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MP 211 - FLUID MECHANICS

2013/2014

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COURSE CONTENT

Building a Fluid Mechanics Vocabulary

Fluid Properties Fluid Statics

Pressure Measurement

Fluid Forces

Newtonian and non-Newtonian fluids

Fluids in Motion

Conservation of Mass: Continuity Equation

Conservation of Energy, Part I: Bernoulli’s Equation

Conservation of Momentum: Momentum Equation

Conservation of Energy, Part II: Energy Equation

Application of the Basic Principles

Hydraulic Modeling

Flow of fluids in circular and non-circular Pipes

Flow of fluids in open channels

Pumps and pump characteristics

Energy Losses

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INTRODUCTION

Fluid mechanics is the study of the behavior of

fluids , either at rest (fluid statics) or in motion (fluid

dynamics)

Fluid is defined as a substance that deforms

continuously when acted on by a shearing stress of

any magnitude.

Fluids can be either liquids or

gases Liquids e.g. Water, gasoline, and

alcohol

Gases e.g. Air, water vapour,

oxygen, nitrogen, methane etc

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LIQUID VS GAS

A liquid takes the shape of thecontainer it is in and forms a

free surface in the presence of

gravity

A gas expands until itencounters the walls of the

container and fills the entire

available space. Gases cannot

form a free surface

Gas and vapor are often used

as synonymous words

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5

HISTORY

Faces of Fluid Mechanics

Archimedes(C. 287-212 BC)

Newton(1642-1727)

Leibniz(1646-1716)

Euler(1707-1783)

Navier(1785-1836)

Stokes(1819-1903)

Reynolds(1842-1912)

Prandtl(1875-1953)

Bernoulli(1667-1748)

Taylor

(1886-1975)

http://www-gap.dcs.st-and.ac.uk/~history/PictDisplay/Taylor_Geoffrey.html

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DIMENSIONS AND UNITS

Any physical quantity canbe characterized bydimensions.

The magnitudes assignedto dimensions are calledunits.

Primary dimensions (orfundamental dimensions)include: mass M , length L,time T , and temperature Ɵ,etc .

By General Conference of Weights and

Measures

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For a wide variety of problems involving fluid

mechanics, only the three basic dimensions, L, T ,

and M are required. Alternatively, L, T , and F could

be used, where F is the basic dimensions of force

Quantity Name of unit SI - Unit symbol

Length (L) meter m

Mass (m) kilogram kg or (N.s2 /m)

Time (T ) second s

Temperature (ɵ) Kelvin K

Force (F ) Newton N or (kg.m/s2 )

Note: equivalent unit of F = ma = kg.m/s 2

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Secondary dimensions (derived dimensions) can be expressed interms of primary dimensions and include: area (A), pressure (P),velocity V , energy E , and volume V, etc...

e.g. Area= length × length =L 2 = m 2

Velocity = Distance/time = L/T = m/s

Unit systems include; English system or U.S. Customary system and

Metric SI (International System)

All theoretically derived equations describing physical phenomenaare dimensional ly homogeneous

Equation:

Dimension:

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Homogeneity of an equation;

general homogeneous equat ions : valid in any system of

units

restr ic ted h omogeneous equat ions: valid only to a

particular system of units

E.g. an equation for the distance, d, traveled by a freely falling

body can be expressed as;

, OR ...........????

Show the dimensions of the above equations!!

What is your observation?

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DIMENSIONS AND UNITS Based on the notational scheme introduced in 1967,

The degree symbol was officially dropped from the absolutetemperature unit,

All unit names were to be written without capitalization even if theywere derived from proper names (Table 1 –1).

However, the abbreviation of a unit was to be capitalized if the unit was

derived from a proper name. For example, the SI unit of force, which isnamed after Sir Isaac Newton (1647 –1723), is newton (not Newton),and it is abbreviated as N.

Also, the full name of a unit may be pluralized, but its abbreviationcannot. For example, the length of an object can be 5 m or 5 meters,not 5 ms or 5 meter.

Finally, no period is to be used in unit abbreviations unless they appearat the end of a sentence. For example, the proper abbreviation ofmeter is m (not m.).

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Unit systemsSI Units

In SI, the units of mass, length, and time are the kilogram (kg),

meter (m), and second (s), respectively.

Bri t ish Gravi tat ional (BG) System

The unit of length is the foot (ft), the time unit is the second (s), the

force unit is the pound (lb)

The unit of mass is slug and it is derived from from Newton’s second

law F = ma

m=F/a = lb/(ft/s2) = lb.s2 /ft = slug

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English Engineering (EE) Units In English or U.S customary system, the units of mass,

length, and time are the pound-mass (lbm), foot (ft), and

second (s), respectively

The unit of force is pound (lb or lbf) Force and mass are defined independently

Special care is required while using these parameters

in conjunction with Newton’s second law

Is “ 1lbm= 1 slug” ????

1lbm = 0.45359 kg

1ft = 0.3048 m

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Force Units and relation between lbm and slug

SI :

EE:

BG :

We call a mass of 32.174 lbm 1 slug

For the EE system, a 1-lb force is defined as that force which givesa 1lbm a standard acceleration of gravity which is taken as

32.2ft/s2

A dimensionally homogeneous Newton’s second law is expressed

as

where, g c is propotionality constrant

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Pressure is defined as the amount of force exerted on a unit area ofa substance

Pam

N

area

force

P

2

Pressure is compressive normal force applied by the fluid to the

surface

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Consider a force, , acting on a 2D region of area

A sitting on x-y plane

Cartesian components:

F

x

y

z

F

F F i F j F k

x y z ( ) ( ) ( )

A

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C ARTESIAN COMPONENTS i

F z

F x

j

k

- Unit vector in x-direction

- Unit vector in y-direction

- Unit vector in z-direction

- Magnitude of in x-direction (tangent to surface) F

F y - Magnitude of in y-direction (tangent to surface)

- Magnitude of in z-direction (normal to surface)

F

F

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F

A shear stress

x( )

- For simplicity, let F y 0

• Pressure and Shear stress

p F

A

normal stress pressure z

( ( ))

• Shear stress and pressure at a point

F

A

x

Alim 0

p F

A

z

A

lim 0

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Pascals’ laws:

Pressure acts uniformly in all directions ona small volume (point) of a fluid

In a fluid confined by solid boundaries,

pressure acts perpendicular to the

boundary – it is a normal force.

More on pressure...

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Furnace duct Pipe or tube

Heat exchanger

Dam

Pressure is a Normal Force

(acts perpendicular to surfaces)

It is also called a Surface Force

Direction of fluid pressure on boundaries

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Unit Definition or

Relationship

1 pascal (Pa) 1 kg m-1

s-2

1 bar 1 x 105 Pa

1 atmosphere (atm) 101,325 Pa

1 torr 1 / 760 atm760 mm Hg 1 atm

14.696 pounds per

sq. in. (psi)

1 atm

Units for Pressure

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Weight, W is a force. It is the gravitational force applied to a body,and its magnitude is determined from Newton’s second law ,

where m is the mass of the body, and g is the local gravitationalacceleration (g is 9.807 m/s2 or 32.174 ft/s2

The weight of a unit volume of a substance is called the specificweight, g and is determined from;

or where r is density, V is volume

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Work , which is a form of energy, can simply be defined asforce times distance; therefore, it has the unit “newton-meter

(N . m),” which is called a joule (J). That is,

A more common unit for energy in SI is the kilojoule (1 kJ =103 J). In the English system, the energy unit is the Btu

(British thermal unit), which is defined as the energy required

to raise the temperature of 1 lbm of water at 68°F by 1°F.

In the metric system, the amount of energy needed to raise

the temperature of 1 g of water at 14.5°C by 1°C is defined

as 1 calorie (cal), and 1 cal = 4.1868 J. The magnitudes of

the kilojoule and Btu are almost identical (1 Btu = 1.0551 kJ).

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Dimensional homogeneity is a valuable tool in checking for errors. Make

sure every term in an equation has the same units.

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Unity conversion ratios are helpful in converting units. Use them.

All nonprimary units ( secondary units ) can be formed by

combinations of primary units. Force units, for example, can be

expressed as

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PROPERTIES OF FLUID

Gases are light and compressible,

Liquids are heavy (by comparison) and relatively incompressible

Syrup flows slowly from a container than water

To quantify these differences certain fluid properties are used

Fluid properties include;

Mass (m) –already discussed!

Pressure – Already discussed!

Density ( ρ )

specific weight (w) specific gravity (sg )

Bulk modulus of elastisity (E)

viscosity i.e. Dynamic or kinematic viscosity

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PROPERTIES OF FLUID

Density, ρ (rho) of a fluid is defined as its mass (m) per unit

volume (V)

The value of ρ can vary widely between different fluids For liquids, variations in pressure and temperature generally

have only a small effect on the value of ρ

Unlike liquids, the density of a gas is strongly influenced by

both pressure and temperature

The maximum density ( ρ = 1000kg/m3 ) of water is reached at

40C

Specific volume, V s of fluid is its volume per unit mass

ρ = m/V (kg/m3 )

V s = V/m =1/ ρ (m3

/kg)

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PROPERTIES OF FLUID

Effect of temperature on the value of ρ of water

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PROPERTIES OF FLUID

Specific weight of a fluid, , (gamma), is defined as its weight perunit volume

The specific weight is related to density through the equation;

Specific gravity of a fluid, SG, is defined as the ratio of the densityof the fluid to the density of water at some specified temperature

Usually the specified temperature is taken as 40C (39.20F)and at thistemperature the density of water 1000kg/m3 or 1.94slug/ft3

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PROPERTIES OF FLUID

Density of Ideal Gases

Gases are highly compressible in comparison to liquid.

Changes in gas density directly related to changes in pressure

and temperature through the equation of ideal or perfect gas

law ;

The gas constant, R , depends on the particular gas and is

related to the molecular weight of the gas by equation;

R = R u /M R u , is universal gas constant

(8.314kJ/kmol K)

M , molecular weight

Where P , is absolute pressure (N/m2),

T , is are absolute temperature (K)

R , is gas constantP = ρRT

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PROPERTIES OF FLUID

Example 1:

A reservoir of oil has a mass of 825kg and a volume of

0.917m3 calculate;

Weight of the reservoir

The density, specific weight and specific gravity of the

oil

Example 2:

Glycerine at 20 0 C has a specific gravity of 1.263. compute its

density and specific weight . Example 3

A gas weighs 20N/m3 at 30 0 C and at an absolute pressure of

350kN/m2. Determine the gas constant and density of the

gas.

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BULK MODULUS OF ELASTICITY

The bulk modulus of elasticity or compressibility refer tothe change in volume (V) of a substance that is

subjected to a change in pressure on it.

Liquids are heavy and very slightly compressible

For liquids, very large change in pressure is required to

produce small change a in volume.

Gases light and compressible –bulk modulus is not

usually applied to gases.

For gases, to determine the change in volume of a gas

with a change in pressure, the principles of

thermodynamics must be applied

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VISCOSITY ( µ )

Different fluids deform at different rates when the sameshear stress (force/area) is applied.

Viscosity, µ is a fluid property that offers resistance to the

shear force or relative motion of fluid

It can be thought as the internal stickiness of a fluid

Representative of internal friction in fluids

Internal friction forces in flowing fluids result from cohesion

and momentum interchange between molecules.

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EFFECT OF TEMPERATURE ON VISCOSITY..

In liquids; viscosity decreases with increasing temperature

When temperature increases the distance between

molecules increases and the cohesive forcedecreases.

In gases;

The contribution to viscosity is more due to

momentum transfer

As temperature increases, more molecules cross over withhigher momentum differences and setting up strong internalshear

Thus, viscosity increases with increasing temperature

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Dynamic (absolute)

viscosity of some common

fluids as a function oftemperature

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MORE ON VISCOSITY...

Viscosity is important, for example,

In determining the amount of fluids that can be

transported in a pipeline during a specific period

of time

determining energy losses associated with

transport of fluids in ducts, channels and pipes

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NO SLIP CONDITION

Because of viscosity, at boundaries (walls) particles

of fluid adhere to the walls, and so the fluid velocity

is zero relative to the wall

Viscosity and associated shear stress may be

explained via the following: flow between no-slip

parallel plates.

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Flow between no-slip parallel plates-each plate has area A

Moving plate

Fixed plate

F U ,

Y

x z

y

F Fi

U Ui

Force induces velocity on top plate. At top plate flow velocity is

F

U

U

At bottom plate velocity is 0

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The velocity induced by moving top plate can be sketched as follows:

y

u y( )

Y

U

u yU

Y y( )

The velocity induced by top plate is expressed as:

u y( ) 0 0

u y Y U ( )

Slope: Take derivertive of the equation gives;The expression is valid

for most common fluids

such as water, gasoline

and airWhere: δβ is rate of shearing strain &

du/dy velocity gradient = slope

slope

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For a large class of fluids, empirically, F AU

Y

More specifically, F AU Y

; is coefficient of vis itycos

Shear stress induced by F is F

A

U

Y

From previous slide, note that

du

dy

U

Y

Thus, shear stress is du

dy

In general we may use this expression to find shear stress at a point

inside a moving fluid. Note that if fluid is at rest this stress is zero because

du

dy 0

F

A shear stress

x( )Recall: definition of Shear stress

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Alternatively: The shearing stress (τ) is directly propotional tothe rate of deformation/shearing strain(δβ)/velocity gradient

==>

The expression is know as Newton’s law of viscosity

Newton’s law of viscosity states that the shear force to beapplied for a deformation rate of (du/dy) over an area A isgiven by,

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du

dy

- Dynamic viscosity (coeff. of viscosity)

Fixed no-slip plate

u y velocity profile( ) ( )

Shear stress due to viscosity at a point:

fluid surface

e.g.: wind-driven flow in ocean

r - kinematic

viscosity

y

NEWTON’S EQUATION OF VISCOSITY

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N EWTONIAN AND NON -N EWTONIAN FLUIDS

Fluids for which the shearing stress is linearly related

to the rate of shearing strain are designated as

Newtonian fluids

Newtonian fluids obey eqn:

Fluids for which the shearing stress is not linearly

related to the rate of shearing strain are designated

as non-Newtonian fluids

Non-Newtonian fluids do not obey the above

equation

Linear variation of shearing stress with rate of shearing strain

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g g

for common fluids

V i ti f h i t ith t f h i t i f

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Variation of shearing stress with rate of shearing strain for

several types of fluids, including common non-Newtonian fluids

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SELF STUDY.....

Surface tension...

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END

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