fluid dynamics class presentation notes.

Notes on Fluid Mechanics

1. FLUID DYNAMICS

2. FLUID DYNAMICS
A fluid in motion is subjected to several forces, which
results in the variation of the acceleration and the
energies involved in the flow of the fluid. The study of
the forces and energies that are involved in the fluid
flow is known as Dynamics of fluid flow.
The various forces acting on a fluid mass may be
classified as:
1. Body or volume forces :-Ex. Weight, Centrifugal
force, magnetic force, Electromotive force etc.
2. Surface forces :-Ex. pressure force, shear or
tangential force etc.
3. Line forces :-Ex. surface tension.

3. FORCES INFLUENCING MOTION OF FLUID
AccordingtoNewton’ssecondlawofmotion,thenetforceFxactingonafluid
elementinthedirectionofxisequaltomassmofthefluidelementmultipliedby
theaccelerationaxinthex-direction.
F = ma…………(1)
variousforcesthatmayinfluencethemotionaredueto
1.Gravityforce(Fg)
2.Pressureforce(Fp)
3.Viscousforce(Fv)
4.Turbulenceforce(Ft)
5.Surfacetensionforce(Fs)
6.Compressibilityforce(Fc)

4. FORCES ACTING ON FLUID IN MOTION:
The various forces that influence the motion of fluid are due to
gravity force(ρg), pressure force(Fp), viscous force(Fv),
turbulence(Ft) and compressibility(Fc).
If a certain mass of fluid in motion is influenced by all the above
forces, then according to Newto’ssecond law of motion :-
The net force along x-direction is given as :-
Fx=M.ax= (Fg)x + ( Fp)x + ( Fv )x + ( Ft )x + ( Fc)x
i) if the net force due to compressibility(Fc) is negligible, the
resulting net force Fx= (Fg) x + (Fp) x + (Fv) x + (Ft) x and the
equation of motions are called Reynolds’s equations of motion.
ii) For flow where (Ft) is negligible, the resulting equations of
motion are known as Navier–Stokes equation.
iii) If the flow is assumed to be ideal, viscous force (Fv) is zero
and the equations of motion are known as Euler’s equation of
motion.

5. According to Newton’s second law of motion, the equation (1) can be written as-
ma = F
ma = Fg+ Fp+ Fv+ Ft + Fs + Fc……….(2)
In most of the problems of the fluids in motion the surface tension forces and the
compressibility forces are not significant. Hence these forces may be neglected .
Therefore equation (2) reduces to
1.Reynold’sequationofmotion-
ma = Fg+ Fp+ Fv+ Ft……………(3)
Itisusefulintheanalysisoftheturbulentflows.
2.Naiver–Stoke’sequation-
forlaminarorviscousflowstheturbulentforcesalsobecomelessthen
ma = Fg+ Fp+ Fv………….(4)
itisusefulintheanalysisofviscousflow.
3.Euler’sequationofmotion-
viscousforcesarelesssignificantandhenceitmaybeneglected
ma = Fg+ Fp…………(5)
itisusefulintheanalysisoffluidflow.

6. Euler's equation of motion
Euler'slawsofmotionareequationsofmotionwhichextendNewton'slawsof
motionforpointparticletorigidbodymotion.
Considertheflowofanidealfluidalongstreamtube.Seperateoutasmallelementof
fluidofcrosssectionalareadA,lengthdSfrommovingfluid.
F= Fg + Fp…….(i)
Nownetpressureforceinthedirectionofflowis
Fp=pdA–(p+dp)dA=-dpdA……..(ii)
Componentofweightofthefluidelementinthedirectionofflowis
Fg = -ρg dA dS cos θ (cos θ= ds dz)
= -ρg dA ds ds dz = -ρg dA dZ….....(iii)

7. mass of fluid element (m)= ρdAdS
accleeration(a)= V dv/ds
Total force F = Fp+ Fg
= -dpdA-ρg dAdZ……….(iv)
F= ma= Fp+ Fg
ρ dAdS. V dv/ds = -dpdA-ρ g dAdZ
Dividebothsidesby–ρdA…….
Finaleq…..dp/ρ+gdZ+vdv=0

8. Bernaulli'sequation
BERNOULLISTHEOREM-Bernoullisprinciplewhichstatesthatforanymassof
flowingliquid,whenthereisacontinuousconnectionbetweenalltheparticlesof
theflowingliquidthetotalenergyremainsthesameateverysectionprovided.
thereisnoadditionorsubtractionsofenergy.Totalenergyisconstant.
Assumption of Bernoulli’s Theorem-
1. Velocity isuniform.
2. Flow is uniform.
3. Flow is steady.
4. Flow isirrational
5. The fluid isincompressible.
6. Flow is caused only bygravity force
7. There is no loss of energy in the flow.
8. Centrifugal force in the curve path is neglected.
9. Frictional, as well as viscous drag forces, are also neglected.
10. Liquid flow in acontinuous stream.

9. ENERGY POSSESSED BY A FLUID PARTICLE –
The different kinds of energy possessed by a fluid particle are :
(i) Potential energy (Datum energy)-Potential Energy of a liquid in motion It is
the energy possessed by a liquid particle by virtue of its position.
(ii) Kinetic energy -Kinetic energy of a liquid in motion It is the energy possessed
by a liquid particle by virtue of its motion or velocity.
(iii) Pressure energy -Pressure energy of a liquid in motion It is the energy
possessed by a liquid particle by virtue of its pressure.
The total energy of a liquid particle in motion is the sum of its potential
energy,Kinetic energy and pressure energy.
p/ρg + V
2
/2g + Z = const.
Z = potential head or Datum head
V
2
/2g = velocity head or kinetic head
p/ρg = pressure head or static head

11. LIMITATIONS OF BERNOULLI’S THEOREM
1.Equation is derived with the assumption that fluid is non viscous, no friction
losses, only gravity, pressure force exists and it is incompressible.
2.Velocity is uniform but in actual practice velocity varies (maximum at
centre) and gradually decreases towards the walls of the pipe.
3.External forces acting on the fluid also affect the fluid flow.
Application of Bernoulli’s theorem
1.Venturimeter
2.Orificemeter
3.Pitottube
4.Rotameter
5.Airplanewings
6.Measuringbloodpressure
7.Swimming

12. VENTURIMETER
Venturimeterisaflowmeasurementinstrument.Here,aconvergingsectionofa
pipeisusedtoincreasetheflowvelocityandacorrespondingpressuredropfrom
whichtheflowrateofthefluidisdeducedbasedonBernoulli'sequation.
Theventurimeterisanapparatususedtofindoutthedischargeofaliquidflowing
atanypointalongapipelinewhenitisrunningfull.
ThreePart-
(i)Convergentcone (ii)Throat (iii)Dovergentcone

13. EXPRESSION FOR THE DISCHARGE THROUGH A VENTURIMETER -
Where-
h = manometric height difference
Q
act. = actual discharge
A
1 = area at section 1
A
2 = area at section 2
C
d = Coefficient of discharge

14. Orifice meter
orificemeterisadeviceusedformeasuringflowrate,forreducingpressureor
forrestrictingflow.
orificeplateisathinplatewithaholeinit,whichisusuallyplacedinapipe.
Whenafluid(whetherliquidorgaseous)passesthroughtheorifice,its
pressurebuildsupslightlyupstreamoftheorifice
[1]
butasthefluidisforcedto
convergetopassthroughthehole,thevelocityincreasesandthefluidpressure
decreases.

15. EXPRESSION FOR THE DISCHARGE THROUGH A ORIFICE METER -
Where-
h = manometricheight difference
Q
act. = actual discharge
D
1 = diameter of pipe at section 1
D
2 = diameter of pipe at section 2
C
d = Coefficient of discharge

17. Rotameter
Arotameterisagaugeformeasuringfluidflowusingagraduatedglasstube
withanenclosedfreefloat.Alsoknownasvariableareaflowmeters,
rotametersareusedtomeasureliquidorgasvolumetricflowratesastheypass
throughthetaperedtubeoftherotameter.

18. Pitot tube
Apitottube(pitotprobe)measuresfluidflowvelocity.Itwasinventedby
aFrenchengineer,HenriPitot,intheearly18thcentury,
[1]
andwasmodifiedtoits
modernforminthemid-19thcenturybyaFrenchscientist,HenryDarcy.
[2]
Itis
widelyusedtodeterminetheairspeedofaircraft;
[3]
thewaterspeedofboats;and
theflowvelocityofliquids,air,andgasesinindustry.

19. Types of Kinetic Energy
1.Radiantenergy
2.Fluidenergy
3.Thermalenergy
4.Soundenergy
5.Electricalenergy
6.Mechanicalenergy

20. Momentum correction factors
Momentumcorrectionfactorisdefinedastheratioofmomentumoftheflowper
secondbasedonactualvelocitytothemomentumoftheflowpersecondbased
onaveragevelocityacrossasection.
Forturbulentflowthemomentumcorrectionfactorisslightlyhigherthanone
nearto1.2andforlaminar,itsvalueis1.33.
Momentumcorrectionfactorwillbedisplayedbythesymbol(β).
Momentum correction factor, β=(Momentum per second based on actual
velocity) / (Momentum per second based on average velocity)

21. Kinetic energy correction factors
Kinetic energy correction factor is basically defined as the ratio of kinetic energy
of the flow per second based on actual velocity across a section to the kinetic
energy of the flow per second based on average velocity across the same
section.
Kinetic energy correction factor will be displayed by the symbol (α)
Kinetic Energy Factor value for a fully developed laminar pipe flow is around 2
Kinetic energy correction factor, α= (K.E. per second based on actual
velocity) / (K.E. per second based on average velocity)