Fluid Mechanics

Fluid MechanicsFluid Mechanics

4th SEMESTER

BS Mechanical Engineering(2009-2013)

4th SEMESTER

BS Mechanical Engineering(2009-2013)

Fluid MechanicsFluid Mechanics

It is the study of fluids either in motion or at rest and subsequent effect of fluids on the boundaries.

It is the study of fluids either in motion or at rest and subsequent effect of fluids on the boundaries.

Fluid Mechanics

Fluid StaticsStudy of Mechanics of Fluids at Rest

KinematicsDeals with Velocities and Streamlines

Fluid DynamicsStudy of Fluids in Motion Considering Forces

Dimensions of mechanics are PRIMARY or DERIVED length time mass force temperature

Dimensions of mechanics are PRIMARY or DERIVED length time mass force temperature

aF m aF m

L

T

MMLT-2

Dimensions and UnitsDimensions and Units

Dimension is measure by which a physical variable is expressed qualitativelyUnit is a particular way of attaching a number to dimension.


Quantity SymbolDimensionsVelocity V LT-1

Acceleration a LT-2

Area A L2

Volume L3

Discharge Q L3T-1

Pressure p ML-1T-2

Gravity g LT-2

Temperature T’ Mass concentration C ML-3

Quantity SymbolDimensionsVelocity V LT-1

Acceleration a LT-2

Area A L2

Volume L3

Discharge Q L3T-1

Pressure p ML-1T-2

Gravity g LT-2

Temperature T’ Mass concentration C ML-3


Quantity Symbol DimensionsDensity ML-3

Specific Weight ML-2T-2

Dynamic viscosity ML-1T-1

Kinematic viscosity L2T-1

Surface tension MT-2

Bulk mod of elasticity E ML-1T-2

Quantity Symbol DimensionsDensity ML-3

Specific Weight ML-2T-2

Dynamic viscosity ML-1T-1

Kinematic viscosity L2T-1

Surface tension MT-2

Bulk mod of elasticity E ML-1T-2

These are _______ properties!fluid

HistoryHistoryArchimedes (B.C.) Postulates of buoyancy

Leonardo da Vinci, 1500 Eq. of conservation of mass for incompressible, 1D, viscous flow

Torricelli, 1644 Torricelli’s law

Mariotte, 1686Invented First Wind tunnel to measure drag

Newton, 1687 Newton’s law of viscosity, Calculus

Daniel Bernoulli, 1738

Pressure Gradient is proportional to acceleration for frictionless flow

Leonhard Euler, 1755 Euler Equations, Bernoulli’s Eqd’Alembert, 1752 d’Alembert’s paradox

Navier 1827, Stokes 1845 Navier Stoke Equation

Prandtl, 1904 Boundary Layer Theory

Fluid Mechanics, cont.Fluid Mechanics, cont.

Fluid Mechanics is Based on

Hydrodynamics:o A Mathematical Subjecto Deals with Ideal Fluid o Results are of Limited Value

Hydraulics:o Based on Experimento Gives Empirical Resultso Confined to Water Only

+ Fluid Mechanics

Fluid Mechanics, cont.Fluid Mechanics, cont.

Fluid Mechanics

Theoreticalo Based on Lawso Mostly Applies to

Ideal Situationso Of Little Practical

Value

Experimentalo Supports the

Theoretical Branch

Obstacles1.Geometry2.Viscosity

Concept of FluidConcept of Fluid

Solid Molecules are Closer Hard in Appearance “Requires a Certain

Amount of Stress Before Deformation Occurs”

Solid Molecules are Closer Hard in Appearance “Requires a Certain

Amount of Stress Before Deformation Occurs”

Fluid Molecules are Apart Squashy in

Appearance “Fluid will deform in

Time to the Slightest Stress”

Fluid Molecules are Apart Squashy in

Appearance “Fluid will deform in

Time to the Slightest Stress”

Distinction b/w Gas and LiquidDistinction b/w Gas and Liquid

Liquid Relatively Closely

Packed Strong Cohesion Forces Tends to Retain

Volume Forms a Free Surface

Liquid Relatively Closely

Packed Strong Cohesion Forces Tends to Retain

Volume Forms a Free Surface

Gas Not Closely Packed Week Cohesion

Forces Does not Retain

Volume Volume is Affected

by Pressure

Gas Not Closely Packed Week Cohesion

Forces Does not Retain

Volume Volume is Affected

by Pressure

ContinuumContinuum

All materials, solid or fluid, are composed of molecules discretely spread and in continuous motion.

However, in dealing with fluid-flow relations on a mathematical basis, it is necessary to replace the actual molecular structure by a hypothetical continuous medium, called the continuum.

All materials, solid or fluid, are composed of molecules discretely spread and in continuous motion.

However, in dealing with fluid-flow relations on a mathematical basis, it is necessary to replace the actual molecular structure by a hypothetical continuous medium, called the continuum.

Continuum, contd.Continuum, contd.

Density = Mass / Unit Volume Molecules are Always in Motion so What is

the Meaning of Density of a Fluid????

Density = Mass / Unit Volume Molecules are Always in Motion so What is

the Meaning of Density of a Fluid????

For Air,10-9 mm3 has 3*107

molecules

Continuum, contd.Continuum, contd.

Fluid in Which Properties Vary Continually and Smoothly

Differential Calculus can be Employed to Analyze the Problems

Fluid in Which Properties Vary Continually and Smoothly

Differential Calculus can be Employed to Analyze the Problems

Flow VelocityFlow Velocity

“Distance Travelled by an Object per Unit Time in a Particular Direction”

It is a Vector Quantity For a Fluid, Velocity is a Point Property It Has Three Components u, v, w V(x, y, z, t) = iu(x, y, z, t ) + jv(x, y, z, t ) +

kw(x, y, z, t )

“Distance Travelled by an Object per Unit Time in a Particular Direction”

It is a Vector Quantity For a Fluid, Velocity is a Point Property It Has Three Components u, v, w V(x, y, z, t) = iu(x, y, z, t ) + jv(x, y, z, t ) +

kw(x, y, z, t )

Viscosity _______ Density _______ Specific weight _______ Relationships between Pressure and volume

Ideal Gas Law __________ Bulk Modulus of Elasticity _______

Vapor Pressure _______ Surface Tension _______

Viscosity _______ Density _______ Specific weight _______ Relationships between Pressure and volume

Ideal Gas Law __________ Bulk Modulus of Elasticity _______

Vapor Pressure _______ Surface Tension _______

PV=nRTPV=nRTKK

PvPv

Fluid PropertiesFluid Properties

Fluid Properties, Contd.Fluid Properties, Contd.

DENSITY:- (Specific Mass) “It is the Mass of a

Substance Per Unit Volume” Density is a Point Property

Specific Weight (weight per unit volume) __________________

DENSITY:- (Specific Mass) “It is the Mass of a

Substance Per Unit Volume” Density is a Point Property

Specific Weight (weight per unit volume) __________________

950960970980990

1000

0 50 100Temperature (C)

Den

sity

(kg

/m3 )

= g = 9806 N/m3

R is the universal gas constant T is in Kelvin P is in N/m2

R is the universal gas constant T is in Kelvin P is in N/m2

Perfect Gas LawPerfect Gas Law

PV nRTN m

8314Kgmol K

R

Use absolute pressure for P and absolute temperature for T

gasP R T2

2

m287

s Kgas p v airgas

RR c c R

M

Fluid Deformation between Parallel Plates

Fluid Deformation between Parallel Plates

Side viewSide view

Force F causes the top plate to have velocity U.Force F causes the top plate to have velocity U.What other parameters control how much force is What other parameters control how much force is required to get a desired velocity?required to get a desired velocity?

Distance between plates (b)Distance between plates (b)

Area of plates (A)Area of plates (A)

F

b

U

Viscosity!Viscosity!

Shear StressShear Stress

M

L T

M

L T

change in velocity with respect to distancechange in velocity with respect to distance

AFAF

2m

N

2m

N

dydu dydu

b

AUF

b

AUF Fb

AU

Fb

AU

2

N s kg

m m×sor

2

N s kg

m m×sor

dimension of

Tangential force per unit area

rate of shear

U

b

U

b

b

Ub

U

s

1

s

1

Rate of angular deformation

units of

dimension of units of2

M

L T

2

M

L T

dimension of units of1

T

1

T

Fluid classification by response to shear stress

Fluid classification by response to shear stress

Newtonian Ideal Fluid Ideal plastic

Newtonian Ideal Fluid Ideal plastic

Shear stress Shear stress

Rat

e of

def

orm

atio

nR

ate

of d

efor

mat

ion

dydu

dydu dydu

Ideal FluidNewtonian

Ideal plastic

1

molasses, tar, 20w-50 oil

Examples of highly viscous fluids ______________________

Fundamental mechanisms Gases - transfer of molecular momentum

Viscosity __________ as temperature increases. Viscosity __________ as pressure increases.

Liquids - cohesion and momentum transfer Viscosity decreases as temperature increases. Relatively independent of pressure (incompressible)

Examples of highly viscous fluids ______________________

Fundamental mechanisms Gases - transfer of molecular momentum

Viscosity __________ as temperature increases. Viscosity __________ as pressure increases.

Liquids - cohesion and momentum transfer Viscosity decreases as temperature increases. Relatively independent of pressure (incompressible)

Fluid ViscosityFluid Viscosity

increases

_______

increases

______________

Example: Measure the viscosity of water

Example: Measure the viscosity of water

The inner cylinder is 10 cm in diameter and rotates at 10 rpm. The fluid layer is 2 mm thick and 10 cm high. The power required to turn the inner cylinder is 50x10-6 watts. What is the dynamic viscosity of the fluid?

The inner cylinder is 10 cm in diameter and rotates at 10 rpm. The fluid layer is 2 mm thick and 10 cm high. The power required to turn the inner cylinder is 50x10-6 watts. What is the dynamic viscosity of the fluid?

Outer Outer cylindercylinder

Thin layer of waterThin layer of water

Inner Inner cylindercylinder

Solution SchemeSolution Scheme

Restate the goal Identify the given parameters and represent the

parameters using symbols Outline your solution including the equations

describing the physical constraints and any simplifying assumptions

Solve for the unknown symbolically Substitute numerical values with units and do the

arithmetic Check your units! Check the reasonableness of your answer

Restate the goal Identify the given parameters and represent the

parameters using symbols Outline your solution including the equations

describing the physical constraints and any simplifying assumptions

Solve for the unknown symbolically Substitute numerical values with units and do the

arithmetic Check your units! Check the reasonableness of your answer

Viscosity Measurement: SolutionViscosity Measurement: Solution

hr

Pt322

hr

Pt322

23-32

6-

s/mN 1.16x10m) (0.1m) (0.05(1.047/s)2

m) (0.002 W)10(50

x 23-32

6-

s/mN 1.16x10m) (0.1m) (0.05(1.047/s)2

m) (0.002 W)10(50

x

tAU

F t

AUF U U A A

thr

F22

thr

F22

P P

thr

P322

thr

P322

r 2rh

Fr

Outer Outer cylindercylinder

Thin layer of waterThin layer of water

Inner Inner cylindercylinder

r = 5 cmt = 2 mmh = 10 cmP = 50 x 10-6 W10 rpm

Role of ViscosityRole of Viscosity

Statics Fluids at rest have no relative motion between

layers of fluid and thus du/dy = 0 Therefore the shear stress is _____ and is

independent of the fluid viscosity Flows

Fluid viscosity is very important when the fluid is moving

Statics Fluids at rest have no relative motion between

layers of fluid and thus du/dy = 0 Therefore the shear stress is _____ and is

independent of the fluid viscosity Flows

Fluid viscosity is very important when the fluid is moving

zerozero

Dynamic and Kinematic Viscosity

Dynamic and Kinematic Viscosity

Kinematic viscosity (__) is a fluid property obtained by dividing the dynamic viscosity (__) by the fluid density

Kinematic viscosity (__) is a fluid property obtained by dividing the dynamic viscosity (__) by the fluid density

3mkg

smkg

3mkg

smkg

[m2/s]

2m

sN

2m

sN

2s

mkgN

2s

mkgN

Bulk Modulus of ElasticityBulk Modulus of Elasticity

Relates the change in volume to a change in pressure changes in density at

high pressure pressure waves

_________ ______ __________

Relates the change in volume to a change in pressure changes in density at

high pressure pressure waves

_________ ______ __________

soundsoundwater hammerwater hammer

/v

dpE

d

2.00

2.05

2.10

2.15

2.20

2.25

2.30

2.35

0 20 40 60 80 100

Temperature (C)

Bul

k M

odul

us o

f el

asti

city

(G

Pa)

Water

vE

a

vEa speed of soundspeed of sound

/v

dpE

dV V

Vapor PressureVapor Pressure

0

1000

2000

3000

4000

5000

6000

7000

8000

0 10 20 30 40

Temperature (C)

Vap

or p

ress

ure

(Pa)

liquid

What is vapor pressure of water at 100°C?101 kPa

Connection with phenomenon called cavitation!

CavitationCavitation

Cavitation DamageCavitation Damage

Video clip of cavitation processVideo clip of cavitation process

Surface TensionSurface Tension

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Fluid Mechanics