmt

mmt

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Kinematic ViscosityKinematic Viscosity

vvDefining kinematic viscosity Defining kinematic viscosity νν= μ/= μ/rr [Ny] [Ny] ððThe dimensions of kinematic viscosity are LThe dimensions of kinematic viscosity are L22/T./T.ððThe units of kinematic viscosity in SI system are The units of kinematic viscosity in SI system are

mm22/s./s.

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FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Fluid Fluid StaticsStaticsvvPressure at a PointPressure at a PointvvBasic Equation for Pressure FieldBasic Equation for Pressure FieldvvPressure variation in a Fluid at RestPressure variation in a Fluid at RestvvBuoyancy, Floating, and StabilityBuoyancy, Floating, and Stability

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Pressure at a Point Pressure at a Point 1/41/4

vvPressure ? Pressure ? ððIndicating the normal force per unit area at a given poIndicating the normal force per unit area at a given po

int acting on a given plane within the fluid mass of intint acting on a given plane within the fluid mass of interest.erest.

vvHow the pressure at a point varies with the orientatioHow the pressure at a point varies with the orientation of the plane passing through the point ?n of the plane passing through the point ?

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Pressure at a Point Pressure at a Point 2/42/4

vvConsider the freeConsider the free--body diagram within a fluid mass. body diagram within a fluid mass. vvIn which there are no shearing stress, the only externaIn which there are no shearing stress, the only externa

l forces acting on the wedge are due to the pressure anl forces acting on the wedge are due to the pressure and the weight.d the weight.

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fig_02_01

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Pressure at a Point Pressure at a Point 3/43/4

vvThe equation of motion (Newton’s second law, F=ma) The equation of motion (Newton’s second law, F=ma) in the y and z direction are,in the y and z direction are,

zSzZ

ySyy

a2

δxδyδzρ2

δxδyδzγ-δxδscosθPδxδyPF

a2

δxδyδzρδxδssinθPδxδzPF

=å -=

=å -=

( )2δyρaPP

2δzγρaPP ySyZSZ =-+=-

qddqdd sinszcossy ==

δx=0δx=0、、δy=0δy=0、、δz=0δz=0 SyZ PPP ==

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Pressure at a Point Pressure at a Point 4/44/4

vvThe The pressure at a pointpressure at a point in a fluid at rest, or in motion, in a fluid at rest, or in motion, is is independent of the directionindependent of the direction as long as there are no as long as there are no shearing stresses present.shearing stresses present.

vvThe result is known asThe result is known as Pascal’s lawPascal’s law named in honor onamed in honor off Blaise PascalBlaise Pascal (1623(1623--1662).1662).

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figun_02_p

039

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Basic Equation for Pressure FieldBasic Equation for Pressure Field

vvTo obtain an basic equation for pressure field in a statTo obtain an basic equation for pressure field in a static fluid.ic fluid.

vvApply NewtonApply Newton’’s second law to a differential fluid mass second law to a differential fluid masss

å = aδmFδððThere are two types of fThere are two types of f

orces acting on the mass orces acting on the mass of fluid: of fluid: surface force ansurface force and body force.d body force. Vm rd=d

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Body Force Body Force on Elementon Element

VgmgFB rd=d=drrr

kzyxkFB

rrddgd-=d-

Vm rd=dWhere ρ is the density.Where ρ is the density.gg is the local gravitationis the local gravitational al acceleration.acceleration.

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Surface Forces Surface Forces 1/41/4

vvNo shear stresses, the only surface force is the pressure force.No shear stresses, the only surface force is the pressure force.

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Surface Forces Surface Forces 2/42/4

vv The pressure at the left faceThe pressure at the left face

vv The pressure at the right facThe pressure at the right facee

vv The pressure force in y direcThe pressure force in y directiontion

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

¶¶

-=÷÷ø

öççè

涶

+-÷÷ø

öççè

涶

-=

( )2

dyypp

2dy

yppyy

yppp LL ¶

¶-=÷

øö

çèæ-

¶¶

+=-¶¶

+=

( )2

dyypp

2dy

yppyy

yppp RR ¶

¶+=÷

øö

çèæ

¶¶

+=-¶¶

+=

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Surface Forces Surface Forces 3/43/4

vvThe pressure force in x directionThe pressure force in x direction

vvThe pressure force in z directionThe pressure force in z direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

¶¶

-=÷øö

çèæ

¶¶

+-÷øö

çèæ

¶¶

-=

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

¶¶

-=÷øö

çèæ

¶¶

+-÷øö

çèæ

¶¶

-=

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Surface Forces Surface Forces 4/44/4

vvThe net surface forces acting on the elementThe net surface forces acting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs ÷÷

ø

öççè

涶

+¶¶

+¶¶

-=++=

kzpj

ypi

xppgradp

rrr

¶¶

+¶¶

+¶¶

=Ñ=

δxδyδzpδxδyδz)(gradpFδ s -Ñ=-=

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General Equation of MotionGeneral Equation of Motion

zyxakzyxzyxp dddrddgdddd rr=-Ñ-

Vd)gp(zyx)gp(FFF BS

r

rrrr

r+-Ñ=dddr+-Ñ=d+d=d

akp rrrg =-Ñ-

The general equation of motion for a fluid in which there are no shearing stresses

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Pressure Variation in a Fluid Pressure Variation in a Fluid at at RestRest

vvFor a fluid at rest For a fluid at rest aa=0=0 0akp ==-Ñ-rr

rg

directionz...0gzp

directiony...0gyp

directionx...0gxp

z

y

x

-=+¶¶

-

-=+¶¶

-

-=+¶¶

-

r

r

r

g-=¶¶

=¶¶

=¶¶

zP

0yP

0xP

gg,0g,0g

z

yx

-=

==

g-=r-= gdzdp

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PressurePressure--Height RelationHeight Relation

vvThe basic pressureThe basic pressure--height relation of static fluid :height relation of static fluid :

vvRestriction:Restriction:ððStatic fluid.Static fluid.ððGravity is the only body force.Gravity is the only body force.ððThe z axis is vertical and upward.The z axis is vertical and upward.

Integrated to determine the Integrated to determine the pressure distribution in a pressure distribution in a static fluidstatic fluid with appropriatewith appropriateboundary conditions.boundary conditions.

How the specific weight varies with z?How the specific weight varies with z?

g-=r-= gdzdp

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Pressure in Incompressible FluidPressure in Incompressible Fluid

vv A fluid with constant densitA fluid with constant density is called an incompressible y is called an incompressible fluid.fluid.

pp11－－ pp22 = γ(z= γ(z22－－zz11)=γh)=γhpp11=γh +p=γh +p22

h= zh= z22－－zz11，，h is the depth of fluid h is the depth of fluid measured downward from the locameasured downward from the location of ption of p22..

This type of pressure distribution This type of pressure distribution is called a hydrostatic distribution.is called a hydrostatic distribution.

g-=r-= gdzdp òò -= 2

1

2

1

z

z

p

pdzdp g

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Example: Example: PressurePressure--Depth RelationshipDepth Relationship

•• Because of a leak in a buried gasoline storage tank, water has sBecause of a leak in a buried gasoline storage tank, water has seeped in to the depth shown in eeped in to the depth shown in Figure. Figure. If the specific gravity of If the specific gravity of the gasoline is SG=0.68. Determine the pressure at the gasolinthe gasoline is SG=0.68. Determine the pressure at the gasolinee--water interface and at the bottom of the tank. Express the prewater interface and at the bottom of the tank. Express the pressure in units of ssure in units of NN//mm22, , NN//mmmm22, and as pressure head in , and as pressure head in metersmetersof water.of water.Determine the pressure at the gasolineDetermine the pressure at the gasoline--water interface and at the bottom of the water interface and at the bottom of the tanktank

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Example SolutionSolution

The pressure at the interface isThe pressure at the interface is

ppoo is the pressure at the free surface of the gasoline.is the pressure at the free surface of the gasoline.

( )20

03

01

/7.34

)2.5)(/9800)(68.0(2

mkNp

pmmNphSGp OH

+=

+=+= g

The pressure at the tank bottomThe pressure at the tank bottom

2

23

12

/5.43/7.34)9.0)(/9800(

22

mkNmkNmmN

php OHOH

=

+=

+= g

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BUOYANCY BUOYANCY 1/21/2

vvBuoyancy: The net vertical force acting on any body Buoyancy: The net vertical force acting on any body which which is immersed in a liquid, or floating on its suris immersed in a liquid, or floating on its surfaceface due to liquid pressure. Fdue to liquid pressure. FBB

vvConsider a body of arbitrary sConsider a body of arbitrary shape, having a volume V, that hape, having a volume V, that is immersed in a fluid,is immersed in a fluid,

vvWe enclose the body in a paraWe enclose the body in a parallelepiped and draw a freellelepiped and draw a free--bobody diagram of parallelpiped dy diagram of parallelpiped with body removed as shown with body removed as shown in (b).in (b).

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BUOYANCY BUOYANCY 2/22/2

[ ]VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

----=-=-

--=

ggg

VgFB r=

FFBB is the force the body is exerting on the fluiis the force the body is exerting on the fluid.d.W is the weight of the shaded fluid volume (pW is the weight of the shaded fluid volume (parallelepiped minus body).arallelepiped minus body).A is the horizontal area of the upper (or lower) A is the horizontal area of the upper (or lower) surface of the parallelepiped.surface of the parallelepiped.

For a For a submerged bodysubmerged body, the buoy, the buoyancy force of the fluid is equal to tancy force of the fluid is equal to the weight of displaced fluidhe weight of displaced fluid

A

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Archimedes’ PrincipleArchimedes’ Principle

vvFor a submerged body, the buoyancy force of the fluiFor a submerged body, the buoyancy force of the fluid is equal to the weight of displaced fluid and is directd is equal to the weight of displaced fluid and is directly vertically upward.ly vertically upward.

vvThe relation reportedly was used by Archimedes in 22The relation reportedly was used by Archimedes in 220 B.C. to determine the gold content in the crown of 0 B.C. to determine the gold content in the crown of King King HieroHiero II.II.

VgFB r=

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Example Example Buoyant force on a submerged objectBuoyant force on a submerged object

The 1.8 N leas fish sinker is attached to a fishing line. The specific The 1.8 N leas fish sinker is attached to a fishing line. The specific gravity of the sinker is SGgravity of the sinker is SGsinkersinker = 11.3. Determine the difference = 11.3. Determine the difference between the tension in the line above and below the sinker.between the tension in the line above and below the sinker.

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Example Example SolutionSolution

0=-+- WFTT BBA

)]/1(1[/

kersin

kersin

kersin

SGWTTSGWF

VSGWVF

BA

B

B

-=-===gg

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FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Fluid Kinematics Fluid Kinematics vvThe Velocity FieldThe Velocity FieldvvThe Acceleration FieldThe Acceleration Field

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Field Representation of flow Field Representation of flow 1/21/2

vvAt a given instant in time, At a given instant in time, any fluid propertyany fluid property (s(such as density, pressure, velocity, and acceleratuch as density, pressure, velocity, and acceleration) ion) can be described as a functions of thecan be described as a functions of the fluifluid’s locationd’s location..

vvThis representation of fluid parameters as funcThis representation of fluid parameters as functions of the spatial coordinates is termed a tions of the spatial coordinates is termed a field field representation of flowrepresentation of flow..

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Field Representation of flow Field Representation of flow 2/22/2

vvThe specific The specific field representationfield representation may be different at dmay be different at different times, so that to describe a fluid flow we must ifferent times, so that to describe a fluid flow we must determine the various parameter determine the various parameter not only as functionnot only as functions of the spatial coordinates but also as a function of s of the spatial coordinates but also as a function of time.time.

vvEXAMPLE: Temperature field EXAMPLE: Temperature field T = T ( x , y , z , t )T = T ( x , y , z , t )vvEXAMPLE: Velocity field EXAMPLE: Velocity field

k)t,z,y,x(wj)t,z,y,x(vi)t,z,y,x(uV ++=

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Velocity FieldVelocity Field

vvThe velocity at any particle in the flow field (the veloThe velocity at any particle in the flow field (the velocity field) is given bycity field) is given by

k)t,z,y,x(wj)t,z,y,x(vi)t,z,y,x(uV ++=

The velocity of a particle is The velocity of a particle is the time rate of change of tthe time rate of change of the position vector for that phe position vector for that particle.article.

dtrdV A

A

rr=

)t,z,y,x(VV =

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Example Example Velocity Field RepresentationVelocity Field Representation

•• A velocity field is given by where VA velocity field is given by where V00 and and llare constants. are constants. At what location in the flow field is the speed eqAt what location in the flow field is the speed equal to ual to VV00?? Make a sketch of the velocity fieldMake a sketch of the velocity field in the first quadrin the first quadrant (xant (x≧≧0, y 0, y ≧≧0) by drawing arrows representing the fluid velo0) by drawing arrows representing the fluid velocity at representative locations.city at representative locations.

( )( )jyix/VV 0rr

lr

-=

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Example Example SolutionSolution

2/12202/1222 )yx(V)wvu(V +=++=l

The x, y, and z components of the velocity are given by u = VThe x, y, and z components of the velocity are given by u = V00xx//ll, v = , v = --VV00y/ y/ ll, and w = 0 so that the fluid speed V, and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius ll ccentered at the origin [(xentered at the origin [(x22 + y+ y22))1/21/2= = ll] as shown in Figure E4.1 ] as shown in Figure E4.1 (a). (a).

The direction of the fluid velocity relative to the x axis is giveThe direction of the fluid velocity relative to the x axis is given in terms of n in terms of θ= arctan(v/u)θ= arctan(v/u) as shown in Figure E4.1 (b) For as shown in Figure E4.1 (b) For this flow this flow

xy

/xV/yV

uvθtan

0

0 -=

-==

l

l

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Velocity FieldVelocity Field

vvMethod of DescriptionMethod of DescriptionvvSteady and Unsteady FlowsSteady and Unsteady Flowsvv1D, 2D, and 3D Flows1D, 2D, and 3D FlowsvvStreamlinesStreamlines

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Methods of DescriptionMethods of Description

vLagrangian method = System methodvEulerian method = Control volume method

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LagraLagranngian Methodgian Method

vvFollowing individual fluid particles as they move.Following individual fluid particles as they move.vvThe fluid particles are tagged or identified.The fluid particles are tagged or identified.vvDetermining how the fluid properties associated with Determining how the fluid properties associated with

these particles change as a function of time.these particles change as a function of time.vv Example:Example: one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a

particular fluid particle A and record that particle’s temperature particular fluid particle A and record that particle’s temperature as it moves about. as it moves about. TTAA = T= TAA (t)(t) The use of may such measuring The use of may such measuring devices moving with various fluid particles would provide the devices moving with various fluid particles would provide the temperature of these fluid particles as a function of time.temperature of these fluid particles as a function of time.

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Eulerian MethodEulerian Method

vvUse the field concept.Use the field concept.vvThe fluid motion is given by completely prescribing tThe fluid motion is given by completely prescribing t

he necessary properties as a functions of space and tihe necessary properties as a functions of space and time.me.

vvObtaining information about the flow in terms of whaObtaining information about the flow in terms of what happens at fixed points in space as the fluid flows pat happens at fixed points in space as the fluid flows past those points.st those points.

vv Example:Example: one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (x,y,z) and record the temperature at that point particular point (x,y,z) and record the temperature at that point as a function of time. as a function of time. T = T ( x , y , z , t )T = T ( x , y , z , t )

2012-12-24

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59

1D, 2D, and 3D Flows1D, 2D, and 3D Flows

vvDepending on Depending on the number of space coordinates requirthe number of space coordinates required to specify the flow field.ed to specify the flow field.

vvAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimendimensional, analysis is based on sional, analysis is based on fewer dimensions is frequfewer dimensions is frequently meaningfulently meaningful..

vvThe complexity of analysis increases considerably witThe complexity of analysis increases considerably with the number of dimensions of the flow field.h the number of dimensions of the flow field.

60

Steady and Unsteady FlowsSteady and Unsteady Flows

vvSteady flow: the properties at every point in a flow fieSteady flow: the properties at every point in a flow field ld do not change with timedo not change with time..

where where ηη represents any fluid propertyrepresents any fluid property..vvUnsteady flow:…. Unsteady flow:…. Change with time.Change with time.ððNonperiodic flow, periodic flow, and truly random floNonperiodic flow, periodic flow, and truly random flo

w.w.ððMore difficult to analyze.More difficult to analyze.

0t

=¶h¶

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61

Streamlines Streamlines 1/21/2

vvStreamline: Line drawn in the flow field so that at a given Streamline: Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every pinstant they are tangent to the direction of flow at every point in the flow field. >>> No flow across a streamline.oint in the flow field. >>> No flow across a streamline.ØØStreamline is everywhere tangent to the velocity field.Streamline is everywhere tangent to the velocity field.ØØIf the flow is steady, nothing at a fixed point changes If the flow is steady, nothing at a fixed point changes

with time, so the streamlines are fixed lines in space.with time, so the streamlines are fixed lines in space.ØØFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape

with time.with time.ØØStreamlines are obtained analytically by integrating thStreamlines are obtained analytically by integrating th

e equations defining lines tangent to the velocity field.e equations defining lines tangent to the velocity field.

62

Streamlines Streamlines 2/2 2/2

ØØFor two dimensional flows the slope of the streamliFor two dimensional flows the slope of the streamline, dy/dx, must be equal to the tangent of the angle ne, dy/dx, must be equal to the tangent of the angle that the velocity vector makes with the x axisthat the velocity vector makes with the x axis

uv

dxdy

= If the velocity field is knoIf the velocity field is known as a function of x and wn as a function of x and y, this equation can be intey, this equation can be integrated to give the equation grated to give the equation of streamlines.of streamlines.

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63

Example: Example: Streamlines for a Given Streamlines for a Given Velocity Velocity FieldField

•• Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional stdimensional steady flow discussed in Example 4.1, eady flow discussed in Example 4.1,

( )( )jyix/VV 0rr

lr

-= ！

Figure E4.2

64

Example Example SolutionSolution

( )( ) x

yx/Vy/V

uv

dxdy

0

0 -=-

==l

l

( ) ( )y/Vvandx/Vu 00 ll -==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

Integrating….Integrating….

ttanconsxlnylnorx

dxy

dy+-=-= òò

The streamline is xy=C, where C is a constantThe streamline is xy=C, where C is a constant

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65

AccelerationAcceleration

vvFor Lagrangian method, the fluid acceleration is descrFor Lagrangian method, the fluid acceleration is described as done in solid body dynamicsibed as done in solid body dynamics

vvFor Eulerian method, the fluid acceleration is describFor Eulerian method, the fluid acceleration is described as function of position and time without actually fed as function of position and time without actually following any particles.ollowing any particles.

)t(aarr

=

)t,z,y,x(aarr

=

66

Acceleration Field Acceleration Field 1/51/5

vvThe acceleration of a fluid particle for use in Newton’The acceleration of a fluid particle for use in Newton’s second law is:s second law is:

vvTThe problem is : Given the velocity fieldhe problem is : Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle,,

dt/Vdarr

=

)t,z,y,x(VVrr

=

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67

Acceleration Field Acceleration Field 2/52/5

vThe velocity of a fluid particle A in space at time t:

vThe velocity of a fluid particle in space at time t+dt:)t,z,y,x(VV AtA

rr=

)dtt,dzz,dyy,dxx(VV AdttA ++++=+

rr

rdr rr+rr

vThe change in the velocity of the particle, in moving from location to , is given by the chain rule:

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA ¶

¶+

¶¶

+¶

¶+

¶¶

=rrrr

r

68

Acceleration Field Acceleration Field 3/53/5

tV

zVw

yVv

xVu

dtVda

wdt

dz,vdt

dy,udt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

¶¶

+¶

¶+

¶¶

+¶

¶==Þ

===

¶¶

+¶

¶+

¶¶

+¶

¶==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

¶¶

+¶¶

+¶¶

+¶¶

=rrrr

r

Valid for any particle…..

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69

Acceleration Field Acceleration Field 4/54/5

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

¶¶

+¶¶

+¶¶

+¶¶

=

¶¶

+¶¶

+¶¶

+¶¶

=

¶¶

+¶¶

+¶¶

+¶¶

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components

70

Acceleration Field Acceleration Field 5/55/5

is termed the material derivative or suis termed the material derivative or substantial derivative.bstantial derivative.

DtVDar

r=

( ) ( ) ( ) ( ) ( )

( ) ( )( )Ñ×+¶

¶=

¶¶

+¶¶

+¶¶

+¶

¶=

Vt

zw

yv

xu

tDtD

r

Where the operatorWhere the operator

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71

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

vvTime derivative: Local derivative.Time derivative: Local derivative.vvIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

tV¶¶r

Local acceleration

vvSpatial derivative: Convective derivative.Spatial derivative: Convective derivative.vvIt represents the fact that a flow property associated wIt represents the fact that a flow property associated w

ith a fluid particle may vary because of the motion of ith a fluid particle may vary because of the motion of the particle from one point in space to another point.the particle from one point in space to another point.

V)V(rv

Ñ× Convective acceleration

72

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

Ñ×+¶¶

=¶¶

+¶¶

+¶¶

+¶¶

=Þ

¶¶

+¶¶

+¶¶

+¶

¶=

r

vThe material derivative concept is very useful in analysis involving various parameter, not just the acceleration.

vFor example, consider a temperature field T=T(x,y,z,t) associated with a given flow. We can apply the chain rule to determine the rate of change of temperature as

2012-12-24

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73

Example: Example: Acceleration from a Given VeAcceleration from a Given Velocity Fieldlocity Field

•• Consider the steady, twoConsider the steady, two--dimensional flow field discussed in dimensional flow field discussed in pprevious revious ExampleExample. . Determine the acceleration field for this flow.Determine the acceleration field for this flow.

( ) )jyix(/VV 0

rrl

r-=

74

Example Example SolutionSolution

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

¶¶

+¶¶

+¶¶

+¶¶

=Ñ×+¶¶

==rrrr

rrrr

r

In general, the acceleration is given byIn general, the acceleration is given by

u = u = ((VV00/ / ll )x and v = )x and v = --(V(V00/ / ll )y)yFor steady, twoFor steady, two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r÷÷ø

öççè

涶

+¶¶

+÷÷ø

öççè

涶

+¶¶

=¶¶

+¶¶

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r÷ø

öçè

æ÷øö

çèæ-÷

øö

çèæ-+÷

øö

çèæ+÷

ø

öçè

æ÷øö

çèæ+÷

øö

çèæ

÷øö

çèæ=

2

20

y2

20

xyVaxVa

ll==

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75

FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Analysis Analysis of Fluid Flowof Fluid Flow

vvFluid Element KinematicsFluid Element KinematicsvvConservation of MassConservation of MassvvConservation of Linear MomentumConservation of Linear Momentum

76

Motion of a Fluid ElementMotion of a Fluid Element

vv Fluid Translation: The element moves from one point to another.Fluid Translation: The element moves from one point to another.vv Fluid Rotation: The element rotates about any or all of the Fluid Rotation: The element rotates about any or all of the x,y,zx,y,z aa

xesxes..vv Fluid Deformation:Fluid Deformation:

__Angular DeformationAngular Deformation: The : The element’s angles between the sides element’s angles between the sides change.change.

__Linear Linear Deformation:TheDeformation:The element’s sides stretch or contract.element’s sides stretch or contract.

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77

Fluid Translation Fluid Translation velocity and accelerationvelocity and acceleration

vvThe velocity of a fluid The velocity of a fluid particle particle can be expressedcan be expressed

vvThe The total accelerationtotal acceleration of the particle is given byof the particle is given bykwjviu)t,z,y,x(VVrrrrv

++==

zVw

yVv

xVu

tV

DtVDa

wdtdz,v

dtdy,u

dtdx

dtdz

zV

dtdy

yV

dtdx

xV

tV

DtVDa

¶¶

+¶¶

+¶¶

+¶¶

==Þ

===

¶¶

+¶¶

+¶¶

+¶¶

==

rrrrrr

rrrrrr

tDVDar

r= is called the material , or substantial derivative.is called the material , or substantial derivative.

Acceleration field

Velocity field

78

Acceleration Acceleration ComponentComponent

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

¶¶

+¶¶

+¶¶

+¶¶

=

¶¶

+¶¶

+¶¶

+¶¶

=

¶¶

+¶¶

+¶¶

+¶¶

=

Rectangular coordinRectangular coordinates systemates system

zVVV

rV

rVV

tVa

zVV

rVVV

rV

rVV

tVa

zVV

rVV

rV

rVV

tVa

zz

zzr

zz

zr

r

rz

2rr

rr

r

¶¶

+q¶

¶+

¶¶

+¶

¶=

¶¶

++q¶

¶+

¶¶

+¶

¶=

¶¶

+-q¶

¶+

¶¶

+¶¶

=

q

qqqqqqq

qq

Cylindrical coordinCylindrical coordinates systemates system

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79

Linear TranslationLinear Translation

vvAll points in the element havAll points in the element have the same velocity (which is e the same velocity (which is only true if there are only true if there are nno veloco velocity gradients), then the elemeity gradients), then the element will simply translate from nt will simply translate from one position to another.one position to another.

80

Linear Deformation Linear Deformation 1/1/33

vvThe shape of the fluid element, described by the angleThe shape of the fluid element, described by the angles at its vertices, remains unchanged, since s at its vertices, remains unchanged, since all right anall right angles continue to be right anglesgles continue to be right angles..

vvA change in the x dimension requires a nonzero value A change in the x dimension requires a nonzero value of of

vvA ……………… y A ……………… y vvA ……………… z A ……………… z z/w ¶¶

y/v ¶¶

x/u ¶¶

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81

Linear Deformation Linear Deformation 2/2/33

vvThe change in length of the sides may produce changThe change in length of the sides may produce change in volume of the element.e in volume of the element.

The change inThe change in )t)(zy(xxuV ddd÷

øö

çèæ d¶¶

=d

The rate at which the The rate at which the ddV is changing per unit volume due tV is changing per unit volume due to gradient o gradient ¶¶u/ u/ ¶¶xx

( )xu

dtVd

V1

¶¶

=d

d

82

If If ¶¶v/ v/ ¶¶y and y and ¶¶w/ w/ ¶¶z are involvedz are involved

Linear DeformationLinear Deformation 33//33

( ) Vzw

yv

xu

dtVd

V1 r

×Ñ=¶¶

+¶¶

+¶¶

=d

d

Volumetric dilatation rate

2012-12-24

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83

Angular Rotation Angular Rotation 1/41/4

δtδαω lim

0δtOA

®

=The angular velocity of line OA

txv

x

txxv

tan d¶¶

=d

dd÷øö

çè涶

=da=da &For small angles

xv

OA ¶¶

=w

yu

OB ¶¶

=w

CWCW

CCWCCW

“-” for CCW

84

Angular Rotation Angular Rotation 2/42/4

( ) ÷÷ø

öççè

涶

-¶¶

=w+w=wyu

xv

21

21

OBOAz

The rotation of the element about the zThe rotation of the element about the z--axis is defined as the axis is defined as the average of the angular veaverage of the angular velocities locities wwOAOA and and wwOBOB of the two mutually perpendicular lines OA and OB.of the two mutually perpendicular lines OA and OB.

÷÷ø

öççè

涶

-¶¶

=wzv

yw

21

x ÷øö

çèæ

¶¶

-¶¶

=xw

zu

21

yw

kji zyxrvvr

w+w+w=wIn vector formIn vector form

2012-12-24

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85

Angular Rotation Angular Rotation 3/43/4

úúû

ù

êêë

é÷÷ø

öççè

涶

-¶¶

+÷øö

çèæ

¶¶

-¶¶

+÷÷ø

öççè

涶

-¶¶

=w

÷øö

çèæ

¶¶

-¶¶

=w÷÷ø

öççè

涶

-¶¶

=w

kyu

xvj

xw

zui

zv

yw

21

xw

zu

21

zv

yw

21

yx

rrrr

÷÷ø

öççè

涶

-¶¶

=wyu

xv

21

z

kyu

xv

21j

xw

zu

21i

zv

yw

21V

21Vcurl

21 rrrrrr

úû

ùêë

鶶

-¶¶

+úûù

êëé

¶¶

-¶¶

+úû

ùêë

鶶

-¶¶

=´Ñ==w

Defining vorticityDefining vorticity V2rr

´Ñ=w=zDefining irrotationDefining irrotation 0V =´Ñ

r

86

Angular Rotation Angular Rotation 4/44/4

kyu

xv

21j

xw

zu

21i

zv

yw

21

wvuzyx

kji

21V

21Vcurl

21

rrr

rrr

rrr

úû

ùêë

鶶

-¶¶

+úûù

êëé

¶¶

-¶¶

+úû

ùêë

鶶

-¶¶

=

¶¶

¶¶

¶¶

=´Ñ==w

2012-12-24

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87

VorticityVorticity

vv Defining Vorticity ζ whichDefining Vorticity ζ which is a measurement of the rotation is a measurement of the rotation of a fluid elementof a fluid element as it moves in the flow field:as it moves in the flow field:

vv In cylindrical coordinates systemIn cylindrical coordinates system::

V21k

yu

xvj

xw

zui

zv

yw

21 rrrrr

´Ñ=úúû

ù

êêë

é÷÷ø

öççè

涶

-¶¶

+÷øö

çèæ

¶¶

-¶¶

+÷÷ø

öççè

涶

-¶¶

=w

VVcurl2rrrr

´Ñ==w=z

÷øö

çèæ

q¶¶

-¶

¶+÷

øö

çèæ

¶¶

-¶¶

+÷øö

çèæ

¶¶

-q¶

¶=´Ñ q

qq r

zzrz

rV

r1

rrV

r1e

rV

zVe

zVV

r1eV

rrrr

88

Angular Deformation Angular Deformation 1/21/2

vv Angular deformation of a particle is given by the sum of the tAngular deformation of a particle is given by the sum of the two angular deformationwo angular deformation

db+da=dg

÷÷ø

öççè

涶

+¶¶

=÷÷ø

öççè

涶

+¶¶

=+

=®® y

uxv

t

tyut

xv

t tt d

dd

ddbdag

dd 00limlim&

Rate of shearing strain or the rate of angular deformationRate of shearing strain or the rate of angular deformation

tyu

y

tyyu

txv

x

txxv

dd

dddbdbd

d

dddada

¶¶

=÷÷ø

öççè

涶

==¶¶

=÷øö

çè涶

== && tan,tan

2012-12-24

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89

txv

x

txxv

tan d¶¶

=d

dd÷øö

çè涶

=da=da &For small angles

90

Angular Deformation Angular Deformation 2/22/2

v The rate of angular deformation in xy plane

v The rate of angular deformation in yz plane

v The rate of angular deformation in zx plane

÷÷ø

öççè

涶

+¶¶

yu

xv

÷÷ø

öççè

涶

+¶¶

zv

yw

÷øö

çèæ

¶¶

+¶¶

zu

xw

2012-12-24

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91

Example: Example: VorticityVorticity•• For a certain twoFor a certain two--dimensional flow field dimensional flow field thth evelocityevelocity is given bis given b

y y

Is this flow Is this flow irrotationalirrotational? ?

j)yx(2ixy4V 22rrr

-+=

92

Example Example SolutionSolution

0wyxvxy4u 22 =-==

0xw

zu

21

0zv

yw

21

y

x

=÷øö

çèæ

¶¶

-¶¶

=w

=÷÷ø

öççè

涶

-¶¶

=w

0yu

xv

21

z =÷÷ø

öççè

涶

-¶¶

=w

This flow is irrotationalThis flow is irrotational

2012-12-24

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93

Conservation of Mass Conservation of Mass 1/51/5

vvWith field representation, the property fields are definWith field representation, the property fields are defined by continuous functions of the space coordinates aed by continuous functions of the space coordinates and time.nd time.

vvTo derive the differential equation for conservation of To derive the differential equation for conservation of mass in rectangular and in cylindrical coordinate systmass in rectangular and in cylindrical coordinate system.em.

vvThe derivation is carried out by applying conservatioThe derivation is carried out by applying conservation of mass to a differential control volume.n of mass to a differential control volume.

The differential form of continuity equation???The differential form of continuity equation???

94

Conservation of Mass Conservation of Mass 2/52/5

vv The CV chosen is an infinitesimal cube with sides of length The CV chosen is an infinitesimal cube with sides of length ddx, x, dd y, and y, and dd z.z.

zyxtpVd

t CV ddd¶¶

=r¶¶ò

( )2x

xuu|u

2dxx

d¶r¶

+r=r÷øö

çèæ+

( )2x

xuu|u

2xx

d¶r¶

-r=r÷øö

çèæ d

-

Net rate of mass Outflow in x-direction

2012-12-24

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95

Conservation of Mass Conservation of Mass 3/53/5

Net rate of mass Net rate of mass Outflow in xOutflow in x--directiondirection

( ) ( ) ( ) zyxxuzy

2x

xuuzy

2x

xuu ddd

¶r¶

=ddúûù

êëé d

¶r¶

-r-ddúûù

êëé d

¶r¶

+r=

Net rate of mass Net rate of mass Outflow in yOutflow in y--directiondirection

( ) zyxyv

ddd¶r¶

=×××××=

Net rate of mass Net rate of mass Outflow in zOutflow in z--directiondirection

( ) zyxzw

ddd¶r¶

=×××××=

96

Conservation of Mass Conservation of Mass 4/54/5

Net rate of mass Outflow

( ) ( ) ( ) zyxzw

yv

xu

dddúû

ùêë

é¶r¶

+¶r¶

+¶r¶

The differential equation for conservation of massThe differential equation for conservation of mass

( ) ( ) ( ) 0Vtz

wyv

xu

t=r×Ñ+

¶r¶

=¶r¶

+¶r¶

+¶r¶

+¶r¶ r

Continuity equationContinuity equation

2012-12-24

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97

Conservation of Mass Conservation of Mass 5/55/5

vvIncompressible fluidIncompressible fluid

vvSteady flowSteady flow

0Vzw

yv

xu

=×Ñ=¶¶

+¶¶

+¶¶ r

0Vz

)w(y

)v(x

)u(=r×Ñ=

¶r¶

+¶r¶

+¶r¶ r

98

Example: Example: Continuity EquationContinuity Equation•• The velocity components for a certain incompressible, steady fThe velocity components for a certain incompressible, steady f

low field arelow field are

Determine the form of the z component, w, required to satisfy Determine the form of the z component, w, required to satisfy the continuity equation.the continuity equation.

?wzyzxyvzyxu 222

=++=++=

2012-12-24

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99

Example Example SolutionSolution0

zw

yv

xu

=¶¶

+¶¶

+¶¶The continuity equationThe continuity equation

)y,x(f2zxz3w

zx3)zx(x2zw

zxyv

z2xu

2

+--=Þ

--=+--=¶¶

+=¶¶

=¶¶

100

Conservation of MassConservation of MassCylindrical Coordinate System Cylindrical Coordinate System 1/31/3

vvThe CV chosen is an infinitesimal cube with sides of lThe CV chosen is an infinitesimal cube with sides of length ength drdr, , rdθrdθ, and , and dzdz..

vvThe net rate of mass flux out through the control surfaThe net rate of mass flux out through the control surfacece

zrzVrV

rVrV zr

r dqddúûù

êëé

¶r¶

+q¶

r¶+

¶r¶

+r q

vv The rate of change of mass inside the cThe rate of change of mass inside the control volumeontrol volume

drdzrdt

qr¶¶

2012-12-24

51

101

Conservation of MassConservation of MassCylindrical Coordinate System Cylindrical Coordinate System 2/32/3

The continuity equationThe continuity equation

By “Del” operatorBy “Del” operator

The continuity equation becomesThe continuity equation becomes

0z

)V()V(r1

r)Vr(

r1

tzr =

¶r¶

+q¶

r¶+

¶r¶

+¶r¶ q

zk

r1e

rer ¶

¶+

q¶¶

+¶¶

=Ñ q

rrr

0Vt

=r×Ñ+¶r¶ r

102

Conservation of MassConservation of MassCylindrical Coordinate SystemCylindrical Coordinate System 3/33/3

vvIncompressible fluidIncompressible fluid

vvSteady flowSteady flow

0Vz

)V()V(r1

r)rV(

r1 zr =×Ñ=

¶¶

+q¶

¶+

¶¶ q

r

0Vz

)V()V(r1

r)Vr(

r1 zr =r×Ñ=

¶r¶

+q¶

r¶+

¶r¶ q

r

2012-12-24

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103

Stream Function Stream Function 1/61/6

vv Streamlines ?Streamlines ? Lines tangent to the instantaneous velocity vectLines tangent to the instantaneous velocity vectors at every point.ors at every point.

vv Stream function ΨStream function Ψ(x,y)(x,y) [Psi] ? Used to represent the velocity [Psi] ? Used to represent the velocity component u(x,y,t) and v(x,y,t) of component u(x,y,t) and v(x,y,t) of a twoa two--dimensional incompredimensional incompressible flow.ssible flow.

vv Define a function ΨDefine a function Ψ(x,y), called the stream function, which rel(x,y), called the stream function, which relates the velocities shown by the figure in the margin asates the velocities shown by the figure in the margin as

xv

yu

¶y¶

-=¶y¶

=

104

Stream Function Stream Function 2/62/6

vv The stream function ΨThe stream function Ψ(x,y) (x,y) satisfies the twosatisfies the two--dimensional fordimensional form of the incompressible continuity equationm of the incompressible continuity equation

vv ΨΨ(x,y) (x,y) ?? Still unknown for a particular problem, but at least Still unknown for a particular problem, but at least we have simplify the analysis by having to determine only one we have simplify the analysis by having to determine only one unknown, unknown, ΨΨ(x,y)(x,y) , rather than the two function u(x,y) and v(x,, rather than the two function u(x,y) and v(x,y).y).

0xyyx

0yv

xu 22

=¶¶y¶

-¶¶y¶

Þ=¶¶

+¶¶

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105

Stream Function Stream Function 3/63/6

vv Another advantage of using stream function is related to the faAnother advantage of using stream function is related to the fact that line along which ct that line along which ΨΨ(x,y) =constant(x,y) =constant are streamlines.are streamlines.

vv How to prove ? From the definition of the streamline that the sHow to prove ? From the definition of the streamline that the slope at any point along a streamline is given bylope at any point along a streamline is given by

uv

dxdy

streamline

=÷øö

Velocity and velocity component along a streamlineVelocity and velocity component along a streamline

106

Stream Function Stream Function 4/64/6

vvThe change of ΨThe change of Ψ(x,y) as we move from one point (x,(x,y) as we move from one point (x,y) to a nearly point (x+dx,y+dy) is given byy) to a nearly point (x+dx,y+dy) is given by

0udyvdx0d

udyvdxdyy

dxx

d

=+->>=y>>

+-=¶y¶

+¶y¶

=y

uv

dxdy

streamline

=÷øö

Along a line of constant ΨAlong a line of constant Ψ

This is the definition for a streamline. Thus, if we know the function Ψ(x,y) we cThis is the definition for a streamline. Thus, if we know the function Ψ(x,y) we can plot lines of constant Ψto provide the family of streamlines that are helpful in an plot lines of constant Ψto provide the family of streamlines that are helpful in visualizing the pattern of flow. There are an infinite number of streamlines that mvisualizing the pattern of flow. There are an infinite number of streamlines that make up a particular flow field, since for each constant value assigned to Ψa streaake up a particular flow field, since for each constant value assigned to Ψa streamline can be drawn.mline can be drawn.

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107

Stream Function Stream Function 5/65/6

vv The actual numerical value associated with a particular streamlThe actual numerical value associated with a particular streamline is not of particular significance, but the change in the valuine is not of particular significance, but the change in the value of Ψe of Ψ is related to the volume rate of flow.is related to the volume rate of flow.

122

1

yyy

yyy

y

y-==

=¶¶

+¶¶

=-=

ò dq

ddxx

dyy

vdxudydq

108

Stream Function Stream Function 6/66/6

vv Thus the volume flow rate between any two streamlines can be Thus the volume flow rate between any two streamlines can be written as the difference between the constant values of Ψ defiwritten as the difference between the constant values of Ψ defining two streamlines.ning two streamlines.

vv The velocity will be relatively high wherever the streamlines aThe velocity will be relatively high wherever the streamlines are close together, and relatively low wherever the streamlines are close together, and relatively low wherever the streamlines are far apart.re far apart.

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109

figun_06_p

274a

110

Stream Function Stream Function Cylindrical Coordinate SystemCylindrical Coordinate System

vvFor a twoFor a two--dimensional, incompressible flow in the rdimensional, incompressible flow in the rθθplane, conservation of mass can be written as:plane, conservation of mass can be written as:

vvThe velocity components can be related to the stream The velocity components can be related to the stream function, Ψ(r,function, Ψ(r,θθ) through the equation) through the equation

rvand

r1vr ¶

y¶-=

q¶y¶

= q

0vr

)rv( r =q¶

¶+

¶¶ q

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111

Example: Example: Stream FunctionStream Function•• The velocity component in a steady, incompressible, two dimeThe velocity component in a steady, incompressible, two dime

nsional flow field arensional flow field are

Determine the corresponding stream function and show on a skDetermine the corresponding stream function and show on a sketch several streamlines. Indicate the direction of glow along tetch several streamlines. Indicate the direction of glow along the streamlines.he streamlines.

4xv2yu ==

112

Example Example SolutionSolution

(y)fx2(x)fy 22

12 +-=y+=y

Cyx2 22 ++-=y

From the definition of the stream functionFrom the definition of the stream function

x4x

vy2y

u =¶y¶

-==¶y¶

=

For simplicity, we set C=0For simplicity, we set C=0

22 yx2 +-=yΨ=0Ψ=0

ΨΨ≠≠0012/

,222

=-±=yy

xyxy

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113

Conservation of Linear MomentumConservation of Linear Momentum

vv Applying Newton’s second law to control volumeApplying Newton’s second law to control volume

( )

amDt

VDm

zVw

yVv

xVu

tVm

tDmVDF

rr

rrrrrr

d=d=

÷÷ø

öççè

涶

+¶¶

+¶¶

+¶¶

d=d

=d

SYSDt

PDFr

r=

For a For a infinitesimal system of mass dminfinitesimal system of mass dm, what’s the , what’s the TThe differential form of linear momentum equation?he differential form of linear momentum equation?

VdVdmVP)system(V)system(Msystem r== òòrrr

114

Forces Acting on Element Forces Acting on Element 1/21/2

v The forces acting on a fluid element may be classified as body forces and surface forces; surface forces include normal forcenormal forces and tangentials and tangential (shear) forces.

kFjFiF

kFjFiF

FFF

bzbybx

szsysx

BS

rrr

rrr

rrr

d+d+d+

d+d+d=

d+d=dSurface forces acting on a fluid Surface forces acting on a fluid element can be described in terelement can be described in terms of normal and shearing stressms of normal and shearing stresses.es.

AFn

tn dd

sd 0lim

®=

AF

t dd

td

1

01 lim®

=AF

t dd

td

2

02 lim®

=

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115fig_06_11

116

Double Subscript Notation for StressesDouble Subscript Notation for Stresses

xytThe direction of thThe direction of the normal to the plae normal to the plane on which the strne on which the stress actsess acts

The direction of the stressThe direction of the stress

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117

Forces Acting on Element Forces Acting on Element 2/22/2

zyxgFzyxgFzyxgF

zyxzyx

F

zyxzyx

F

zyxzyx

F

zbz

yby

xbx

zzyzxzsz

zyyyxysy

zxyxxxsx

dddr=d

dddr=ddddr=d

ddd÷÷ø

öççè

æ¶s¶

+¶t¶

+¶t¶

=d

ddd÷÷ø

öççè

æ¶t¶

+¶s¶

+¶t¶

=d

ddd÷÷ø

öççè

æ¶t¶

+¶t¶

+¶s¶

=d

Equation of MotionEquation of Motion

118

Equation of MotionEquation of Motion

These are the differential equations of motion for anyThese are the differential equations of motion for anyfluid fluid satisfying the continuum assumptionsatisfying the continuum assumption..How to solve u,v,w ?How to solve u,v,w ?

÷÷ø

öççè

涶

+¶¶

+¶¶

+¶¶

r=¶s¶

+¶t¶

+¶t¶

+r

÷÷ø

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涶

+¶¶

+¶¶

+¶¶

r=¶t¶

+¶s¶

+¶t¶

+r

÷÷ø

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涶

+¶¶

+¶¶

+¶¶

r=¶t¶

+¶t¶

+¶s¶

+r

zww

ywv

xwu

tw

zyxg

zvw

yvv

xvu

tv

zyxg

zuw

yuv

xuu

tu

zyxg

zzyzxzz

zyyyxyy

zxyxxxx

zzyyxx maFmaFmaF d=dd=dd=d

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119

Inviscid FlowInviscid Flow

vv Shearing stresses develop in a moving fluid because of the viscShearing stresses develop in a moving fluid because of the viscosity of the fluid.osity of the fluid.

vv For some common fluid, such as air and water, the viscosity is For some common fluid, such as air and water, the viscosity is small, and therefore it seems reasonable to assume that under ssmall, and therefore it seems reasonable to assume that under some circumstances we may be able to simply neglect the effect ome circumstances we may be able to simply neglect the effect of viscosity.of viscosity.

vv Flow fields in which the shearing stresses are assumed to be neFlow fields in which the shearing stresses are assumed to be negligible are said to be inviscid, nonviscous, or frictionless.gligible are said to be inviscid, nonviscous, or frictionless.

zzyyxxp s=s=s=-Define the pressure, p, as the negative of the normal stressDefine the pressure, p, as the negative of the normal stress

120

Euler’s Equation of Motion Euler’s Equation of Motion

vUnder frictionless conditionfrictionless condition, the equations of motion are reduced to Euler’s Equation:Euler’s Equation:

÷÷ø

öççè

涶

+¶¶

+¶¶

+¶¶

r=¶¶

-r

÷÷ø

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涶

+¶¶

+¶¶

+¶¶

r=¶¶

-r

÷÷ø

öççè

涶

+¶¶

+¶¶

+¶¶

r=¶¶

-r

zww

ywv

xwu

tw

zpg

zvw

yvv

xvu

tv

ypg

zuw

yuv

xuu

tu

xpg

z

y

x

÷÷ø

öççè

æÑ×+

¶¶

r=Ñ-r V)V(tVpg

rrr

r

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121

Bernoulli Equation Bernoulli Equation 1/31/3

vvEuler’s equation for steady flowEuler’s equation for steady flow along a streamline isalong a streamline is

V)V(pgrrr

Ñ×r=Ñ-r

( ) ( ) ( )VVVV21VV

rrrrrr´Ñ´-×Ñ=Ñ×

zgg Ñ-=r

)()(2

VVVVpzgrrrr

´Ñ´-×Ñ=Ñ-Ñ- rrr

Selecting the coordinate system with the zSelecting the coordinate system with the z--axis vertical so axis vertical so that the acceleration of gravity vector can be expressed asthat the acceleration of gravity vector can be expressed as

Vector identity ….Vector identity ….

122

Bernoulli Equation Bernoulli Equation 2/32/3

( ) ( )[ ] sdVVsdzgsdV21sdp 2 rrrvrr

×´Ñ´=×Ñ+×Ñ+×rÑ

( )VVrr

´Ñ´ perpendicular to perpendicular to Vr( ) ( )VVzgV

21p 2

rr´Ñ´=Ñ+Ñ+

rÑ

sdr×

With With kdzjdyidxsdrrsr

++=

dpdzzpdy

ypdx

xpsdp =

¶¶

+¶¶

+¶¶

=×Ñr

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123

figun_06_p

280b

124

Bernoulli Equation Bernoulli Equation 3/33/3

( ) 0gdzVd21dp 2 =++

r

ttanconsgz2

Vdp 2

=++rò

ttanconsgz2

Vp 2

=++r

( ) 0sdzgsdV21sdp 2 =×Ñ+×Ñ+×

rÑ rrr

Integrating …Integrating …

For steady inviscid, incompressible fluid ( commonly called ideFor steady inviscid, incompressible fluid ( commonly called ideal fluids) al fluids) along a streamlinealong a streamline

Bernoulli equationBernoulli equation

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125

Irrotational Flow Irrotational Flow 1/41/4

vIrrotation ? The irrotational condition is

[In rectangular coordinates system

[In cylindrical coordinates system

0V =´Ñr

0xw

zu

zv

yw

yu

xv

=¶¶

-¶¶

=¶¶

-¶¶

=¶¶

-¶¶

0Vr1

rrV

r1

rV

zV

zVV

r1 rzrz =

q¶¶

-¶

¶=

¶¶

-¶¶

=¶¶

-q¶

¶ qq

126

Irrotational Flow Irrotational Flow 2/42/4

vvA general flow field would not be irrotational flow.A general flow field would not be irrotational flow.vvA special uniform flow field is an example of an irrotA special uniform flow field is an example of an irrot

ation flowation flow

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127

Irrotational Flow Irrotational Flow 3/43/4

v A general flow fieldðð A solid body is placed in a uniform stream of fluid. Far away fA solid body is placed in a uniform stream of fluid. Far away f

rom the body remain uniform, and in this far region the flow is rom the body remain uniform, and in this far region the flow is irrotational. irrotational.

ðð The flow around the body remains irrotational except very neThe flow around the body remains irrotational except very near the boundary.ar the boundary.

ðð Near the boundary the veloNear the boundary the velocity changes rapidly from zcity changes rapidly from zero at the boundary (noero at the boundary (no--slislip condition) to some relativp condition) to some relatively large value in a short diely large value in a short distance from the boundary.stance from the boundary. Chapter 9Chapter 9

128

Irrotational Flow Irrotational Flow 4/44/4

v A general flow fieldðð Flow from a large reservoir enters a pipe through a streamlined Flow from a large reservoir enters a pipe through a streamlined

entrance where the velocity distribution is essentially uniform. entrance where the velocity distribution is essentially uniform. Thus, at entrance the flow is irrotational. (b)Thus, at entrance the flow is irrotational. (b)

ðð In the central core of the pipe the flow remains irrotational for In the central core of the pipe the flow remains irrotational for some distance.some distance.

ðð The boundary layer will develop along the wall and grow in thiThe boundary layer will develop along the wall and grow in thickness until it fills the pipe.ckness until it fills the pipe. Viscous forces are dominantViscous forces are dominant

Chapter 8Chapter 8

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129

Bernoulli Equation for Bernoulli Equation for IrrotationalIrrotational Flow Flow 1/31/3

vv The Bernoulli equation forThe Bernoulli equation for steady, incompressible, and inviscsteady, incompressible, and inviscid flowid flow isis

vv The equation can be applied betweenThe equation can be applied between any two points on the sany two points on the same streamlineame streamline. . In general,In general, the value of the constant will vathe value of the constant will vary from streamline to streamlinery from streamline to streamline..

vv Under additionalUnder additional irrotational conditionirrotational condition, , the Bernoulli equatiothe Bernoulli equation ?n ? Starting with Euler’s equation in vector formStarting with Euler’s equation in vector form

( ) ( )VVVV21kgp1V)V(

rrrrrrr´Ñ´-×Ñ=-Ñ

r-=Ñ×

ttanconsgz2

Vp 2

=++r

ZERO Regardless of the direction of dsZERO Regardless of the direction of ds

130

Bernoulli Equation for Bernoulli Equation for IrrotationalIrrotational Flow Flow 2/32/3

vWith irrotationalirrotational condition

( ) ( ) kgp1V21VV

21 2

rrr-Ñ

r-=Ñ=×Ñ

0V =´Ñr

rdr×

( )

( ) ( ) 0gdzVd21dpgdzdpVd

21

rdkgrdp1rdV21

22

2

=++r

>>-r

-=>>

×-×Ñr

-=×Ñ vrvv

( ) ( )VVVV21kgp1V)V(

rrrrrrr´Ñ´-×Ñ=-Ñ

r-=Ñ×

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131

Bernoulli Equation for Bernoulli Equation for IrrotationalIrrotational Flow Flow 3/33/3

Integrating for incompressible flowIntegrating for incompressible flow

This equation is valid between any two points in a steaThis equation is valid between any two points in a steady, incompressible, inviscid, and irrotational flow.dy, incompressible, inviscid, and irrotational flow.

ttanconsgz2

Vp 2

=++r

ttancongz2

Vdp 2

=++rò

2

222

1

211 z

g2Vpz

g2Vp

++g

=++g

132

Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure1/51/5

vvEach term in the Bernoulli equation can be interpreteEach term in the Bernoulli equation can be interpreted as a form of pressure.d as a form of pressure.

ððpp is the actual thermodynamic pressure of the fluid as is the actual thermodynamic pressure of the fluid as it flows. To measure this pressure, one must move aloit flows. To measure this pressure, one must move along with the fluid, thus being “ng with the fluid, thus being “staticstatic” relative to the m” relative to the moving fluid. Hence, it is termed the oving fluid. Hence, it is termed the static pressurestatic pressure … … seen by the fluid particle as it movesseen by the fluid particle as it moves. .

Cz2

Vp2

=g+r+ Each term can be interpreted as a form of pressure

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133

Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure2/52/5

ððThe static pressureThe static pressure is measured in a flowing fluid usinis measured in a flowing fluid using a wall pressure “tap”, or a static pressure probe.g a wall pressure “tap”, or a static pressure probe.

hhhphp 34133131 g=g+g=+g= ---The static pressureThe static pressureððggzz is termed the is termed the hydrostatic phydrostatic p

ressureressure. It is not actually a pr. It is not actually a pressure but does represent the essure but does represent the change in pressure possible dchange in pressure possible due to potential energy variatiue to potential energy variations of the fluid as a result of ons of the fluid as a result of elevation changes.elevation changes.

134

Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure3/53/5

ððrrVV22/2/2 is termed the is termed the dynamic pressuredynamic pressure. It can be interp. It can be interpreted as the pressure at the end of a small tube insertereted as the pressure at the end of a small tube inserted into the flow and pointing upstream. d into the flow and pointing upstream. After the initial After the initial transient motion has died out, the liquid will fill the tutransient motion has died out, the liquid will fill the tube to a height of H.be to a height of H.

vvThe fluid in the tube, including that at its tip (2), will The fluid in the tube, including that at its tip (2), will be stationary. That is, Vbe stationary. That is, V22=0, or point (2) is a stagnatio=0, or point (2) is a stagnation point.n point. 2

112 V21pp r+=Stagnation pressure

Static pressure Dynamic pressureDynamic pressure

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135

Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure4/54/5

vvThere is a stagnation point on any stationary body thaThere is a stagnation point on any stationary body that is placed into a flowing fluid. Some of the fluid flowt is placed into a flowing fluid. Some of the fluid flows “over” and some “under” the object.s “over” and some “under” the object.

vvThe dividing line is termed the The dividing line is termed the stagnation streamlinstagnation streamlinee and terminates at the stagnation point on the body.and terminates at the stagnation point on the body.

vvNeglecting the elevation efNeglecting the elevation effects, fects, the stagnation pressthe stagnation pressure is the largest pressure ure is the largest pressure obtainable along a given sobtainable along a given streamlinetreamline..

stagnation pointstagnation point

136

Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure5/55/5

vvThe The sum of the static pressure, dynamic pressure, sum of the static pressure, dynamic pressure, and hydrostatic pressure is termed the total pressuand hydrostatic pressure is termed the total pressurere..

vvThe Bernoulli equation is a statement that the total prThe Bernoulli equation is a statement that the total pressure remains constant along a streamline.essure remains constant along a streamline.

ttanconspz2

Vp T

2

==g+r+ Constant along a streamline

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137

The PitotThe Pitot--static Tube static Tube 1/51/5

r-=>>

r=-

==»

r+==

/)pp(2V

2/Vpp

pppzz

2/Vppp

43

243

14

41

232

vvKnowledge of the values of the static and sKnowledge of the values of the static and stagnation pressure in a fluid implies that thtagnation pressure in a fluid implies that the fluid speed can be calculated.e fluid speed can be calculated.

vvThis is This is the principle on which the Pitotthe principle on which the Pitot--sstatic tube is based.tatic tube is based.

Static pressureStatic pressure

Stagnation pressureStagnation pressure

138

fig_03_07b

2012-12-24

70

139

fig_03_e06a

140

fig_03_07a

2012-12-24

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141

Application of Bernoulli Equation Application of Bernoulli Equation 1/21/2

vvThe Bernoulli equation can be appliedThe Bernoulli equation can be applied between any tbetween any two points on a streamline providedwo points on a streamline provided that the other that the other ththree restrictionsree restrictions are satisfied. The result isare satisfied. The result is

2

22

21

21

1 z2Vpz

2Vp g+

r+=g+

r+

Restrictions : Steady flow.Restrictions : Steady flow.Incompressible flow.Incompressible flow.Frictionless flow.Frictionless flow.Flow along a streamline.Flow along a streamline.

142

Flowrate Measurement Flowrate Measurement inin pipes 1/5pipes 1/5

vvVarious flow meters are Various flow meters are governed by the governed by the BernouBernoulli and continuity equatilli and continuity equationsons..

2211

222

211

VAVAQ

V21pV

21p

==

r+=r+

( )[ ]2

12

212 )A/A(1

pp2AQ-

-=

r

The theoretical flowrateThe theoretical flowrate

Typical devices for measuring flowrate in pipesTypical devices for measuring flowrate in pipes

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143

Example: Example: VenturiVenturi MeterMeter•• Kerosene (SG = 0.85) flows through the Venturi meter shown iKerosene (SG = 0.85) flows through the Venturi meter shown i

n Figure E3.11 with flowrates between 0.005 and 0.050 mn Figure E3.11 with flowrates between 0.005 and 0.050 m33/s. /s. Determine the range in pressure difference, Determine the range in pressure difference, pp11 –– pp22, needed to , needed to measure these flowrates.measure these flowrates.

Known Q, Determine pKnown Q, Determine p11--pp22

144

Example Example SolutionSolution1/21/2

33O2H kg/m850)kg/m1000(85.0SG ==r=r

22

2

A2])A/A(1[Q

pp2

12

21

-r=-

For steady, inviscid, and incompressible flow, the relationship betFor steady, inviscid, and incompressible flow, the relationship between flowrate and pressure ween flowrate and pressure

The density of the flowing fluidThe density of the flowing fluid

The area ratioThe area ratio

36.0)m10.0/m006.0()D/D(/AA 221212 ===

( )[ ]2

12

212 )A/A(1

pp2AQ-r

-= Eq. 3.20Eq. 3.20

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145

Example Example SolutionSolution2/22/2

The pressure difference for the The pressure difference for the smallest flowratesmallest flowrate isis

The pressure difference for the The pressure difference for the largest flowratelargest flowrate isis

22

22

21 ])m06.0)(4/[(2)36.01()850)(05.0(pp

p-

=-

kPa116N/m1016.1 25 =´=

kPa16.1N/m1160])m06.0)(4/[(2

)36.01()kg/m850()/sm005.0(pp2

22

2323

21

==p

-=-

kPa116-ppkPa16.1 21 ££

146

Viscous FlowViscous Flow

vTo incorporate viscous effects into the differential analysis of fluid motion

XXStressStress--Deformation Relationship Deformation Relationship aaaaaa

÷÷ø

öççè

涶

+¶¶

+¶¶

+¶¶

r=¶s¶

+¶

t¶+

¶t¶

+r

÷÷ø

öççè

涶

+¶¶

+¶¶

+¶¶

r=¶

t¶+

¶

s¶+

¶

t¶+r

÷÷ø

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涶

+¶¶

+¶¶

+¶¶

r=¶t¶

+¶

t¶+

¶s¶

+r

zww

ywv

xwu

tw

zyxg

zvw

yvv

xvu

tv

zyxg

zuw

yuv

xuu

tu

zyxg

zzyzxzz

zyyyxyy

zxyxxxx

General equation of motioGeneral equation of motionn

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147

StressStress--Deformation Relationship Deformation Relationship 1/21/2

vvThe stresses must be eThe stresses must be expressed in terms of thxpressed in terms of the velocity and pressure e velocity and pressure field.field.

÷÷ø

öççè

涶

+¶¶

m=t=t

÷øö

çèæ

¶¶

+¶¶

m=t=t

÷÷ø

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涶

+¶¶

m=t=t

¶¶

m+×Ñm--=s

¶¶

m+×Ñm--=s

¶¶

m+×Ñm--=s

zv

yw

zu

xw

yu

xv

zw2V

32p

yv2V

32p

xu2V

32p

zyyz

zxxz

yxxy

zz

yy

xx

r

r

r

Cartesian coordinates

148

StressStress--Deformation Relationship Deformation Relationship 2/22/2

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çèæ

¶¶

+¶¶

m=t=t

÷øö

çèæ

q¶¶

+¶¶

m=t=t

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æq¶

¶+÷

øö

çèæ

¶¶

m=t=t

¶¶

m+-=s

÷øö

çèæ +

q¶¶

m+-=s

¶¶

m+-=s

qqq

qqq

qqq

rv

zv

vr1

zv

vr1

rv

rr

zv2p

rvv

r12p

rv2p

zrzrrz

zzz

rrr

zzz

r

rrr

Introduced into the differentialIntroduced into the differentialequation of motion….equation of motion….

Cylindrical polar coordinates

2012-12-24

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149

The NavierThe Navier--Stokes Equations Stokes Equations 1/51/5

vvThese obtained equations of motion are called the NaThese obtained equations of motion are called the Naviervier--Stokes Equations.Stokes Equations.

úû

ùêë

é÷øö

çèæ ×Ñ-

¶¶

m¶¶

+úúû

ù

êêë

é÷÷ø

öççè

涶

+¶¶

m¶¶

+úû

ùêë

é÷øö

çèæ

¶¶

+¶¶

m¶¶

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The NavierThe Navier--Stokes Equations Stokes Equations 2/52/5

Cylindrical polar coordinatesCylindrical polar coordinates

2012-12-24

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151

The NavierThe Navier--Stokes Equations Stokes Equations 3/53/5

vvUnderUnder incompressible flow with constant viscosity incompressible flow with constant viscosity conditionsconditions, , the Navierthe Navier--Stokes equations are reduced tStokes equations are reduced to:o:

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152

The NavierThe Navier--Stokes Equations Stokes Equations 4/54/5

vvUndeUnder r frictionless conditionfrictionless condition, , the equations of motion the equations of motion are reduced toare reduced to Euler’s EquationEuler’s Equation::

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y

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2012-12-24

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153

The NavierThe Navier--Stokes Equations Stokes Equations 5/55/5

vvThe NavierThe Navier--Stokes equations apply to both laminar anStokes equations apply to both laminar and turbulent flow, but for turbulent flow each velocity d turbulent flow, but for turbulent flow each velocity component fluctuates randomly with respect to time acomponent fluctuates randomly with respect to time and this added complication makes an analytical solutind this added complication makes an analytical solution intractable.on intractable.

vvThe exact solutions referred to are for laminar flows iThe exact solutions referred to are for laminar flows in which the velocity is either independent of time (sten which the velocity is either independent of time (steady flow) or dependent on time (unsteady flow) in a ady flow) or dependent on time (unsteady flow) in a wellwell--defined manner.defined manner.

154

Some Simple Solutions for Some Simple Solutions for Viscous Viscous Incompressible Incompressible FluidsFluids

vv A principal difficulty in solving the A principal difficulty in solving the NavierNavier--Stokes equations is Stokes equations is because of their nonlinearity arising from the because of their nonlinearity arising from the convective accelconvective acceleration termseration terms..

vv There are no general analytical schemes for solving nonlinear There are no general analytical schemes for solving nonlinear partial differential equations.partial differential equations.

vv There are a few special cases for which the convective acceleraThere are a few special cases for which the convective acceleration vanishes. In these cases exact solution are often possible.tion vanishes. In these cases exact solution are often possible.

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155

Steady, Laminar Flow between Fixed PSteady, Laminar Flow between Fixed Parallel Plates arallel Plates 1/51/5

vvThe NavierThe Navier--Stokes equations reduce toStokes equations reduce to

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156

Steady, Laminar Flow between Fixed PSteady, Laminar Flow between Fixed Parallel Plates arallel Plates 2/52/5

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IntegratingIntegrating

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Steady, Laminar Flow between Fixed PSteady, Laminar Flow between Fixed Parallel Plates arallel Plates 3/53/5

vvWith the boundary conditions u=0 at y=With the boundary conditions u=0 at y=--h u=0 at yh u=0 at y=h=h

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m=Velocity distributionVelocity distribution

158

Steady, Laminar Flow between Fixed PSteady, Laminar Flow between Fixed Parallel Plates arallel Plates 4/54/5

vvShear stress distributionShear stress distribution

vvVolume flow rate Volume flow rate

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12

2012-12-24

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159

Steady, Laminar Flow between Fixed PSteady, Laminar Flow between Fixed Parallel Plates arallel Plates 5/55/5

vvAverage velocityAverage velocity

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160

Couette Flow Couette Flow 1/31/3

vvSince only theSince only the boundary conditions have changedboundary conditions have changed, , tthere ishere is no need to repeat the entire analysisno need to repeat the entire analysis of the of the “both plates stationary” case.“both plates stationary” case.

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Couette Flow Couette Flow 2/32/3

vvThe boundary conditions for the moving plate case arThe boundary conditions for the moving plate case aree

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162

Couette Flow Couette Flow 3/33/3

v Simplest type of Couette flow

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This flow can be approximated by the flow between closely spaced concentric cylinder is fixed and the other cylinder rotates with a constant angular velocity.

Flow in the narrow gap of a journal bearing.

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ioi-wm=t

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Plane Plane CouetteCouette FlowFlow•• A wide moving belt passes through a A wide moving belt passes through a

container of a viscous liquid. The belt container of a viscous liquid. The belt moves vertically upward with a constmoves vertically upward with a constant velocity, Vant velocity, V00, as illustrated in Figur, as illustrated in Figure E6.9(a). Because of viscous forces te E6.9(a). Because of viscous forces the belt picks up a film of fluid of thiche belt picks up a film of fluid of thickness h. Gravity tends to make the flukness h. Gravity tends to make the fluid drain down the belt. Use the Navier id drain down the belt. Use the Navier Stokes equations to determine an exprStokes equations to determine an expression for the average velocity of the ession for the average velocity of the fluid film as it is dragged up the belt. fluid film as it is dragged up the belt. Assume that the flow is laminar, steadAssume that the flow is laminar, steady, and fully developed.y, and fully developed.

164

Example Example SolutionSolution1/21/2

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xp

=¶¶

=¶¶

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, and for steady flow, so thatv=v(x)v=v(x)

2

2

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=

dxdv

cxdxdv

xy

1

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165

Example Example SolutionSolution2/22/2

mg

-=hc1

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22 cx

2hx

2v +

mg

-mg

=

0xatVv 0 ==

02 Vx

2hx

2v +

mg

-mg

=

166

Steady, Laminar Flow in Circular Tubes Steady, Laminar Flow in Circular Tubes 1/51/5

vvConsider the flow through a horizontal circular tube oConsider the flow through a horizontal circular tube of radius R.f radius R.

0z

v0v,0v zr =

¶¶

Þ== q ( )rvv zz =

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Steady, Laminar Flow in Circular TubesSteady, Laminar Flow in Circular Tubes 2/52/5

Navier Navier –– Stokes equation reduced to Stokes equation reduced to

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168

Steady, Laminar Flow in Circular TubesSteady, Laminar Flow in Circular Tubes 3/53/5

At r=0, the velocity vAt r=0, the velocity vzz is finite. At r=R, the velocity vis finite. At r=R, the velocity vzz is zero.is zero.

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zp

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m-==

( )22z Rr

zp

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Velocity distributionVelocity distribution

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Steady, Laminar Flow in Circular Tubes Steady, Laminar Flow in Circular Tubes 4/54/5

vvThe shear stress distributionThe shear stress distribution

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rx

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12

170

Steady, Laminar Flow in Circular Tubes Steady, Laminar Flow in Circular Tubes 5/55/5

vvAverage velocityAverage velocity

vvPoint of maximum velocityPoint of maximum velocity

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=p

==8

pRRQ

AQV

2

2average

0dr

dv z = at r=0at r=0

2

max

zaverage

2

max Rr1

vvV2

4pRv ÷

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çèæ-=Þ=

mD

-=l

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Steady, Axial, Laminar Flow in an Annulus Steady, Axial, Laminar Flow in an Annulus 1/21/2

Boundary conditionsBoundary conditionsvvzz = 0 , at r = r= 0 , at r = roovvzz = 0 , at r = r= 0 , at r = rii

For steady, laminar flow in circular tubesFor steady, laminar flow in circular tubes

?c?ccrlncrzp

41v 2121

2z ++÷

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m=

172

Steady, Axial, Laminar Flow in an Annulus Steady, Axial, Laminar Flow in an Annulus 2/22/2

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ù

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é -+-÷

øö

çè涶

m=

oio

2o

2i2

o2

z rrln

)r/rln(rrrr

zp

41v

The volume rate of flowThe volume rate of flow

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é ---÷

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mp

-=p= ò )r/rln()rr(rr

zp

8dr)r2(vQ

io

22i

2o4

i4

or

rz

o

i

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2/1

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2i

2o

m )r/rln(2rrr

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rvz =¶

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The velocity distributionThe velocity distribution